## 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 382

given , wherefore also the angle EAD is given . therefore because the straight

line DA is drawn to the given point A in the straight line EF

makes with it a given angle EAD ; d . 32. Dat . AD is given d in position . 1 31 .

given , wherefore also the angle EAD is given . therefore because the straight

line DA is drawn to the given point A in the straight line EF

**given in position**, andmakes with it a given angle EAD ; d . 32. Dat . AD is given d in position . 1 31 .

Page 385

engle HCC , for it is equal to the

AD which is

the ...

engle HCC , for it is equal to the

**given**angle ADC ; HG is**given**a . 29. I. in**position**b ; but it is**given**also in magnitude , because it is equal to b . 32. Dat .AD which is

**given**in magnitude . therefore because G one of the extremities ofthe ...

Page 386

From the given point A let the straight line AED be drawn to the two parallel

straight lines FG , BC , and let the ratio of the feg . ments AE , AD be given ; if one

of the parallels BC be

From the given point A let the straight line AED be drawn to the two parallel

straight lines FG , BC , and let the ratio of the feg . ments AE , AD be given ; if one

of the parallels BC be

**given in position**, the other FG is also**given in position**. Page 387

into which the straight line GEF is cut by the three paraileis , be given ; the third

parallel HK is

perpendicular to CD , meeting HK in N. because LM is drawn from the given point

L to CD ...

into which the straight line GEF is cut by the three paraileis , be given ; the third

parallel HK is

**given in position**. In AB take a given point L , and draw LMperpendicular to CD , meeting HK in N. because LM is drawn from the given point

L to CD ...

Page 446

DC is drawn to the given point D in the straight line BD given in c . 32. Dat .

position in the given angle BDC , DC is

circumference ABC is

P. XCIV .

DC is drawn to the given point D in the straight line BD given in c . 32. Dat .

position in the given angle BDC , DC is

**given in position**, and the d . 28. Dat.circumference ABC is

**given in position**, therefore the point C is given . 91 . PROP. XCIV .

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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.