## 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 329

3 .

the half of AB , wherefore the

AL , that is of KG . add to each the square of of KE , therefore the square b of ...

3 .

**rectangle**C , D , from the determination , is not greater than the square of ALthe half of AB , wherefore the

**rectangle**EA , AH is not greater than the square ofAL , that is of KG . add to each the square of of KE , therefore the square b of ...

Page 330

Book VI . given

exceeding by a square . Draw AE , BF at right angles to AB , on the contrary sides

of it , and make AE equal to C , and BF equal to D. join EF and bisect it in G ...

Book VI . given

**rectangle**, it is required to apply a**rectangle**to AB equal to C , D ,exceeding by a square . Draw AE , BF at right angles to AB , on the contrary sides

of it , and make AE equal to C , and BF equal to D. join EF and bisect it in G ...

Page 438

add the

AB together with the

add the

**rectangle**CB , BD , and the square of BC becomes equal to the square ofAB together with the

**rectangle**CB , BD . thercfore this**rectangle**, that is the given**rectangle**HG , GL is the excess of the squares of BC , AB . from the point A ... Page 440

than GH , GL , that is than the square of AC ; wherefore twice the

AF is less than the square of AC , and ... BC and complete the

ABCD ; ABCD is the

...

than GH , GL , that is than the square of AC ; wherefore twice the

**rectangle**CA ,AF is less than the square of AC , and ... BC and complete the

**rectangle**G K HLABCD ; ABCD is the

**rectangle**which was to be found . draw BE perpendicular to...

Page 441

becaufe the angle ABL is equal to EFG , the triangle ABL is equi , angular to EFG

. and the parallelogram AC , that is the

BC as ( the straight line AL to AB , that is as EG to EF , that is as ) the

becaufe the angle ABL is equal to EFG , the triangle ABL is equi , angular to EFG

. and the parallelogram AC , that is the

**rectangle**AL , BC is to the**rectangle**AB ,BC as ( the straight line AL to AB , that is as EG to EF , that is as ) the

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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.