## 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 112

Book v . than that of the second , the multiple of the third is also greater than that

of the

N. B. “ When four magnitudes are proportionals , it is usually expressed by ...

Book v . than that of the second , the multiple of the third is also greater than that

of the

**fourth**. VI . Magnitudes which have the fame ratio are called proportionals .N. B. “ When four magnitudes are proportionals , it is usually expressed by ...

Page 113

... to the

shewn in the 16th Prop . of this 5th Book . ... Inversion ; when there are four

proportionals , and it is inferred , that the second is to the first , as the

third .

... to the

**fourth**; or that the first is to the third , as the fecond to the**fourth**. as isshewn in the 16th Prop . of this 5th Book . ... Inversion ; when there are four

proportionals , and it is inferred , that the second is to the first , as the

**fourth**to thethird .

Page 116

IF F the first magnitude be the same multiple of the second that the third is of the

then shall the first together with the fifth be the same multiple of the second , that ...

IF F the first magnitude be the same multiple of the second that the third is of the

**fourth**, and the fifth the same multiple of the second that the fixth is of the**fourth**;then shall the first together with the fifth be the same multiple of the second , that ...

Page 121

Book V. PROP . Ą .: THEOR . If the first of four magnitudes has to the second , the

Sec N. same ratio which the third has to the

the second , the third is also greater than the

Book V. PROP . Ą .: THEOR . If the first of four magnitudes has to the second , the

Sec N. same ratio which the third has to the

**fourth**; then if the firit be greater thanthe second , the third is also greater than the

**fourth**; and if equal , equal ; if less ... Page 141

IF F the first has to the second the same ratio which the See Ni third has to the

the first and fifth together shall have to the second , the same ratio which the third

...

IF F the first has to the second the same ratio which the See Ni third has to the

**fourth**; and the fifth to the second the same ratio which the sixth has to the**fourth**;the first and fifth together shall have to the second , the same ratio which the third

...

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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.