## 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, 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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Side 47

If a straight line be bisected , and produced to any point , the square of the whole

line thus produced and the square of the part of it produced , are together double

of the square of

If a straight line be bisected , and produced to any point , the square of the whole

line thus produced and the square of the part of it produced , are together double

of the square of

**half**the line bisected , and of the square of the line made up of ... Side 200

LEMMA I. Which is the first proposition of the tenth book , and is necessary to

some of the propositions of this book . ar 10 21 { If from the greater of two unequal

magnitudes , there be taken more than its

than ...

LEMMA I. Which is the first proposition of the tenth book , and is necessary to

some of the propositions of this book . ar 10 21 { If from the greater of two unequal

magnitudes , there be taken more than its

**half**, and from the remainder morethan ...

Side 391

the square of the radius is to the square of the tangent of

opposite to the base . Q. E. D. PROP . VII . FIG . 12. 13 . In a plane triangle , the

base is to the sum of the sides , as the difference of the sides is to the sum or

difference ...

the square of the radius is to the square of the tangent of

**half**the angle BACopposite to the base . Q. E. D. PROP . VII . FIG . 12. 13 . In a plane triangle , the

base is to the sum of the sides , as the difference of the sides is to the sum or

difference ...

Side 413

THE rectangle contained by

of two arches , is equal to the rectangle contained by the sines of

and

THE rectangle contained by

**half**of the radius , and the excess of the versed sinesof two arches , is equal to the rectangle contained by the sines of

**half**the sum ,and

**half**the difference of the same arches . Let AB , AC be any two arches , and ... Side 414

and the excess of the sides , and the sine of the arch , which is

of the same , to the square of the sine of

ABC be a spherical triangle , of which the two sides are AB , BC , and base AC ...

and the excess of the sides , and the sine of the arch , which is

**half**the differenceof the same , to the square of the sine of

**half**the angle opposite to the base . LetABC be a spherical triangle , of which the two sides are AB , BC , and base AC ...

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The Elements of Euclid: Viz, the First Six Books, Together with the Eleventh ... Robert Simson,Euclid Euclid Ingen forhåndsvisning tilgjengelig - 2018 |

### Vanlige uttrykk og setninger

added altitude angle ABC angle BAC base centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter divided double draw drawn equal equal angles equiangular equimultiples Euclid excess fore four fourth given angle given in position given in species given magnitude given ratio given straight line gles greater Greek half join less likewise manner meet multiple Note opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition pyramid Q. E. D. PROP radius reason rectangle rectangle contained rectilineal figure remaining right angles segment shown sides similar sine solid solid angle sphere square square of AC taken THEOR third triangle ABC wherefore whole

### Populære avsnitt

Side 45 - Ir a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Side 41 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Side 54 - Ir any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Let ABC be a circle, and A, B any two points in the circumference ; the straight line drawn from A to B shall fall within the circle.

Side 18 - ABD, the less to the greater, which is impossible ; therefore BE is not in the same straight line with BC.

Side 10 - From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line: it is required to draw from the point A a straight line equal to BC.

Side 8 - Let it be granted that a straight line may be drawn from any one point to any other point.

Side 256 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Side 129 - 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.

Side 23 - At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, and A...

Side 20 - ANY two angles of a triangle are together less than two right angles.