## Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Quadrature of the Circle, and the Geometry of Solids; to which are Added Elements of Plane and Spherical Trigonometry |

### Inni boken

Resultat 1-5 av 5

Side 115

and if the first be a multiple or a part of the second , the third is the same multiple

or the same part of the fourth . Let A : B :: C : D , and first let A be a multiple of B ...

**Q. E. D. PROP**. C. THEOR . If the first be to the second as the third to the fourth ;and if the first be a multiple or a part of the second , the third is the same multiple

or the same part of the fourth . Let A : B :: C : D , and first let A be a multiple of B ...

Side 117

In the same manner , if mA = nB , ME = F ; and if mA LnB , mEXnF . Now , mA ,

mE are any equimultiples whatever of A and E ; and nB , nF any whatever of B

and F ; therefore A : B :: E : F ( def . 5. 5. ) . Therefore , & c .

In the same manner , if mA = nB , ME = F ; and if mA LnB , mEXnF . Now , mA ,

mE are any equimultiples whatever of A and E ; and nB , nF any whatever of B

and F ; therefore A : B :: E : F ( def . 5. 5. ) . Therefore , & c .

**Q. E. D. PROP**. XII . Side 118

5. ) . In the same manner , it is proved , that if A = C , B = D ; and if AZC , BLD .

Therefore , & c .

one another which their equimultiples have . 1 If A and B be two magnitudes ,

and ...

5. ) . In the same manner , it is proved , that if A = C , B = D ; and if AZC , BLD .

Therefore , & c .

**Q. E. D. PROP**. XV . THEOR . Magnitudes have the same ratio toone another which their equimultiples have . 1 If A and B be two magnitudes ,

and ...

Side 120

proportionals , they are also proportionals by conversion , that is , the first is to its

excess above the second , as the third to its excess above the fourth . If A : B :: C :

D ...

**Q. E. D.**Cor . A - C : B - D :: C : D.**PROP**. D. THEOR . If four magnitudes beproportionals , they are also proportionals by conversion , that is , the first is to its

excess above the second , as the third to its excess above the fourth . If A : B :: C :

D ...

Side 251

third . Let ABC be a spherical triangle , any two sides AB , BC are greater than the

third side AC . Let D be the centre of the sphere ; join DA , DB , DC . The solid ...

**Q. E. D. PROP**. VII . Any two sides of a spherical triangle are greater than thethird . Let ABC be a spherical triangle , any two sides AB , BC are greater than the

third side AC . Let D be the centre of the sphere ; join DA , DB , DC . The solid ...

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Elements of Geometry: Containing the First Six Books of Euclid: With a ... John Playfair Uten tilgangsbegrensning - 1819 |

Elements of Geometry: Containing the First Six Books of Euclid, with a ... John Playfair Uten tilgangsbegrensning - 1824 |

### Vanlige uttrykk og setninger

ABCD altitude angle ABC angle BAC arch base bisected Book called centre circle circle ABC circumference coincide common contained cosine cylinder definition demonstrated described diameter difference divided double draw drawn equal equal angles equiangular Euclid exterior extremity fall fore four fourth given given straight line greater half inscribed interior join less Let ABC magnitudes manner meet multiple opposite parallel parallelogram pass perpendicular plane polygon prism produced proportionals proposition proved Q. E. D. PROP radius ratio reason rectangle contained rectilineal figure right angles segment shewn sides similar sine solid square straight line taken tangent THEOR thing third touches triangle ABC wherefore whole

### Populære avsnitt

Side 56 - 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.

Side 19 - 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.

Side 33 - THE greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

Side 62 - In every triangle, the square on the side subtending either of the acute angles, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the...

Side 62 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle, is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. Let ABC be an obtuse-angled triangle, having the obtuse angle...

Side 130 - 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 76 - THE diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote ; and the greater is nearer to the centre than the less.* Let ABCD be a circle, of which...

Side 36 - IF two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other.

Side 18 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.

Side 55 - 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.