## The Elements of Euclid, containing the first six books, with a selection of geometrical problems. To which is added the parts of the eleventh and twelfth books which are usually read at the universities. By J. Martin |

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Side 161

respect of quantity . ” 4 . Magnitudes are said to have a

when the less can be multiplied so as to exceed the other . 5 . The first of four ...

**Ratio**is a mutual relation of two magnitudes of the same kind to one another , inrespect of quantity . ” 4 . Magnitudes are said to have a

**ratio**to one another ,when the less can be multiplied so as to exceed the other . 5 . The first of four ...

Side 168

Likewise , if the first has the same

fourth , then also any equimultiples whatever of the first . and third shall have the

same

Likewise , if the first has the same

**ratio**to the second , which the third has to thefourth , then also any equimultiples whatever of the first . and third shall have the

same

**ratio**to the second and fourth ; and in like manner , the first and the third ... Side 175

PROPOSITION 8. - Theorem . Of two unequal magnitudes , the greater has a

greater

has a greater

PROPOSITION 8. - Theorem . Of two unequal magnitudes , the greater has a

greater

**ratio**to any other magnitude than the less has ; and the same magnitudehas a greater

**ratio**to the less of two other magnitudes , than it has to the greater . Side 177

Let C have the same

equal to B. Construction . For , if they are not equal , one of them must be greater

than the other ; let A be the greater ; therefore , as was shown in Prop . 8 , there is

...

Let C have the same

**ratio**to each of the magnitudes A and B. Then A shall beequal to B. Construction . For , if they are not equal , one of them must be greater

than the other ; let A be the greater ; therefore , as was shown in Prop . 8 , there is

...

Side 178

Let A have to C a greater

A D C F в . EConstruction . For , because A has a greater

C , there are some equimultiples of A and B , and some multiple of C such ( V.

Def ...

Let A have to C a greater

**ratio**than B has to C. Then A shall be greater than B.A D C F в . EConstruction . For , because A has a greater

**ratio**to C , than B has toC , there are some equimultiples of A and B , and some multiple of C such ( V.

Def ...

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The Elements of Euclid, Containing the First Six Books, with a Selection of ... Euclides Ingen forhåndsvisning tilgjengelig - 2016 |

### Vanlige uttrykk og setninger

ABCD AC is equal alternate angle ABC angle ACB angle BAC base base BC bisected centre circle ABC circumference common compounded constr Construction contained Demonstration describe diameter divided double draw equal angles equiangular equimultiples exterior angle extremities fall figure four fourth given point given straight line greater half inscribed interior join less Let ABC likewise magnitudes manner meet multiple opposite angle parallel parallelogram pass perpendicular plane polygon Problem produced proportionals proved Q.E.D. PROPOSITION ratio reason rectangle rectangle contained rectilineal figure remaining angle right angles segment shown sides similar square square on AC taken third touches the circle triangle ABC unequal wherefore whole

### Populære avsnitt

Side 4 - If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...

Side 230 - If two triangles, which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another, the remaining sides shall be in a straight line. Let...

Side 110 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Side 207 - ... triangles which have one angle in the one equal to one angle in the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

Side 267 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.

Side 197 - 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 21 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.

Side 61 - If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section. Let the straight line AB be divided into two equal parts...

Side 30 - ... twice as many right angles as the figure has sides ; therefore all the angles of the figure together with four right angles, are equal to twice as many right angles as the figure has sides.