## The Elements of Euclid: Also the Book of Euclid's Data ...cor. viz. The first six books, together with the eleventh and twelfth |

### Inni boken

Resultat 1-5 av 9

Side 269

... the fame vertices with the cones , it may be

the first cone has to each in the other , taking them in the same order , the

triplicate ratio of that which the side AK has to the side EM ; that is , which AC has

to EG ...

... the fame vertices with the cones , it may be

**demonstrated**that each pyramid inthe first cone has to each in the other , taking them in the same order , the

triplicate ratio of that which the side AK has to the side EM ; that is , which AC has

to EG ...

Side 285

In the same manner it may be

sphere less than ABC , the triplicate ratio of that which EF has to BC . Nor can the

sphere ABC have to any sphere greater than DEF , the triplicate ratio of that ...

In the same manner it may be

**demonstrated**that the sphere DEF has not to anysphere less than ABC , the triplicate ratio of that which EF has to BC . Nor can the

sphere ABC have to any sphere greater than DEF , the triplicate ratio of that ...

Side 300

... Apagogic Demonstrations , and sometimes ignorantly enough ; not being

aware that there are some things that cannot be

this the present Proposition is a very clear instance , as no direct

can ...

... Apagogic Demonstrations , and sometimes ignorantly enough ; not being

aware that there are some things that cannot be

**demonstrated**any other way . ofthis the present Proposition is a very clear instance , as no direct

**demonstration**can ...

Side 307

Prop . or its

be greater “ than M , L is greater than N , " & c ... Wherefore these words «

because it is

should have ...

Prop . or its

**demonstration**. for when he fays because it is**demonstrated**that if Kbe greater “ than M , L is greater than N , " & c ... Wherefore these words «

because it is

**demonstrated**, " & c . are wholly foreign to his design , and heshould have ...

Side 308

B. V. There are two cases of this Proposition , of which only the first and simplest

is

5.

B. V. There are two cases of this Proposition , of which only the first and simplest

is

**demonstrated**in the Greek . and it ... Proposition for the same reason . the**demonstration**of the other case is now added , becaufe both of them , as also the5.

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added alſo altitude angle ABC angle BAC baſe 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 radius reaſon rectangle rectangle contained remaining right angles ſame ſecond ſegment ſhall ſhewn ſides ſimilar ſolid ſphere ſquare ſquare of AC taken THEOR theſe third thro triangle ABC wherefore whole

### Populære avsnitt

Side 157 - D ; wherefore the remaining angle at C is equal to the remaining angle at F ; Therefore the triangle ABC is equiangular to the triangle DEF. Next, let each of the angles at C, F be not less than a right angle ; the triangle ABC is also, in this case, equiangular to the triangle DEF.

Side 24 - ... be equal, each to each ; and also the third angle of the one to the third angle of the other. Let ABC, DEF be two triangles which have the angles ABC, ECA equal to the angles DEF, EFD, viz.

Side 45 - 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 73 - THE straight line drawn at right angles to the diameter of a circle, from the extremity of...

Side 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.

Side 81 - ELF. Join BC, EF ; and because the circles ABC, DEF are equal, the straight lines drawn from their centres are equal: therefore the two sides BG, GC, are equal to the two EH, HF ; and the angle at G is equal to the angle at H ; therefore the base BC is equal (4.

Side 323 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Side 80 - EA : and because AD is equal to DC, and DE common to the triangles ADE, CDE, the two sides AD, DE are equal to the two CD, DE, each to each ; and the angle ADE is equal to the angle CDE, for each of them is a right angle ; therefore the base AE is equal (4.

Side 36 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.