## The Elements of Euclid: Viz. the First Six Books, with the Eleventh and Twelfth. In which the Corrections of Dr. Simson are Generally Adopted, But the Errors Overlooked by Him are Corrected, and the Obscurities of His and Other Editions Explained. Also Some of Euclid's Demonstrations are Restored, Others Made Shorter and More General, and Several Useful Propositions are Added. Together with Elements of Plane and Spherical Trigonometry, and a Treatise on Practical Geometry |

### Inni boken

Resultat 1-5 av 7

Side v

But , indeed , this Proposition , which makes the roth Definition of the rith Book , is

not true universally , except in the

figures is contained by no more than three plane angles ; for , in other

...

But , indeed , this Proposition , which makes the roth Definition of the rith Book , is

not true universally , except in the

**case**in which each of the solid angles of thefigures is contained by no more than three plane angles ; for , in other

**cases**, two...

Side 272

BAD , B oppofite contained (

less than BAD , if one of them B , C be of diff . affection , or C. less than the

supplement of BAD , if B and C be of the same affection : In other

ambiguous .

BAD , B oppofite contained (

**case**3. ) ... In other**cases**BAC is ambiguous . ... isless than BAD , if one of them B , C be of diff . affection , or C. less than the

supplement of BAD , if B and C be of the same affection : In other

**cases**it isambiguous .

Side 273

In other

sum of BD , DC ; otherwise it is their difference . AB , BC , and one of R : cos . B ::

tan . AB : tan . BD B , two sides the other (

In other

**cases**, DC is ambiguous . If B and C be of the same affection , BC is thesum of BD , DC ; otherwise it is their difference . AB , BC , and one of R : cos . B ::

tan . AB : tan . BD B , two sides the other (

**case**2. ) , and the difference of 7 and ... Side 278

... found that two points of one of them coincide with two points of anotber , in

which

than from their uniformity : and it is probable , that Euclid understood his tenth

axiom ...

... found that two points of one of them coincide with two points of anotber , in

which

**case**the coinciding of the lines follows more immediately from the axiom ,than from their uniformity : and it is probable , that Euclid understood his tenth

axiom ...

Side 296

There is therefore only one

one of the equimultiples of the antecedents is less than a multiple of its

consequent , and the other not less than the fame multiple of its confequent : and

in this ...

There is therefore only one

**case**in which the ratios are not the same , viz . whenone of the equimultiples of the antecedents is less than a multiple of its

consequent , and the other not less than the fame multiple of its confequent : and

in this ...

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### Vanlige uttrykk og setninger

ABC is equal ABCD alſo altitude angle ABC angle BAC arch baſe becauſe biſect Book caſe centre circle circumference common contained cylinder definition demonſtrated deſcribed diameter difference diſtance divided double draw drawn equal equal angles equiangular equimultiples fall fame fides figure firſt folid fore four fourth greater half inches join leſs Let ABC magnitudes mean meaſure meet oppoſite parallel parallelogram parallelopiped paſs perpendicular plane priſm PROB produced PROP proportionals propoſition proved radius rectangle rectangle contained rectilineal remaining right angles ſame ſame multiple ſame reaſon ſecond ſegment ſhall ſides ſimilar ſolid ſquare ſtraight line ſum taken tangent THEOR theſe third touches triangle triangle ABC twice uſe Wherefore whole

### Populære avsnitt

Side 30 - 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 142 - 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 13 - Let it be granted that a straight line may be drawn from any one point to any other point.

Side 30 - ... then shall the other sides 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, BCA equal to the angles DEF, EFD, viz.

Side 72 - 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 57 - If then the sides of it, BE, ED are equal to one another, it is a square, and what was required is now done: But if they are not equal, produce one of them BE to F, and make EF equal to ED, and bisect BF in G : and from the centre G, at the distance GB, or GF, describe the semicircle...

Side 145 - AC the same multiple of AD, that AB is of the part which is to be cut off from it : join BC, and draw DE parallel to it : then AE is the part required to be cut off.

Side 48 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.

Side 35 - F, which is the common vertex of the triangles ; that is, together with four right angles. 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.

Side 10 - 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.