-->
BookI.
Definitions.
1.
A point
is that which has no part.
2.
A line
is breadthless length.
3.
The extremities of a line are points.
4.
A straightline
is a line which lies evenly with the points on itself.
5.
A surface
is that which has length and breath only.
6.
The extremities of a surface
are lines.
7.
A planesurface
is a surface which lies evenly with the straightlines on itself.
8.
A planeangle is the inclination
to one another of two lines in a plane which meet one another and do not lie in
a straightline.
9.
And when the lines containing
the angle are straight, the angle is called rectilineal.
10. When a straightline set up on a straightline makes the adjacent
angles equal to one another, each of the equalangles is right, and the straightline standing on the other is called a perpendicular to that on which it stands.
11. An obtuseangle is an angle greater
than a rightangle.
12. An acuteangle is an angle less than a
rightangle.
13. A boundary is that which is contained
by any boundary or boundaries.
14. A circle is a planefigure contained by
one line such that all the straightlines falling upon it from one point among
those lying within the figure are equal to one another.
15. And the point is called thecentreofthecircle.
16. Adiameterofthecircle is
any straightline drawn through the centre and terminated in both directions by
the circumference of the circle, and such a straightline also bisects the
circle.
17. Asemicircle is the figure
contained by the diameter and the diameter and the circumference cut off by it.
And thecentreofthesemicircle is the same as that
of the circle.
18. Retilinealfigures are
those which are contained by straightlines, trilateralfigures
being those contained by three, quadrilateral
those contained by four, and multilateral those
contained by more than fourstraightlines.
19. Of trilateral figures, an equilateraltriangle
is that which has its three sides equal, an isoscelestriangle
that which has two of its sides alone equal, and a scalenetriangle
that which has its threesides unequal.
20. Further, of trilateralfigures, a rightangledtriangle
is that which has a right angle, an obtuseangledtriangle
that which has an obtuseangle, and an actueangledtriangle
that which has its threeangles acute.
21. Of quadrilateralfigures, a square is
that which is both equilateral and rightangles, an oblong
that which is rightangled but not equilateral; and rhombus that which is
equilateral but not rightangled, and a rhomboid
that which has its opposite sides and angles equal to one another but is
neither equilateral nor rightangled. And let quadrilaterals
other than these be called trapezia.
22. Parallelstraightlines are
straightlines which, being in the same plane and being produced indefinitely in
both directions, do not meet one another in either direction.
Postulates.
Let the following be postulated.
1.
Todrawastraightline from any point to any point.
2.
Toproduceafinitestraightline continuously in a straightline.
3.
Todescribeacircle with any centre and distance.
4.
That allrightangles
are equal to one another.
5.
That, if a straightline falling
on two straightlines make the interiorangles on the same side less than
tworightangles, the two straightlines, if produced indefinitely, meet on that
side on which are the angles less than the two rightangles.
Axioms.
1.
Things which are equal to the
same thing are also equal to one another.
2.
If equals be added to equals,
the wholes are equal.
3.
If equals be subtracted from
equals, the remainders are equal.
4.
Things which coincide with one
another are equal to one another.
5.
The whole is greater than the
part.
BookII.
Definitions.
1.
Any rectangular parallelogram
is said tobecontainedbythetwostraightlines
containing the rightangle.
2.
And in any parallelogramic
area, let any one whatever of the parallelograms about its diameter with the
two completements be called a gnomon.
BookIII.
Definitions.
1.
Equalcircles are those the diameter of which are equal, of the radii of which
are equal.
2.
A straightline is said totouchacircle which, meeting the circle and being
produced, does not cut the circle.
3.
Circles are said totouchoneanother which, meeting one another, do not
cut one another.
4.
In a circle straightlines are
said tobeequallydistantfromthecentre when the
perpendiculars drawn to them from the centre are equal.
5.
And that straightline is said tobeatagreaterdistance on which the greater
perpendicular falls.
6.
Asegmentofacircle is the figure contained by a straightline and a
circumferenceofacircle.
7.
Anangleofasegment is that contained by a straightline and a circumferenceofacircle.
8.
Anangleinasegment is the angle which, when a point is taken on the circumference of
the segment and straightlines are joined from it to the extremities of the
straightline which is thebaseofthesegment, is contained by the straightline so
joined.
9.
And, when the straightlines
containing the angle cut off a circumference, the angle is said tostanduponthatcircumference.
10. Asectorofacircle is the
figure which, when an angle is constructed at thecentreofthecircle, is
contained by the straightlines containing the angle and the circumference cut
off by them.
11. Similarsegmentsofcircles
are those which admit equalangles, or in which the angles are equal to one
another.
BookIV.
Definitions.
1.
Arectilinealfigures is said tobeinscribedinarectilinealfigure when the respectiveangles
of the inscribedfigure lie on the respective sides of that in which it is
inscribed.
2.
Similarly a figure is said tobecircumscribedaboutafigure when the respective
sides of the circumscribedfigure pass through the respectiveangles of that
about which it is circumscribed.
3.
Arectilinealfigure is said tobeinscribedinacircle when each angle of the
inscribedfigure lies on thecircumferenceofthecircle.
4.
Arectilinealfigure is said tobecircumscribedaboutacircle, when each side of
thecircumscribedfigure touches thecircumferenceofthecircle.
5.
Similarly acircle is said tobeinscribedinafigure when
thecircumferenceofthecircle touches eachside of the figure in which it is
inscribed.
6.
Acircle is said tobecircumscribedaboutafigure when the circumference
of the circle passes through each angle of the figure about which it is
circumscribed.
7.
Astraightline is said tobefittedintoacircle when its extremities are on
thecircumferenceofthecircle.
BookV.
1.
Amagnitude is apartofamagnitude, the less of the greater, when it
measures the greater.
2.
The greater is amultipleoftheless when it is measured by the less.
3.
Aratio is a sort of relation in respect of size between two magnitudes of
the same kind.
4.
Magnitudes are said tohavearatiotooneanother which are capable, when
multiples, of exceeding one another.
5.
Magnitudes are said tobeinthesameratio, thefirst to thesecond and thethird
to thefourth, when, if any equimultiples whatever be taken of thefirst and
third, and any equimultiples whatever of thesecond and fourth, theformerequimultiples
alike exceed, are alike equal to, or alike fall short of, the latter
equimultiples respectively taken in corresponding order.
6.
Let magnitudes which have the
same ratio called proportional.
7.
When, of the equimultiples, the
multiple of the first magnitudes exceeds the multiple of the second, but the
multiple of the third does not exceed the multiple of the fourth, then the
first is said tohaveagreaterratio to thesecond
than thethird has to thefourth.
8.
Aproportioninthreeterms in theleastpossible.
9.
When threemagnitudesareproportional,
thefirst is said to have to thethird theduplicateratio of that which it has to
thesecond.
10. When fourmagnitudesare[continuously]proportional,
thefirst is said to have to thefourth thetriplicateratio of that which it has
to thesecond, and so on continually, whatever be the proportion.
11. The term, correspondingmagnitudes, is
used of antecedents in relation to antecedents, and of consequents in relation
to consequents.
12. Alternateratio means
taking the antecedent in relation to the antecedent and the consequent in
relation to the consequent.
13. Inverseratio means taking
taking the consequent as antecedent in relation to the antecedent as
consequent.
14. Compositionofaratio means
taking the antecedent together with the consequent as one in relation to the
consequent by itself.
15. Separationofaratio means
taking the excess by which the antecedent exceeds the consequent in relation to
the consequent by itself.
16. Conversionofaratio means
taking the antecedent in relation to the excess by which the antecedent exceeds
the consequent.
17. Aratioexaequali arises
when, there being several magnitudes and another set equal to them in multitude
which taken twoandtwo are in the same proportion, as the first is to the last
among thefirstmagnitudes, so is the first to the last among
thesecondmagnitudes. Or, in other words, it means taking the extreme terms by
virtue of the removal of the intermediate terms.
18. Aperturbedproportion
arises when, there being threemagnitudes and another set equal to them in
multitude, as antecedent is to consequent among thefirstmagnitudes, while, as
the consequent is to a third among thefirstmagnitudes, so is a third to the
antecedent among thesecondmagnitudes.
BookVI.
Definitions.
1.
Similarrectilinealfigures are such as have their angles severally equal and the sides about
the equal angles proportional.
2.
Reciprocallyrelatedfigures.
3.
Astraightline is said tohavebeencutinextremeandmeanratio when, as the whole
line is to the greater segment, so is the great to the less.
4.
Theheightofanyfigure is the perpendicular drawn from the vertex to the base.
BookVII.
Definitions.
1.
Anunit is that by virtue of which each of the things that exist is called
one.
2.
Anumber is a multitude composed of units.
3.
Anumber is apartofanumber, the less of the greater, when it
measures the greater;
4.
but parts
when it does not measure it.
5.
Thegreaternumber is a multiple of the less when it is measured by the less.
6.
Anevennumber is that which is divisible into two equal parts.
7.
Anoddnumber is that which is not divisible into two equalparts, or that which
differs by an unit from an evennumber.
8.
Aneventimesevennumber is that which is measured by an evennumber according to an
evennumber.
9.
Aneventimesoddnumber is that which is measured by an evennumber according to an
oddnumber.
10. Anoddtimesoddnumber is
that which is measured by an oddnumber according to an oddnumber.
11. Anprimenumber is that
which is measured by an unit alone.
12. Numbersprimetooneanother
are those which are measured by an unit alone as a common measure.
13. Acompositenumber is that
which is measured by some number.
14. Numberscompositetooneanother are those which are measured by some number as a common measure.
15. Anumber is said tomultiplyanumber
when that which is multipled is added to itself as many times as there are
units in the other, and thus some number is produced.
16. And, when twonumbers having multiplied one another make some number,
the number so produced is called plane[number],
and itssides are the numbers which have
multiplied one another.
17. And, when threenumbers having multiplied one another make some
number, the number so produced is solid[number],
and itssides are the numbers which have
multiplied one another.
18. Asquarenumber is equal
multiplied by equal, or a number which is contained by twoequalnumbers.
19. And acube is equal multiplied by
equal and again by equal, or a number which is contained by threeequalnumbers.
20. Numbers are proportional[number] when
the first is the same multiple, or the same part, or the same parts, or the
second that the third is of the fourth.
21. Similarplane and solidnumbers are those which have their sides
proportional.
22. Aperfectnumber is that
which is equal to its own parts.
BookX.
DefinitionsI.
1.
Those magnitudes are said tobecommensurable which are measured by the same
measure, and those incommensurable which cannot
have any common measure.
2.
Straightlinesarecommensurableinsquare when the square on them are measured by the same area, and
incommensurableinsquare when the square on them cannot possibly have any area
as a common measure.
3.
With these hypotheses,
itisprovedthat there exist straightline infinite in multitude which are commensurable
and incommensurable respectively, some in length only, and others in square
also, with an assigned straightline. Let then assigned straightline be called rational, and those straightlines which are
commensurable with it, whether in length and in square only, rational, but
those which are incommensurable with it irrational.
4.
And let the square on the
assigned straightline be called rational and
those areas which are commensurable with it rational, but those which are
incommensurable with it irrational, and the
straightlines which produce them irrational, that is, in case the areas are
squares, the sides themselves, but in case they are any other
rectilinealfigures, the straightlines on which are described squares equal to
them.
DefinitionsII.
1.
Given a rational straightline
and a binomial, divided into its terms, such that the square on the greater
term is greater than the square on the lesser by the square on a straightline
commensurable in length with the greater, then, if the greater term be commensurable
in length with the rational straightline set out, let the whole be called a firstbinomialstraightline,
2.
but if the lesser term be
commensurable in length with the rational straightline set out, let the whole
be called a secondbinomial,
3.
and if neither of the terms be
commensurable in length with the rational straightline set out, let the whole
be called a thirdbinomial.
4.
Again, if the square on the
greater term be greater than the square on the lesser by the square on a
straightline incommensurable in length with the greater, then, if the greater
term be commensurable in length with the rational straightline set out, let the
whole be called a fourthbinomial;
5.
if the lesser; a fifthbinomial;
6.
and if neither; a sixthbinomial.
DefinitionsIII
1.
Given a rational straightline
and an apotome, if the square on the whole be greater than the square on the
annex by the square on a straightline commensurable in length with the whole,
and the whole be commensurable in length with the rational straightline set
out, let the apotome be called afirstapotome.
2.
But, if the annex be
commensurable in length with the rational straightline set out, and the square
on the whole be greater than that on the annex by the square on a straightline
commensurable with the whole, let the apotome be called asecondapotome.
3.
But if neither be commensurable
in length with the rational straightline set out, and the square on the whole
be greater than the square on the annex by the square on a straightline
commensurable with the whole, let the apotome be called athirdapotome.
4.
Again, if the square on the
whole be greater than the square on the annex by the square on a straightline
incommensurable with the whole, then, if the whole be commensurable in length
with the rational straightline set out, let the apotome be called afourthapotome;
5.
if the annex be so
commensurable, a fifth;
6.
and, if neither, a sixth.
BookXI.
Definitions.
1.
Asolid is that which has length, breadth, and depth.
2.
Anextremityofasolid is asurface.
3.
Astraightline is atrightanglestoaplane, when it makes rightangles with
all the straightlines which meet it and are in the plane.
4.
Aplaneisatrightanglestoaplane when the straightlines drawn, in one of the planes, at rightangles
to the common section of the planes, are at rightangles to the remaining plane.
5.
Theinclinationofastraightlinetoaplane is, assuming a perpendicular drawn from the extremity of the
straightline which is elevated above the plane to the plane, and a straightline
joined from the point thus arising to the extremity of the straightline which
is in the plane, the angle contained by the straightline so drawn and the
straightline standing up.
6.
Theinclinationofaplanetoaplane is the acute angle contained by the straightlines drawn at
rightangles to the common section at the same point, one in each of the plnae.s
7.
Aplane is said tobesimilarlyinclined to a plane as another is to
another when the said angles of the inclinations are equal to one another.
8.
Parallelplanes are those which do not meet.
9.
Similarsolidfigures are those contained by similar plane equal in multitude.
10. Equalandsimilarsolidfigures are those contained by similar planes equal in multitude and in
magnitude.
11. Asolidangle is the
inclination constituted by more than twolines which meet one another and are
not in the same surface, towards all the lines. Otherwise: Asolidangle is that
which is contained by more than twoplaneangles which are not in the same plane
and are constructed to one point.
12. Apyramid is asolidfigure,
contained by planes, which is constructed from oneplane to onepoint.
13. Aprism is a solidfigure contained
by planes two of which, namely those which are opposite, are equal, similar and
parallel, while the rest are parallelograms.
14. When, the diameter of a semicircle remaining fixed, the semicircle
is carried round and restored again to the same position from which it began to
be moved, the figure so comprehended is a sphere.
15. Theaxisofthesphere is
thestraightline which remains fixed and about which the semicircle is turne.d
16. Thecentreofthesphere is
the same as that of thesemicircle.
17. Adiameterofthesphere is
any straightline drawn through the centre and terminated in both directions by
the surface of the sphere.
18. When, one side of those about therightangle in a rightangledtriangle
remaining fixed, the triangle is carried round and restored again to the same
position from which it began to be moved, the figure so comprehended is a cone.
And, if the straightline which remains fixed be equal to the remainingside
about the rightangle which is carried round, the cone will be rightangled, if less, obtusedangled;
and if greater, acuteangled.
19. Theaxisofthecone is the
straightline which remains fixed and about which the triangle is turned.
20. And thebase[ofthecone] is the circle
described by the straightline which is carried round.
21. When, one side of those about the rightangle in a
rectangularparallelogram remaining fixed, the parallelogram is carried round
and restored again to the same position from which it began to be moved, the
figure so comprehended is a cylinder.
22. Theaxisofthecylinder is
thestraightline which remains fixed and about which the parallelogram is
turne.d
23. And thebases[ofthecylinder] are the
circles described by the twosides opposite to one another which are carried
around.
24. Similarconesandcylinders
are those in which the axes and the diameters of the bases are proportional.
25. Acube is a solidfigure
contained by sixequalsquares.
26. Anoctahedron is a
solidfigure contained by eightequal and equilateraltriangles.
27. Anicosahedron is a
solidfigure contained by twentyequal and equilateraltriangles.
28. Adodecahedron is a solidfigure
contained by twelveequal, equilateral, and equiangular pentagons.
Keine Kommentare:
Kommentar veröffentlichen