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The Propositions of the Fifth and Sixth Books of Euclid as expressed by
Prop I Theor.
If any number of magnitudes be
equimultiples of as many, each of each, what
multiple soever any one of them is of
its part, the same multiple shall all
the first magnitudes be of all the
other.
Prop A Theor.
If the first of the four magnitudes has
to the second the same ratio which
the third has to the fourth, then if
the first be greater than the second
the third is also greater than the fourth
and if equal, equal: if less, less.
Prop IX Theor.
Magnitudes which have the
same ratio to the same magnitude
are equal to one another: and
those to which the same magnitude
has the same ratio are equal to one
another.
Prop XV Theor.—
Magnitudes have the same ratio
to one another which their
equimultiples have.
Prop XX Theor.—
If there be three magnitudes and other
three, which taken two and two have
the same ratio, if the first be greater
than the third the fourth shall be
greater than the sixth: and if equal,
equal, and if less, less—
Prop F Theor.
Ratios which are compounded of
the same Ratios are the same
with one another.
---page break---
Prop II Theor.—
If the first magnitude, be the same
multiple of the second that the third is of the
fourth, and the fifth the same multiple
of the second that the sixth is of the fourth:
then shall the first together with the
fifth, be the same multiple of the second
that the third together with the sixth is
of the fourth.—
Prop B Theor.
If four magnitudes are
proportionals, they are proportionals
also when taken inversely.
Prop X Theor.—
That magnitude which has a greater
ratio than another has unto the
same magnitude, is the greater of
the two. and that magnitude to
which the same has a greater
ratio than it has unto another
magnitude is the lesser of the two.
Prop XVI Theor—
If four magnitudes of the same
kind be proportionals they shall also
be proportional when taken alternately.
Prop XXI Theor.
If there be three magnitudes, and other
three, which have the same ratio taken
two and two but in a cross order: if
the first magnitude be greater than the
third the fourth shall be greater
than the sixth; and if equal, equal,
and if less, less.—
Prop G Theor.—
If several ratios be the same with
several ratios, each to each: the
ratio which is compounded of ratios
which are all the same with the first
ratios, each to each, is the same with
the ratio compounded of ratios which are
the same with the other ratios, each to each.
---page break---
Prop III Theor.
If the first be the same multiple of the
second which the third is of the fourth:
and if of the first and third there be taken
equimultiples, these shall be equimultiples,
the one of the second and the
other of the fourth.—
Prop C Theor.
If the first be the same multiple
of the second, or the same part of it
that the third is of the fourth:
the first is to the second as the
third is to the fourth.
Prop XI Theor.
Ratios that are the same to the
same Ratio, are the same to one
another.—
Prop XVII Theor.
If magnitudes taken jointly be proportionals,
when taken separately they shall
also be proportionals: that is, if two
magnitudes together have to one of them
the same ratio which two others have
to one of these, the remaining one
of the first two shall have to the other
the same ratio which the remaining one
of the last two has to the other of these.
Prop XXII Theor.
If there be any number of magnitudes,
and as many others, which taken
two and two in order have the same
ratio. the first shall have to the last
of the first magnitudes, the same
ratio which the first of the others
has to the last. NB. This is usually cited
by the words. "ex aequali" or "ex acquo."
Prop H Theor.
If a ratio compounded of several ratios be the same with a ratio c
of any other ratios, and if any of the first ratios, or a ratio comp
of the first, be the same with one of the last ratios or with the
pounded of any of the last: then the ratio compounded of
maining ratios of the first or the remaining ratio of
if but one remain, is the same with the ratio compou
those remaining of the last or with the remaining ratio
last.
---page break---
Prop IV. Theor
If the first of four, has the s
to the second which the third
fourth: then any equimultip
of the first and third shall have
ratio to any equimultiples of
and fourth, viz. the equimultip
first, shall have the same ratio
the second which the equimultipl
third has to that of the fourth.
Prop D Theo
If the first be to the second
third to the fourth and
be a multiple or part of
the third is the same mu
part of the fourth.
Prop XII The
If any number of magni
proportionals, as of
cedents is to its consequ
shall all the Antecedents
all the Consequents.
Prop XVIII The
If magnitudes taken separate
tionals, they shall also be prop
when taken jointly, that is, if
to the second as the third to
the first and second together
the second as the third and
ther to the fourth.—
Prop XXIII Theor
If there be any number of mag
as many others, which taken tw
a cross order, have the same
first shall have to the last
first magnitudes the same
the first of the others has to the
NB This is usually cited by
exaequali in proportione
or ex acquo perturbato.
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the propositions of the fifth and sixth books of euclid as expressed by simson those written with red ink are added by himself |
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