ed by Simson those written with red ink are added by himself
-of four, has the same ratio
which the third has to the
ny equimultiples whatever
third shall have the same
uimultiples of the second
the equimultiple of the
the same ratio to that of
the equimultiple of the
that of the fourth.
p D Theor.
to the second as the
fourth and if the first
le or part of the second
the same multiple or
p XII Theor -ber of magnitudes be
ls, as of the Ante-
its consequent, so
Antecedents be to
XVIII Theor -taken separate be proper
all also be proportionals
tly, that is, if the first be
the third to the fourth
cond together shall be to
the third and fourth toge-
number of magnitudes &
which taken two & two in
ave the same ratio: the
ave to the last of the
tudes the same ratio which
other has to the last.
sually cited by the words
with a ratio compounded
a ratio compounded of any
tios or with the ratiocom-
pounded of the re-
ratio of the first
ratio compounded of
maining ratio of the
Prop V Theor.
If one magnitude be the same multiple of another
which a magnitude taken from the first is of a
magnitude taken from the other: the remainder
shall be the same multiple of the remainder
that the whole is of the whole.
Prop VII Theor.
Equal magnitudes have the same ratio
to the same magnitude: and the same has
the same ratio to equal magnitudes.
Prop XIII Theor.
If the first has to the second the same Ratio which
the third has to the fourth, but the third to the fourth
a greater ratio than the fifth has to the sixth:
the first shall also have to the second a greater
ratio than the fifth has to the sixth.
Prop XIX Theor.
If a whole magnitude be to a whole as a magnitude
taken from the first is to a magnitude taken from
the other: the remainder shall be to the remainder
as the whole to the whole.
Prop XXIV Theor.
If the first has to the second the same ratio which the
third has to the fourth and the fifth to the second the
same ratio which the sixth has to the fourth; the
first and fifth together shall have to the second
the same ratio which the third and sixth
together have to the fourth.
Prop K. Theor.
If there be any number of ratios, and any number of other ratios such, that the ratio compounded of ratios which
are the same with the first ratios, each to each, is the same with the ratio compounded of ratios which are the same
each to each with the last ratios: and if one of the first ratios is the ratio which is compounded of ratios which are the
same with several of the first ratios each to each, be the same with one of the last ratios or with the ratios
compounded of ratios which are the same each, to each, with several of the last ratios: then the ratio compounded of
ratios which are the same with the remaining Ratios of the first each to each, or the remaining ratio of the first
if but one remain: is the same with the ratio compounded with ratios which are the same with those remaining
of the last each to each, or with the remaining ratio of the last.—
Prop VI Theor.
If two magnitudes be equimultiples of two others,
and if Equimultiples of these be taken from the
first two the remainders are eithr equal to these
others or equimultiples of them.
Prop VIII Theor.
Of unequal magnitudes the greater has a
greater to ratio to the same than the less has:
and the same magnitude has a greater ratio to
the less than it has to the greater.
Prop XIV Theor. If the first has to the second the same ratio, which
the third has to the fourth: then if the first be greater
than the third the Second shall be greater than
the fourth, if equal, equal, and if less, less.
Prop E Theor.
If four magnitudes be proportionals, they are also
proportional by Conversion: that is the first is to its
excess above the second as the third is to its
excess above the fourth.—
Prop XXV Theor.
If four magnitudes are proportionals, the greatest
and least of them together shall have the same
are greater than the other two together.—
Identifier: | JB/135/072/004
"JB/" can not be assigned to a declared number type with value 135.
the propositions of the fifth and sixth books of euclid as expressed by simson those written with red ink are added by himself
sir samuel bentham
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