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<head>The Propositions of the Fifth and Sixth Books of Euclid as expressed by</head><p><head>Prop I Theor</head><lb/>If any number of magnitudes be<lb/>equimultiples of as many, each of each, what<lb/>multiple soever any one of them is of<lb/>its part, the same multiple shall all<lb/>the first magnitudes be of all the<lb/>other.</p>-----<p><head>Prop A Theor</head><lb/>If the first of the four magnitudes has<lb/>to the second the same ratio which<lb/>the third has to the fourth, then if<lb/>the first be greater than the second<lb/>the third is also greater than the fourth<lb/>and if equal, equal: if less, less.</p>-----<p><head>Prop IX Theor.</head><lb/>Magnitudes which have the<lb/>same ratio to the same magnitude<lb/>are equal to one another: and<lb/>those to which the same magnitude<lb/>has the same ratio are equal to one<lb/>another.</p>-----<p><head>Prop XV Theor. -</head><lb/>Magnitudes have the same ratio<lb/>to one another which their<lb/>equimultiples have.</p>-----<p><head>Prop XX Theor -</head><lb/>If there be three magnitudes and other<lb/>three, which taken two and two have<lb/>the same ratio, if the first be greater<lb/>than the third the fourth shall be<lb/>greater than the sixth: and if equal,<lb/>equal, and if less, less —</p>-----<p><head>Prop F Theor.</head><lb/>Ratios which are compounded of<lb/>the same Ratios are the same<lb/>with one another.</p><pb/><p><head>Prop II Theor. -</head><lb/>If the first magnitude, be the same<lb/>multiple of the second that the third is of the<lb/>fourth, and the fifth the same multiple<lb/>of the second that the sixth is of the fourth:<lb/>then shall the first together with the<lb/>fifth, be the same multiple of the second<lb/>that the third together with the sixth is<lb/>of the fourth. —</p>-----<p><head>Prop B Theor.</head><lb/>If four magnitudes are<lb/> | <head>The Propositions of the Fifth and Sixth Books of Euclid as expressed by</head><p><head>Prop I Theor</head><lb/>If any number of magnitudes be<lb/>equimultiples of as many, each of each, what<lb/>multiple soever any one of them is of<lb/>its part, the same multiple shall all<lb/>the first magnitudes be of all the<lb/>other.</p>-----<p><head>Prop A Theor</head><lb/>If the first of the four magnitudes has<lb/>to the second the same ratio which<lb/>the third has to the fourth, then if<lb/>the first be greater than the second<lb/>the third is also greater than the fourth<lb/>and if equal, equal: if less, less.</p>-----<p><head>Prop IX Theor.</head><lb/>Magnitudes which have the<lb/>same ratio to the same magnitude<lb/>are equal to one another: and<lb/>those to which the same magnitude<lb/>has the same ratio are equal to one<lb/>another.</p>-----<p><head>Prop XV Theor. -</head><lb/>Magnitudes have the same ratio<lb/>to one another which their<lb/>equimultiples have.</p>-----<p><head>Prop XX Theor -</head><lb/>If there be three magnitudes and other<lb/>three, which taken two and two have<lb/>the same ratio, if the first be greater<lb/>than the third the fourth shall be<lb/>greater than the sixth: and if equal,<lb/>equal, and if less, less —</p>-----<p><head>Prop F Theor.</head><lb/>Ratios which are compounded of<lb/>the same Ratios are the same<lb/>with one another.</p><pb/><p><head>Prop II Theor. -</head><lb/>If the first magnitude, be the same<lb/>multiple of the second that the third is of the<lb/>fourth, and the fifth the same multiple<lb/>of the second that the sixth is of the fourth:<lb/>then shall the first together with the<lb/>fifth, be the same multiple of the second<lb/>that the third together with the sixth is<lb/>of the fourth. —</p>-----<p><head>Prop B Theor.</head><lb/>If four magnitudes are<lb/>proportionals, they are proportionals<lb/>also when taken inversely.</p>-----<p><head>Prop X Theor. -</head><lb/>That magnitude which has a greater<lb/>ratio than another has unto the<lb/>same magnitude, is the greater of<lb/>the two. and that magnitude to<lb/>which the same has a greater<lb/>ratio than it has unto another<lb/>magnitude is the lesser of the two.</p>-----<p><head>Prop XVI Theor -</head><lb/>If four magnitudes of the same<lb/>kind be proportionals they shall also<lb/>be proportional when taken alternately.</p>-----<p><head>Prop XXI Theor.</head><lb/>If there be three magnitudes, and other<lb/>three, which have the same ratio taken<lb/>two and two but in a cross order: if<lb/>the first magnitude be greater than the<lb/>third the fourth shall be greater<lb/>than the sixth; and if equal, equal,<lb/>and if less, less. —</p>-----<p><head>Prop G Theor -</head><lb/>If several ratios be the same with<lb/>several ratios, each to each: the<lb/>ratio which is compounded of ratios<lb/>which are all the same with the first<lb/>ratios, each to each, is the same with<lb/>the ratio compounded of ratios which are<lb/>the same with the other ratios, each to each.</p><pb/> | ||
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---
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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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private material |
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sir samuel bentham |
[[watermarks::[tall thin motif with prince of wales feathers] icv]] |
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