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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> | <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/>proportional, they are proportionals<lb/>also when taken inversely.</p> | ||
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
proportional, they are proportionals
also when taken inversely.
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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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