Barrow, Isaac
,
Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur
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<
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<
s
xml:id
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echoid-s13652
"
xml:space
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preserve
">_Notandum_ eſt eſſe _p_ {2 _m_ - 2 _n_/ } = {ZD {_m_/ }/AD {2 _n_ - _m_/ }}. </
s
>
<
s
xml:id
="
echoid-s13653
"
xml:space
="
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">& </
s
>
<
s
xml:id
="
echoid-s13654
"
xml:space
="
preserve
">_q_ {2 _m_ - 2 _n_/ } =
<
lb
/>
{ZG {_m_/ }/AG {2 _n_ - _m_/ }}. </
s
>
<
s
xml:id
="
echoid-s13655
"
xml:space
="
preserve
">unde erit {ZD {_m_/ }/AD {2 _n_ - _m_/ }} &</
s
>
<
s
xml:id
="
echoid-s13656
"
xml:space
="
preserve
">lt; </
s
>
<
s
xml:id
="
echoid-s13657
"
xml:space
="
preserve
">{ZG {_m_/ }/AG {2 _n_ - _m_/ }}. </
s
>
<
s
xml:id
="
echoid-s13658
"
xml:space
="
preserve
">quare {ZD {_m_/ }/AD {2 _n_ - _m_/ }} eſt mi-
<
lb
/>
<
note
position
="
left
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xlink:label
="
note-0280-01
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xlink:href
="
note-0280-01a
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xml:space
="
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">Fig. 149.</
note
>
nimum.</
s
>
<
s
xml:id
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echoid-s13659
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xml:space
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</
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<
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<
s
xml:id
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echoid-s13660
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xml:space
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">_Exemp_. </
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<
s
xml:id
="
echoid-s13661
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xml:space
="
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">1. </
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>
<
s
xml:id
="
echoid-s13662
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xml:space
="
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">Sit _n_ = 2; </
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>
<
s
xml:id
="
echoid-s13663
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xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s13664
"
xml:space
="
preserve
">_m_ = 3; </
s
>
<
s
xml:id
="
echoid-s13665
"
xml:space
="
preserve
">erit _p_
<
emph
style
="
sub
">2</
emph
>
= {ZD
<
emph
style
="
sub
">3</
emph
>
/AD}. </
s
>
<
s
xml:id
="
echoid-s13666
"
xml:space
="
preserve
">= {BD
<
emph
style
="
sub
">6</
emph
>
/AD
<
emph
style
="
sub
">4</
emph
>
}. </
s
>
<
s
xml:id
="
echoid-s13667
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xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s13668
"
xml:space
="
preserve
">
<
lb
/>
_p_ = {BD
<
emph
style
="
sub
">3</
emph
>
/ADq}. </
s
>
<
s
xml:id
="
echoid-s13669
"
xml:space
="
preserve
">= {ZDq/BD}. </
s
>
<
s
xml:id
="
echoid-s13670
"
xml:space
="
preserve
">Item AD = CA.</
s
>
<
s
xml:id
="
echoid-s13671
"
xml:space
="
preserve
"/>
</
p
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<
p
>
<
s
xml:id
="
echoid-s13672
"
xml:space
="
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">2. </
s
>
<
s
xml:id
="
echoid-s13673
"
xml:space
="
preserve
">Sit _n_ = 3; </
s
>
<
s
xml:id
="
echoid-s13674
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s13675
"
xml:space
="
preserve
">_m_ = 4; </
s
>
<
s
xml:id
="
echoid-s13676
"
xml:space
="
preserve
">erit _p_
<
emph
style
="
sub
">2</
emph
>
= {ZD
<
emph
style
="
sub
">4</
emph
>
/AD
<
emph
style
="
sub
">2</
emph
>
} vel p = {ZD
<
emph
style
="
sub
">2</
emph
>
/AD}
<
lb
/>
= {BD
<
emph
style
="
sub
">4</
emph
>
/AD
<
emph
style
="
sub
">3</
emph
>
} = {ZD
<
emph
style
="
sub
">3</
emph
>
/BD
<
emph
style
="
sub
">2</
emph
>
}. </
s
>
<
s
xml:id
="
echoid-s13677
"
xml:space
="
preserve
">Item AD = 2 CA.</
s
>
<
s
xml:id
="
echoid-s13678
"
xml:space
="
preserve
"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s13679
"
xml:space
="
preserve
">3. </
s
>
<
s
xml:id
="
echoid-s13680
"
xml:space
="
preserve
">Sit _n_ = 5, & </
s
>
<
s
xml:id
="
echoid-s13681
"
xml:space
="
preserve
">_m_ = 8; </
s
>
<
s
xml:id
="
echoid-s13682
"
xml:space
="
preserve
">erit _p_ {6/ } = {ZD
<
emph
style
="
sub
">8</
emph
>
/AD
<
emph
style
="
sub
">2</
emph
>
}. </
s
>
<
s
xml:id
="
echoid-s13683
"
xml:space
="
preserve
">vel _p_
<
emph
style
="
sub
">3</
emph
>
= {ZD
<
emph
style
="
sub
">4</
emph
>
/AD}
<
lb
/>
= {BD
<
emph
style
="
sub
">8</
emph
>
/AD
<
emph
style
="
sub
">5</
emph
>
} = {ZD
<
emph
style
="
sub
">5</
emph
>
/BD
<
emph
style
="
sub
">2</
emph
>
}. </
s
>
<
s
xml:id
="
echoid-s13684
"
xml:space
="
preserve
">Item AD = {2/3} CA.</
s
>
<
s
xml:id
="
echoid-s13685
"
xml:space
="
preserve
"/>
</
p
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<
p
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<
s
xml:id
="
echoid-s13686
"
xml:space
="
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">Quoniam in his _Cyclometriam_ attigi, quid ſi obiter eò ſpectantia
<
lb
/>
_Theoremata_, quæ ad manum, paucula ſubjunxero? </
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>
<
s
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="
echoid-s13687
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xml:space
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preserve
">præſternatur au-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0280-02
"
xlink:href
="
note-0280-02a
"
xml:space
="
preserve
">Fig. 150.</
note
>
tem autem hoc χαυολικὸν:</
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>
<
s
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echoid-s13688
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"/>
</
p
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<
p
>
<
s
xml:id
="
echoid-s13689
"
xml:space
="
preserve
">Sit curva quæpiam AGB, cujus axis AD, & </
s
>
<
s
xml:id
="
echoid-s13690
"
xml:space
="
preserve
">ad hunc ordinatæ
<
lb
/>
rectæ BD, GE; </
s
>
<
s
xml:id
="
echoid-s13691
"
xml:space
="
preserve
">Habebit curva AB ad curvam AG majorem rati-
<
lb
/>
onem quàm recta BD ad rectam GE.</
s
>
<
s
xml:id
="
echoid-s13692
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xml:space
="
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</
p
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<
p
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<
s
xml:id
="
echoid-s13693
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xml:space
="
preserve
">Nam ducatur recta GH ad AD parallela: </
s
>
<
s
xml:id
="
echoid-s13694
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xml:space
="
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">ſecentúrque recta B H
<
lb
/>
punctis Y, & </
s
>
<
s
xml:id
="
echoid-s13695
"
xml:space
="
preserve
">recta GE punctis Z in particulas indefinitè multas; </
s
>
<
s
xml:id
="
echoid-s13696
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xml:space
="
preserve
">pér-
<
lb
/>
que puncta Y, Z ducantur rectæ YM, YN, ZO, ZP ad AD paral-
<
lb
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lelæ: </
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>
<
s
xml:id
="
echoid-s13697
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xml:space
="
preserve
">curvam interſecantes punctis M, N, O, P; </
s
>
<
s
xml:id
="
echoid-s13698
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xml:space
="
preserve
">per quæ ducantur
<
lb
/>
rectæ MR, NS, OT, PV ad BD parallelæ. </
s
>
<
s
xml:id
="
echoid-s13699
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xml:space
="
preserve
">Eſtque angulus BM Y
<
lb
/>
(ut è ſuperius oſtenſis liquet) minor angulo NGS, unde MB. </
s
>
<
s
xml:id
="
echoid-s13700
"
xml:space
="
preserve
">B Y
<
lb
/>
&</
s
>
<
s
xml:id
="
echoid-s13701
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xml:space
="
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">gt; </
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<
s
xml:id
="
echoid-s13702
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xml:space
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">GN. </
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<
s
xml:id
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xml:space
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">NS. </
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<
s
xml:id
="
echoid-s13704
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xml:space
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">Similíque de cauſa eſt NM. </
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<
s
xml:id
="
echoid-s13705
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xml:space
="
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">MR &</
s
>
<
s
xml:id
="
echoid-s13706
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xml:space
="
preserve
">gt; </
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>
<
s
xml:id
="
echoid-s13707
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xml:space
="
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">GN. </
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>
<
s
xml:id
="
echoid-s13708
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xml:space
="
preserve
">NS. </
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>
<
s
xml:id
="
echoid-s13709
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xml:space
="
preserve
">
<
note
symbol
="
*
"
position
="
left
"
xlink:label
="
note-0280-03
"
xlink:href
="
note-0280-03a
"
xml:space
="
preserve
">Vid. _Append_.
<
lb
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Lect. XII.</
note
>
quare conjunctè eft BM + MN + NG. </
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<
s
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echoid-s13710
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xml:space
="
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">BY + MR + NS &</
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>
<
s
xml:id
="
echoid-s13711
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xml:space
="
preserve
">gt;
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s13712
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xml:space
="
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">GN. </
s
>
<
s
xml:id
="
echoid-s13713
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xml:space
="
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">NS; </
s
>
<
s
xml:id
="
echoid-s13714
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xml:space
="
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">hoc eſt arc. </
s
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<
s
xml:id
="
echoid-s13715
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xml:space
="
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">GB. </
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<
s
xml:id
="
echoid-s13716
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xml:space
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">BH &</
s
>
<
s
xml:id
="
echoid-s13717
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xml:space
="
preserve
">gt; </
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>
<
s
xml:id
="
echoid-s13718
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xml:space
="
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">GN. </
s
>
<
s
xml:id
="
echoid-s13719
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xml:space
="
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">NS. </
s
>
<
s
xml:id
="
echoid-s13720
"
xml:space
="
preserve
">rurſus (è diſcur-
<
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/>
ſu conſimili) ratio GN ad NS major eſt ſingulis rationibus OG ad
<
lb
/>
GZ, OP ad PT, & </
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>
<
s
xml:id
="
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xml:space
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">AP ad PV; </
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>
<
s
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="
echoid-s13722
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xml:space
="
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">idcircoq; </
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>
<
s
xml:id
="
echoid-s13723
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xml:space
="
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">junctè eſt GN. </
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<
s
xml:id
="
echoid-s13724
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xml:space
="
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">NS &</
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<
s
xml:id
="
echoid-s13725
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xml:space
="
preserve
">gt; </
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>
<
s
xml:id
="
echoid-s13726
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="
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">
<
lb
/>
arc. </
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<
s
xml:id
="
echoid-s13727
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xml:space
="
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">AG. </
s
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<
s
xml:id
="
echoid-s13728
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xml:space
="
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">GE. </
s
>
<
s
xml:id
="
echoid-s13729
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xml:space
="
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">quapropter erit GB. </
s
>
<
s
xml:id
="
echoid-s13730
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xml:space
="
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">BH &</
s
>
<
s
xml:id
="
echoid-s13731
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xml:space
="
preserve
">gt; </
s
>
<
s
xml:id
="
echoid-s13732
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xml:space
="
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">AG. </
s
>
<
s
xml:id
="
echoid-s13733
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xml:space
="
preserve
">GE. </
s
>
<
s
xml:id
="
echoid-s13734
"
xml:space
="
preserve
">permutan-
<
lb
/>
doque GB. </
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<
s
xml:id
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echoid-s13735
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xml:space
="
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">AG &</
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>
<
s
xml:id
="
echoid-s13736
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xml:space
="
preserve
">gt; </
s
>
<
s
xml:id
="
echoid-s13737
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xml:space
="
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">BH. </
s
>
<
s
xml:id
="
echoid-s13738
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xml:space
="
preserve
">GE. </
s
>
<
s
xml:id
="
echoid-s13739
"
xml:space
="
preserve
">quare componendo eſt AB. </
s
>
<
s
xml:id
="
echoid-s13740
"
xml:space
="
preserve
">A G
<
lb
/>
&</
s
>
<
s
xml:id
="
echoid-s13741
"
xml:space
="
preserve
">gt; </
s
>
<
s
xml:id
="
echoid-s13742
"
xml:space
="
preserve
">BD. </
s
>
<
s
xml:id
="
echoid-s13743
"
xml:space
="
preserve
">GE.</
s
>
<
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="
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xml:space
="
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