Vitruvius
,
De architectura libri decem
,
1567
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illa h i. </
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>
<
s
xml:id
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"
xml:space
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preserve
">quæ lineam a c, tangat, in ſigno, i, a quo ſigno cadat æquidiſtans ipſi a f, & </
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>
<
s
xml:id
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echoid-s20385
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xml:space
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">in ſignum d,
<
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terminet, & </
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>
<
s
xml:id
="
echoid-s20386
"
xml:space
="
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">ſit illa i d. </
s
>
<
s
xml:id
="
echoid-s20387
"
xml:space
="
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">His poſitis, ut facilius res explicetur, uocabimus lineas illas a b. </
s
>
<
s
xml:id
="
echoid-s20388
"
xml:space
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">f &</
s
>
<
s
xml:id
="
echoid-s20389
"
xml:space
="
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">, h i, d
<
lb
/>
c, primas æquidiſtantes, ſed lincas a f. </
s
>
<
s
xml:id
="
echoid-s20390
"
xml:space
="
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">g h. </
s
>
<
s
xml:id
="
echoid-s20391
"
xml:space
="
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">i d. </
s
>
<
s
xml:id
="
echoid-s20392
"
xml:space
="
preserve
">ſecundas æquidiſtantes, ſimiliter duo erunt maiora
<
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/>
trigona, quorum primum erit a b c, cuius angulus b, rectus est. </
s
>
<
s
xml:id
="
echoid-s20393
"
xml:space
="
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">aliud uero trigonum est a f c, & </
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>
<
s
xml:id
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echoid-s20394
"
xml:space
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preserve
">il-
<
lb
/>
lud primum, hoc ſecundum trigonum appellabimus. </
s
>
<
s
xml:id
="
echoid-s20395
"
xml:space
="
preserve
">in primo inerant trigona, quæ a primis æquidiſtantibus
<
lb
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fiunt, & </
s
>
<
s
xml:id
="
echoid-s20396
"
xml:space
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">ſunt g f c. </
s
>
<
s
xml:id
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echoid-s20397
"
xml:space
="
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">i h e. </
s
>
<
s
xml:id
="
echoid-s20398
"
xml:space
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">c d e. </
s
>
<
s
xml:id
="
echoid-s20399
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20400
"
xml:space
="
preserve
">hæc quoniam æquis angulis conſtant,ut patet ex vigeſimanona
<
lb
/>
primi, proportione reſpondentia latera, habent, ut ex quarta ſexti elicitur. </
s
>
<
s
xml:id
="
echoid-s20401
"
xml:space
="
preserve
">Similiter quoniam ſecunda trigona,
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s20402
"
xml:space
="
preserve
">ſecundis æquidistantibus lineis facta, æqualibus angulis conſtant, ſine dubio latera quoque habebunt mu
<
lb
/>
tua ratione comparabilia. </
s
>
<
s
xml:id
="
echoid-s20403
"
xml:space
="
preserve
">Igitur quemadmodum in primis trigonis æquidistantibus proportione ſibi reſpon-
<
lb
/>
dent eæ lineæ, ita ut ſicuti ſe habet a e, ad e g, itaſe habent b e, ad e f. </
s
>
<
s
xml:id
="
echoid-s20404
"
xml:space
="
preserve
">ita in ſecundis æquidistan-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.305-01
"
xlink:href
="
note-514.01.305-01a
"
xml:space
="
preserve
">10</
note
>
tibus,ut ſe habent a e, ad e g, ſic ſe habet f e, ad e h, & </
s
>
<
s
xml:id
="
echoid-s20405
"
xml:space
="
preserve
">iterum quemadmodum in primis, ut ſe
<
lb
/>
habet f e. </
s
>
<
s
xml:id
="
echoid-s20406
"
xml:space
="
preserve
">ad e h, ita g e, ad e i, ita in ſecundis parallelis, ut ſe habebit g e, ad e i, ita ſeha-
<
lb
/>
bebit b e, ad e d, ſunt igitur continuo proportionales a b, f g, h i, c d, quoniam ſicut eſt b c,
<
lb
/>
ad e f, ita eſt a b. </
s
>
<
s
xml:id
="
echoid-s20407
"
xml:space
="
preserve
">ad f g, &</
s
>
<
s
xml:id
="
echoid-s20408
"
xml:space
="
preserve
">ſicuti ſe habet f c, ad c h, ita ſe habet f g, ad h i, & </
s
>
<
s
xml:id
="
echoid-s20409
"
xml:space
="
preserve
">uti ſe ha-
<
lb
/>
bet g e, ad e d, ita h i, ad c d. </
s
>
<
s
xml:id
="
echoid-s20410
"
xml:space
="
preserve
">Propoſitis igitur duabus rectis a b, & </
s
>
<
s
xml:id
="
echoid-s20411
"
xml:space
="
preserve
">c d, duas medias in-
<
lb
/>
uenimus ſcilicet f g, & </
s
>
<
s
xml:id
="
echoid-s20412
"
xml:space
="
preserve
">h i, quod facere propoſitum fuit. </
s
>
<
s
xml:id
="
echoid-s20413
"
xml:space
="
preserve
">Hæc Eratoſthenis demonstratio est, & </
s
>
<
s
xml:id
="
echoid-s20414
"
xml:space
="
preserve
">licet
<
lb
/>
uelit ipſe lineas a b, & </
s
>
<
s
xml:id
="
echoid-s20415
"
xml:space
="
preserve
">c d, rectas eſſe ſupra lineam b d, non tamen eſt dubitandum, quin eadem con-
<
lb
/>
cluſio colligatur quocunque modo utraque linea cadat in b d, modo ſimiles utraque faciat angulos, & </
s
>
<
s
xml:id
="
echoid-s20416
"
xml:space
="
preserve
">ſi-
<
lb
/>
mili ſint ratione æquidis̃tantes. </
s
>
<
s
xml:id
="
echoid-s20417
"
xml:space
="
preserve
">Quoniam res omnis in eo ſita est, quod ex his trigonorum angulos æquales ha
<
lb
/>
bentium,latera proportione ſibi reſpondent. </
s
>
<
s
xml:id
="
echoid-s20418
"
xml:space
="
preserve
">Quareſiplures duabus medias inter duas datas a b, & </
s
>
<
s
xml:id
="
echoid-s20419
"
xml:space
="
preserve
">c d,
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.305-02
"
xlink:href
="
note-514.01.305-02a
"
xml:space
="
preserve
">20</
note
>
inuenire mens fuerit, eodem modo plures parallelas, ſiue ex primis, ſiue ex ſecundis formabimus. </
s
>
<
s
xml:id
="
echoid-s20420
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xml:space
="
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">Organum
<
lb
/>
uerò quo demonſtrationis præhabitæ uſus haberi poſſit huiuſmodi fabrica struitur. </
s
>
<
s
xml:id
="
echoid-s20421
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xml:space
="
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">Lamnam æream; </
s
>
<
s
xml:id
="
echoid-s20422
"
xml:space
="
preserve
">uel li-
<
lb
/>
gneam tabellam ſume quadrangularem oblongam rectis angulis quadratam, ſitq; </
s
>
<
s
xml:id
="
echoid-s20423
"
xml:space
="
preserve
">ea in exemplum a b d
<
lb
/>
c, ſi deformare duas medias ratione comparabiles duabus datis, uelimus,aliquot lamnellas ſuper eam opor-
<
lb
/>
tebit aptare hoc modo. </
s
>
<
s
xml:id
="
echoid-s20424
"
xml:space
="
preserve
">Sume lamnellas tenues ex aliqua materia ſolida quadrangulares, unam ex ijs in media
<
lb
/>
tabula obfirmato ita, ut mouerinequeat, ſitq́;</
s
>
<
s
xml:id
="
echoid-s20425
"
xml:space
="
preserve
">ea e f g h, & </
s
>
<
s
xml:id
="
echoid-s20426
"
xml:space
="
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">in ſignis e, & </
s
>
<
s
xml:id
="
echoid-s20427
"
xml:space
="
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">f, affixis habeatregulas,
<
lb
/>
ut circumuolui poſſin in ſuis axiculis. </
s
>
<
s
xml:id
="
echoid-s20428
"
xml:space
="
preserve
">Eſto regula una e m, altera f n. </
s
>
<
s
xml:id
="
echoid-s20429
"
xml:space
="
preserve
">Sed lamnella alia ſit k d e,
<
lb
/>
ita in maiori lamna collocata ut admoueri poſſit uerſus obfirmatam lamnellam e f g h, atque etiam ab ea
<
lb
/>
amoueri, ita ut latera ſemper habeat æquidiſtantia lateri f h, ſitq́ in ea etiam regula in ſigno k,quæitem
<
lb
/>
conuolui poſſit, & </
s
>
<
s
xml:id
="
echoid-s20430
"
xml:space
="
preserve
">ſit illa k o, quæ cum alijsregulis ſcilicet e m, & </
s
>
<
s
xml:id
="
echoid-s20431
"
xml:space
="
preserve
">f n, ita ponatur, ut omnes ſint pa-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.305-03
"
xlink:href
="
note-514.01.305-03a
"
xml:space
="
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">30</
note
>
rallelæ, & </
s
>
<
s
xml:id
="
echoid-s20432
"
xml:space
="
preserve
">earum communes ſectiones in lineis a g, f h, & </
s
>
<
s
xml:id
="
echoid-s20433
"
xml:space
="
preserve
">l, ſigno, ſint in eademrecta m n l o, ſi-
<
lb
/>
militer a m, ſitæqualis d k. </
s
>
<
s
xml:id
="
echoid-s20434
"
xml:space
="
preserve
">quoniam a m, uel minimo diſcrimineaccedit d k. </
s
>
<
s
xml:id
="
echoid-s20435
"
xml:space
="
preserve
">Hæc, cum ita ſint ex-
<
lb
/>
pedita inter duas rectas a b, & </
s
>
<
s
xml:id
="
echoid-s20436
"
xml:space
="
preserve
">c d, dantur duæ mediæ comparabiles ratione continua, quæſunt e n,
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s20437
"
xml:space
="
preserve
">f o, quod manifeſtum eſt ob ſupradictas rationes. </
s
>
<
s
xml:id
="
echoid-s20438
"
xml:space
="
preserve
">Cæterum ſi ſorte duæ propoſitælineæ, quales ſunt ſ,
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s20439
"
xml:space
="
preserve
">t, quibus opus ſit medias comparabiles inuenire, non erunt æquales lineis in organo poſitis, quæſunt a
<
lb
/>
b, & </
s
>
<
s
xml:id
="
echoid-s20440
"
xml:space
="
preserve
">e r. </
s
>
<
s
xml:id
="
echoid-s20441
"
xml:space
="
preserve
">efficiatur ut quemadmodum ſe habet ſ, ad t, itaſe habeat a b, ad r d, efficietur autem
<
lb
/>
ſi lamella k d e, prore proprius lamellam obfirmatam admouebitur, uel retrocedet in pari ſemper, & </
s
>
<
s
xml:id
="
echoid-s20442
"
xml:space
="
preserve
">
<
lb
/>
æquidis̃tanti laterum ratione. </
s
>
<
s
xml:id
="
echoid-s20443
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20444
"
xml:space
="
preserve
">quoniam ipſi a b. </
s
>
<
s
xml:id
="
echoid-s20445
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20446
"
xml:space
="
preserve
">r d, quæ ſunt in organo duæ mediæ compara-
<
lb
/>
biles inuentæ ſunt, conſequenter ipſi s. </
s
>
<
s
xml:id
="
echoid-s20447
"
xml:space
="
preserve
">ipſi t. </
s
>
<
s
xml:id
="
echoid-s20448
"
xml:space
="
preserve
">propoſitis, duæ erunt mediæ mediæ comparabiles inuentæ. </
s
>
<
s
xml:id
="
echoid-s20449
"
xml:space
="
preserve
">quo
<
lb
/>
igitur maiori artificio elaboratum organum feceris, eo facilius duas inuenies medias proportione conti-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.305-04
"
xlink:href
="
note-514.01.305-04a
"
xml:space
="
preserve
">40</
note
>
nua reſpondentes, ideo capita mobilium lamnellarum in ſuos utrinque canales immittentur, ita ut per
<
lb
/>
lamnam decurrant leniter, & </
s
>
<
s
xml:id
="
echoid-s20450
"
xml:space
="
preserve
">æquabiliter. </
s
>
<
s
xml:id
="
echoid-s20451
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20452
"
xml:space
="
preserve
">Quod ſi plures quàm duas medias inuenire quis uelit,
<
lb
/>
lamnellarum, & </
s
>
<
s
xml:id
="
echoid-s20453
"
xml:space
="
preserve
">regularum additamento id commode faciet. </
s
>
<
s
xml:id
="
echoid-s20454
"
xml:space
="
preserve
">animaduertendum autem oportere lon-
<
lb
/>
gas eſſe regulas, ut cum opus fuerit lamnellas abſcedere a media, regulæ ad ſectiones linearum prædi-
<
lb
/>
ctas pertingere poſſint, tangantq́;</
s
>
<
s
xml:id
="
echoid-s20455
"
xml:space
="
preserve
">ſuperius organi latus regulæ omnes, e m, f x, r u, immo ( ut me-
<
lb
/>
lius dicam ) tam longæ ſint, quantum eſſet diagonia lamnellæ obfirmatæ e f g h. </
s
>
<
s
xml:id
="
echoid-s20456
"
xml:space
="
preserve
">Atque hæc innentio fuit
<
lb
/>
Eratoſthenis.</
s
>
<
s
xml:id
="
echoid-s20457
"
xml:space
="
preserve
">Sed ut facilior uſus inſrumenti appareat,nec ſolum inter duas duas,ſed inter duas plures compa-
<
lb
/>
rabiles inuenire poſſimus.</
s
>
<
s
xml:id
="
echoid-s20458
"
xml:space
="
preserve
">Sint duærectæ a b. </
s
>
<
s
xml:id
="
echoid-s20459
"
xml:space
="
preserve
">c d. </
s
>
<
s
xml:id
="
echoid-s20460
"
xml:space
="
preserve
">incidant ambæ in rectam, & </
s
>
<
s
xml:id
="
echoid-s20461
"
xml:space
="
preserve
">æquidiſtent,tantumq́
<
lb
/>
addatur lineæ c d. </
s
>
<
s
xml:id
="
echoid-s20462
"
xml:space
="
preserve
">ut æqua ſit lineæ a b. </
s
>
<
s
xml:id
="
echoid-s20463
"
xml:space
="
preserve
">cuius ſummum ſit e. </
s
>
<
s
xml:id
="
echoid-s20464
"
xml:space
="
preserve
">Ab a, uſque ad e, ducatur linea,& </
s
>
<
s
xml:id
="
echoid-s20465
"
xml:space
="
preserve
">
<
lb
/>
ita claudatur ſuperficies a b c e. </
s
>
<
s
xml:id
="
echoid-s20466
"
xml:space
="
preserve
">Partiatur inde b c. </
s
>
<
s
xml:id
="
echoid-s20467
"
xml:space
="
preserve
">in partes tres, quarum una ſit ubi f. </
s
>
<
s
xml:id
="
echoid-s20468
"
xml:space
="
preserve
">ultra f ali-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.305-05
"
xlink:href
="
note-514.01.305-05a
"
xml:space
="
preserve
">50</
note
>
quanto amplius notetur g. </
s
>
<
s
xml:id
="
echoid-s20469
"
xml:space
="
preserve
">ut a puncto b. </
s
>
<
s
xml:id
="
echoid-s20470
"
xml:space
="
preserve
">ad g. </
s
>
<
s
xml:id
="
echoid-s20471
"
xml:space
="
preserve
">ſit pluſquam tertia pars lineæ b c. </
s
>
<
s
xml:id
="
echoid-s20472
"
xml:space
="
preserve
">ſimili ratione in li-
<
lb
/>
nea a c. </
s
>
<
s
xml:id
="
echoid-s20473
"
xml:space
="
preserve
">notetur ſignum, quod ab a distet quantum g. </
s
>
<
s
xml:id
="
echoid-s20474
"
xml:space
="
preserve
">ab ipſo b. </
s
>
<
s
xml:id
="
echoid-s20475
"
xml:space
="
preserve
">et ſit illud h. </
s
>
<
s
xml:id
="
echoid-s20476
"
xml:space
="
preserve
">nectatur g, cum
<
lb
/>
a, & </
s
>
<
s
xml:id
="
echoid-s20477
"
xml:space
="
preserve
">h. </
s
>
<
s
xml:id
="
echoid-s20478
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20479
"
xml:space
="
preserve
">a. </
s
>
<
s
xml:id
="
echoid-s20480
"
xml:space
="
preserve
">cum d. </
s
>
<
s
xml:id
="
echoid-s20481
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20482
"
xml:space
="
preserve
">g h. </
s
>
<
s
xml:id
="
echoid-s20483
"
xml:space
="
preserve
">ſecet a d in i. </
s
>
<
s
xml:id
="
echoid-s20484
"
xml:space
="
preserve
">ſimiliter ſecetur tantum de a b, quantum est à g.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s20485
"
xml:space
="
preserve
">ad i. </
s
>
<
s
xml:id
="
echoid-s20486
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20487
"
xml:space
="
preserve
">ſpatium illud ſit b k. </
s
>
<
s
xml:id
="
echoid-s20488
"
xml:space
="
preserve
">ducaturq́ linea ab i ad k. </
s
>
<
s
xml:id
="
echoid-s20489
"
xml:space
="
preserve
">donec tangat lineam g a. </
s
>
<
s
xml:id
="
echoid-s20490
"
xml:space
="
preserve
">in l. </
s
>
<
s
xml:id
="
echoid-s20491
"
xml:space
="
preserve
">Quo-
<
lb
/>
niam ergo ex trigeſiſma tertia primi elementoruma b. </
s
>
<
s
xml:id
="
echoid-s20492
"
xml:space
="
preserve
">æquidiſtat g i h, & </
s
>
<
s
xml:id
="
echoid-s20493
"
xml:space
="
preserve
">ex poſitione g i, & </
s
>
<
s
xml:id
="
echoid-s20494
"
xml:space
="
preserve
">b k. </
s
>
<
s
xml:id
="
echoid-s20495
"
xml:space
="
preserve
">
<
lb
/>
ſunt æquales, ſequitur b g. </
s
>
<
s
xml:id
="
echoid-s20496
"
xml:space
="
preserve
">æquidiſtare i l. </
s
>
<
s
xml:id
="
echoid-s20497
"
xml:space
="
preserve
">Præterea de lineis g c. </
s
>
<
s
xml:id
="
echoid-s20498
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20499
"
xml:space
="
preserve
">h c. </
s
>
<
s
xml:id
="
echoid-s20500
"
xml:space
="
preserve
">auferantur æquales par-
<
lb
/>
tes duæ uerſus i l. </
s
>
<
s
xml:id
="
echoid-s20501
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20502
"
xml:space
="
preserve
">ſint illæ g m. </
s
>
<
s
xml:id
="
echoid-s20503
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20504
"
xml:space
="
preserve
">h n. </
s
>
<
s
xml:id
="
echoid-s20505
"
xml:space
="
preserve
">iunganturq́; </
s
>
<
s
xml:id
="
echoid-s20506
"
xml:space
="
preserve
">i m, & </
s
>
<
s
xml:id
="
echoid-s20507
"
xml:space
="
preserve
">m n, ex præallegata propoſitione
<
lb
/>
g l. </
s
>
<
s
xml:id
="
echoid-s20508
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20509
"
xml:space
="
preserve
">m i. </
s
>
<
s
xml:id
="
echoid-s20510
"
xml:space
="
preserve
">erunt æquidiſtantes, & </
s
>
<
s
xml:id
="
echoid-s20511
"
xml:space
="
preserve
">eadem ratione g h, & </
s
>
<
s
xml:id
="
echoid-s20512
"
xml:space
="
preserve
">m n. </
s
>
<
s
xml:id
="
echoid-s20513
"
xml:space
="
preserve
">ſecet quoq; </
s
>
<
s
xml:id
="
echoid-s20514
"
xml:space
="
preserve
">linea m n. </
s
>
<
s
xml:id
="
echoid-s20515
"
xml:space
="
preserve
">lineam a
<
lb
/>
d. </
s
>
<
s
xml:id
="
echoid-s20516
"
xml:space
="
preserve
">in o. </
s
>
<
s
xml:id
="
echoid-s20517
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20518
"
xml:space
="
preserve
">ſumatur de b k tantum,quantum eſt m o. </
s
>
<
s
xml:id
="
echoid-s20519
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20520
"
xml:space
="
preserve
">ſit pars illa b p. </
s
>
<
s
xml:id
="
echoid-s20521
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s20522
"
xml:space
="
preserve
">ab o. </
s
>
<
s
xml:id
="
echoid-s20523
"
xml:space
="
preserve
">uerſus p. </
s
>
<
s
xml:id
="
echoid-s20524
"
xml:space
="
preserve
">du-
<
lb
/>
catur linea, donec tangat i m. </
s
>
<
s
xml:id
="
echoid-s20525
"
xml:space
="
preserve
">in q. </
s
>
<
s
xml:id
="
echoid-s20526
"
xml:space
="
preserve
">Si ergo linea m e, erit æqualis lineæ o q. </
s
>
<
s
xml:id
="
echoid-s20527
"
xml:space
="
preserve
">bene erit. </
s
>
<
s
xml:id
="
echoid-s20528
"
xml:space
="
preserve
">Sedſi m c. </
s
>
<
s
xml:id
="
echoid-s20529
"
xml:space
="
preserve
">
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.305-06
"
xlink:href
="
note-514.01.305-06a
"
xml:space
="
preserve
">60</
note
>
</
s
>
</
p
>
</
div
>
</
div
>
</
text
>
</
echo
>