Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secūde partis
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25
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aggregato ex .b.c. / quod fuit probandum </
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>
<
s
xml:id
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N1232C
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xml:space
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preserve
">Sed iam
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probo / facta tali variatione aggregatum ex .a.
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d. componitur ex duobus equalibus adequate il-
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lis duobus ex quibus adequate componitur ag-
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gregatum ex .b.c. / quia facta tali variatione .a. ef-
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ficit̄̄ eq̈le ipſi b. et d. efficit̄̄ eq̈le ipſi .c. / vt ↄ̨ſtat: igit̄̄
<
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/>
facta tali variatiõe aggregatū ex a.d. ↄ̨ponit̄̄ ade
<
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te ex duobus aqualibus illis duobus puta .b.c. ex
<
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quibus componitur adequate aggregatum ex .b.
<
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/>
c. / quod fuit oſtendēdum. </
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>
<
s
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N12341
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xml:space
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">Et ſic patet prima pars
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</
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<
s
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N12345
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xml:space
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">Secūda pars probatur: et ſint a.b.c.d. quattuor
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numeri a.d. circūſtantes .b. vero et .c. intermedii et
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diſtet .a. ab .b.g. differētia et .c. excedat .d. / tunc dico /
<
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ſi aggregatū ex .b.c. eſt equale aggregato ex .a.
<
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/>
d.b.c. equaliter diſtant ab .a.d. </
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>
<
s
xml:id
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N12350
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xml:space
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">Quod ſic proba-
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tur / quia .a diſtat a.b.g. differentia: et .c.a.d. diſtat
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eadē differētia. </
s
>
<
s
xml:id
="
N12357
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xml:space
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">igitur illi intermedii equaliter di
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ſtãt ab illis extremis. </
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<
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N1235C
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xml:space
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">Probatur minor / quia ſi .c.
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non eadem differentia diſtat a.d. ſicut a. ab .b. ca-
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pio / igitur vnum terminū qui ſit .f. a quo .c. diſtet
<
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/>
eadē differentia qua .a. diſtat ab .b. / et tunc ex prio
<
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/>
ri parte aggregatuꝫ ex a. et .f. eſt equale aggrega
<
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/>
to ex .b.c. et per te aggregatum ex .a.d. eſt ēt equa-
<
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/>
le aggregato ex .b.c: igitur aggregatum ex .a.f. eſt
<
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/>
equale aggregato ex .a.d. / patet conſequentia ꝑ il
<
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/>
laꝫ dignitatē que eidē tertio equantur inter ſe ſūt
<
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equalia. </
s
>
<
s
xml:id
="
N12371
"
xml:space
="
preserve
">et vltra aggregatum ex .a.f. eſt equale ag
<
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/>
gregato ex .a.d. / ergo ſequitur / eodeꝫ cõmuni dē
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/>
pto puta a. reſidua manebunt equalia videlicet .f.
<
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et .d. et .c. diſtat .g. differētia qua a. diſtat ab .b. ab
<
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ipſo .f. / ergo .c. diſtat .g. differentia ab ipſo .d. / et ſic
<
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b.c. equaliter diſtant ab .a.d. numeris circunſtan-
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tibus / quod fuit probandum. </
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>
<
s
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N12380
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xml:space
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">Patet tamen conſe
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quentia / quia que ſunt equalia qualiter diſtant a
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quouis tertio
<
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xlink:href
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xml:id
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xml:space
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">īueſtigat̄̄
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itas ſe
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cūde con
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cluſionis
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Iordanꝰ
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.1. ele.</
note
>
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Hec cõcluſio in propria forma in
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ſtantiam patitur: ſed ſic poſita eſt / quia ita poni
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tur a iordano primo elementorum. </
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>
<
s
xml:id
="
N12394
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xml:space
="
preserve
">Nam iſti nu-
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meri .8.8. equaliter diſtãt ab his duobus .4.4. in
<
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iſta ſerie .4.8.8.4. / et tamen extrema coniūcta nõ
<
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equantur mediis. </
s
>
<
s
xml:id
="
N1239D
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xml:space
="
preserve
">Item iſti duo numeri .4.1. equa
<
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liter diſtant ab his duobus extremis .8.5. in iſta
<
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ſeries .8.4.1.5. / et tamen medii iuncti non equãtur
<
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extremis coniunctis / vt conſtat. </
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>
<
s
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="
N123A6
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xml:space
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">Item illi numeri .
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4. et .4. coniuncti equantur his numeris ſimul iun
<
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ctis .4. et .4. / et tamen duo intermedii non equali
<
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ter diſtant a duobus extremis: quia non diſtant.
<
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<
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position
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left
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xlink:href
="
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xml:id
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xml:space
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">Senſus
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ſecūde cõ
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cluſionis</
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</
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xml:space
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">¶ Intellige igitur concluſionē in ſenſu in quo ma
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thematici eam intelligunt. </
s
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<
s
xml:id
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xml:space
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">puta / ſi duo nume-
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ri equaliter diſtēt a duobus numeris extrimis ita
<
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primus excedat ſecundum eadē differentia qua
<
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/>
tertius quartum: vel primus excedatur a ſecundo
<
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ea differentia qua tertius exceditur a quarto illi
<
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intermedii ſimul iuncti extremis copulatis equã-
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tur. </
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<
s
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xml:space
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"> ſi intermedii ab extremis diſtãtes ſimul iū
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cti extremis equantur ab extremis eos equidiſta
<
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re neceſſe eſt.
<
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position
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xml:id
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xml:space
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">Primu
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correlari
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um.</
note
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</
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<
s
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xml:space
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">¶ Ex hac concluſione ſequitur arith
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metice medietatis diſiūcte quattuor terminis ab
<
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ſolute extrema ſimul iuncta collectis medii equa
<
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ri.
<
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position
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note-0025-04
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xml:id
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N12431
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xml:space
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">tertia ꝓ-
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prietas
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medieta
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tis arith
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metice.</
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</
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<
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xml:space
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">Et hec eſt tertia ꝓprietas mediedatis arithme
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tice. </
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<
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">Patet hoc correlarium facile ex precedēti cõ
<
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cluſione </
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<
s
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N123EE
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xml:space
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">Nam ſi quattuor termini proportionen
<
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tur arithmetice et diſiiuncte ea differētia que erit
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inter primū et ſecundum. erit inter tertium et quar
<
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tū </
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>
<
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="
N123F7
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xml:space
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">Quare medii equaliter diſtabunt ab extremis
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coniunctis / igitur mediis equabuntur externa col
<
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lecta iuxta doctrinam concluſionis. </
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>
<
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="
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xml:space
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">Et dixi notã-
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Capitulum ſecundum
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ter in correlario. </
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<
s
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">quattuor terminis quia ſi ponã
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tur plures termini non oportet illud verificari.</
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</
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<
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xml:space
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">Quare inconſiderate aliqui illam proprietatem
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abſolute ponūt. </
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<
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xml:space
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">Patet enim inſtantia in his ter
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minis .2.5.7.11.1.4. manifeſtum eſt enim / aggre
<
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gatum ex extremis minus eſt aggregato ex inter-
<
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mediis. </
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>
<
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="
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">Imo implicat aggregatum ex extremis
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equari omnibus itermediis ſimul ſumptis cum
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ſunt plures termini quattuor: quoniam ſuper ag
<
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gregatum ex extermis puta ex primo et vltimo ad
<
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dequatur aggregato ex ſecūdo et penultimo. </
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<
s
xml:id
="
N12459
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xml:space
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">ergo
<
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non aggregato ex omnibus intermediis quia il-
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lud erit maiꝰ. </
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<
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xml:id
="
N12460
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xml:space
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">Si autem velis dicere ꝓprietatē il-
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lam intelligi / aggregatum ex ṗmo et vltimo ade
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quatur aggregato ex ſecūdo et penultimo: et etiã
<
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equatur aggregato ex tertio et ante penultimo .etc̈ /
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patet hoc eſſe falſum in datis terminis. </
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<
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">Nã in il-
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lis duo et .14. conſtituunt .1.6. tertius tñ et ante pe
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nultimus puta .7. et .10. conſtituunt .1.7. / igitur.</
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>
</
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<
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">Secūduꝫ
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correlari
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um.</
note
>
<
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<
s
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xml:space
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">¶ Sequitur ſecundo / poſitis quattuor terminis
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proportionabilibus arithmetice ſiue cõiuncte ſi-
<
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ue diſiuncte aggregatum ex primo et vltimo ē me
<
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dietas aggregati ex omnibus ſimul et etiam ag-
<
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gregatum ex ſecūdo. </
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>
<
s
xml:id
="
N12486
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xml:space
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">et tertio eſt medietas totius
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aggregati ex omnibus ſimul. </
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<
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xml:id
="
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xml:space
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">Patet / quia illa ag
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gregata ſunt eq̈lia ex cõcluſione et adequate com
<
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ponunt aggregatū ex omnibus illis quattuor ter
<
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minis: igitur vtrum illorū aggregatum eſt me-
<
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dietas aggregati ex omnibus illis terminis ſimĺ
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ſumptis / quod fuit probãdum.
<
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xlink:href
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note-0025-05
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xml:id
="
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xml:space
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">Tertium
<
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correlari
<
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um.</
note
>
</
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<
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xml:space
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">¶ Sequitur tertio /
<
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poſitis ſex terminis ſi octo.
<
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xlink:href
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note-0025-06
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xml:id
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xml:space
="
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">Cal. ḋ 10
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ele.</
note
>
</
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<
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xml:id
="
N124A7
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xml:space
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">ſiue .10. et in quo-
<
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cun numero pari cõtinuo proportionabilibus
<
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arithmetice. </
s
>
<
s
xml:id
="
N124AE
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xml:space
="
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">aggregatum ex primo et vltimo et ag
<
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gregatum ex ſecundo et penultimo et aggregatū
<
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ex tertio et ante penultimo / et ſic conſequenter eſt
<
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/>
pars aliquota aggregati ex omnibus illis ter-
<
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minis denominata a numero ſubduplo ad nume-
<
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rum parem in quo conſtituuntur tales termini. </
s
>
<
s
xml:id
="
N124BB
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xml:space
="
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">vt
<
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ſi ſint ſex termini aggregatum ex primo et ſexto et
<
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etiam aggregatum ex ſecundo et quinto et ex ter-
<
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tio et quarto eſt vna tertia aggregati ex omnibus
<
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/>
illis ſex terminis: et ſi fuerint octo talia aggrega
<
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/>
ta erunt quarte / q2 quarta denominatur a nume-
<
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ro ſubduplo ad numerum octonarium. </
s
>
<
s
xml:id
="
N124CA
"
xml:space
="
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">Proba-
<
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tur hoc / et ſint ſex termini .a.b.d.c.e.f. ↄ̨tinuo arith
<
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metice proportionabiles. </
s
>
<
s
xml:id
="
N124D1
"
xml:space
="
preserve
">et arguitur ſic / aggrega
<
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/>
tum ex a.f. eſt equale aggregato ex .b.e. / vt patet ex
<
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/>
concluſione / quia illa extrema equaliter diſtãt ab
<
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/>
illis mediis et eadem ratione aggregatum ex .c.d
<
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/>
eſt equale aggregato ex b.e. / igitur ibi ſūt tria ag
<
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/>
gregata omnino equalia: et illa componunt ag-
<
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/>
gregatum ex omnibus illis .6. adequate: igitur qḋ
<
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/>
libet illorum aggregatorum eſt vna tertia totius
<
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</
s
>
<
s
xml:id
="
N124E3
"
xml:space
="
preserve
">Et iſto modo probabis quando fuerint octo ter-
<
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/>
mini / quia inuenies ibi quattuor aggregata equa
<
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/>
lia: et quando decem inuenies quin. </
s
>
<
s
xml:id
="
N124EA
"
xml:space
="
preserve
">Et ſic dein-
<
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/>
ceps inuenies talia aggregata equalia in ſubdu
<
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/>
plo numero ad numerum terminorum: quoniam
<
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/>
ſemper pro quolibet tali aggregato capis duos
<
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/>
terminos / et per conſequens dualitatem illorum
<
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/>
terminorum. </
s
>
<
s
xml:id
="
N124F7
"
xml:space
="
preserve
">Modo in quolibet numero pari in
<
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/>
duplo pauciores dualitates reperiūtur quam vni
<
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/>
tates. </
s
>
<
s
xml:id
="
N124FE
"
xml:space
="
preserve
">Et ſic patet correlarium.
<
note
position
="
right
"
xlink:href
="
note-0025-07a
"
xlink:label
="
note-0025-07
"
xml:id
="
N12547
"
xml:space
="
preserve
">Quartū
<
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/>
correlari
<
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/>
um.</
note
>
</
s
>
<
s
xml:id
="
N12506
"
xml:space
="
preserve
">¶ Sequitur quar
<
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/>
to / ſint quattuor termini non continuo propor-
<
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/>
tionabiles arithmetice continuo tamen minores
<
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/>
et minores continuo ſe excedētes minori et mino- </
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>
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