Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Table of Notes
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Secūde partis
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24
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tione ſequitur / ſi duo numeri ſe habentes in ali
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qua proportione acquirãt ↄ̨tinuo partes aliquo
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tas eiuſdem denominationis: ſemper manebunt
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in eadem proportione. </
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<
s
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N12144
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xml:space
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">Patet / q2 vter illorū eq̈
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lem proportionem acquirit. </
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<
s
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N12149
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xml:space
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">Patet / quia ſi vter
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illorum numerorum illas partes aliquotas eiuſ-
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dem denominationis deperderet eq̈lē ꝓportionē
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deꝑderet / vt patet ex ſuppoſitione: igitur quando
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acquirit equalem acquirit.</
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</
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<
s
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">Duodecima ſuppoſitio. </
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<
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xml:space
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">Si aliquid
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componitur ex duobus ſiue equalibus ſiue īequa
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libus: et quantum deperdit vnum illorum tantuꝫ
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acquirit reliquum: compoſitum ex illis nichil ac-
<
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quirit vel deperdit ſed ſemper manet equale. </
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<
s
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xml:space
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">Et
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hanc peto quia nota eſt ex ſe.</
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<
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xml:id
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xml:space
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">cal. de in
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duc. gra-
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ſum et de
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mo. 10.</
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<
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">Prima concluſio </
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<
s
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xml:space
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">Omne compoſitū
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ex duobus inequalibus inter que eſt mediuꝫ eſt du
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plum ad medium inter illa vt compoſitum ex .4. et
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2. eſt duplum ad ternarium numerum qui mediat
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inter illos </
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<
s
xml:id
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N12181
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xml:space
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">Probatur / ſint a.c. duo īequalia .a ma
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ius et .c. minus et ſit .b. medium inter .a.c. compoſi
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tum ex a.c. ſit .d. / tunc dico / .d. eſt duplum ad .b.
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</
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>
<
s
xml:id
="
N12189
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xml:space
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">Quod ſic probo / quia cū .b. ſit medium: equali dif
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ferentia diſtat ab extremis ex prima ſuppoſitiõe /
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capio igitur illam differentiã ſiue exceſſum qua .a
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excedit b. / et addo illam .c. / et manifeſtum eſt / .a. et
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b. manēt equalia: et ſimiliter .c. et .b. quia ipſi .c. ad
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dictus eſt exceſſꝰ / quo excedebatur a.b. / igitur ag-
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gregatum ex .a. et .c. componitur ex duobus equa
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lidus .b. adequate. </
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<
s
xml:id
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N1219A
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xml:space
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">igitur tale aggregatum eſt du
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plum ad .b. et tale aggregatum eſt .d. / igitur d. eſt
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duplum ad .b. et .d. eſt in tantum quantum erat añ
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variationem .a.c. / vt patet ex vltima ſuppoſitione /
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igitut .d. ante variationem a.c. eſt duplum ad .b. /
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quod fuit probandum.
<
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xml:space
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">p̄mū cor-
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relarium</
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</
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<
s
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xml:space
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">¶ Ex hac concluſione ſequi
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tur: mediū inter duo inequalia eſt medietas ag
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gregati ex eis. </
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<
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">Patet / quia eſt ſubdupluꝫ / ergo me
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dietas.
<
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xlink:href
="
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xml:id
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xml:space
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">Secūduꝫ
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correlari
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um.</
note
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</
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<
s
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N121BD
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xml:space
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">¶ Sequitur ſecūdo / medietas aggrega
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ti ex duobus inequalibus inter que eſt mediuꝫ: eq̈
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liter ab vtro illorum diſtat. </
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<
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xml:id
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N121C4
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xml:space
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">Probatur / q2 medi
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etas illorum eſt equalis medio inter illa / vt patet
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ex precedenti correlario: ergo ſequitur / equali-
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ter diſtat ab vtro. </
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<
s
xml:id
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N121CD
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xml:space
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">cum mediuꝫ ſit / equaliter di
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ſtat ab extremis / vt patet ex prima ſuppoſitione.
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<
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xml:id
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xml:space
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">Tercium
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correlari
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um.</
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>
¶ Sequitur tertio / omnis numerus circū ſe poſi
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torum numerorum et equaliter ab eo diſtantium
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eſt medietas. </
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<
s
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">Quod ſi eoruꝫ fuerit medietas illos
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ab eo eque diſtare conueniet. </
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">Probatur / ſint .a.c.
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duo numeri inter quos mediat .b. ſit aggregatū
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ex .a.c.d. / tunc .b. eſt medietas ipſius .d. / vt patet ex
<
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ṗmo correlario et ſi .b. eſt medietas aggregati .a.c.
<
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equaliter diſtat ab .a. et .c. / vt patet ex ſecundo cor-
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relario / ergo .a.c. equaliter diſtant .a.b.
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">Quartū
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correlari
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um.</
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</
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<
s
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">¶ Sequi-
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tur quarto / cõiuncte arithmetice medietatis me
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diis terminus extremorum ſimul iunctorum ē me
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dietas: vt captis his terminis .a.bc. continuo ꝓ-
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portionabilibꝰ arithmetice .b. medius terminus
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eſt medietas aggregati ex .a.c. </
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<
s
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N12201
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">Patꝫ ex primo cor
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relario
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">prima 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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">Et hec ſit prima ꝓprietas arithmetice me
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dietatis </
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<
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">Et intelligas hanc proprietatem quan-
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do tales termini continuo proportionaabiles hac ꝓ
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portionalitate fuerint impares: vel quantitates
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continue. </
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<
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">Alias plerū non inuenires medium in
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ter tales terminos. </
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<
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">ſicut inter .2.3.4.5
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xml:id
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correlari
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um.</
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</
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<
s
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">¶ Sequitur
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quinto / diſpoſitis .3. terminis continuo ꝓportio
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Capitulum ſecundum
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nabilibꝰ arithmetice: aggregatū ex maxīo termīo
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et mīmo ē due tertie aggregati ex illis tribꝰ termi
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nis: et diſpoſitis .5. continuo proportionalibus
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arithmetice aggregatum ex maximo et minimo ē
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due quinte:
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xlink:href
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xml:id
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">Secūda
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ꝓprietas
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medietaſ
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arithme-
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tice.</
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et etiam aggregatum ex ſecūdo termi
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no et quarto eſt due quinte: et poſitis .7. aggrega
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tum ex maximo et minimo eſt due ſeptime ſimili-
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ter aggregatum ex ſecundo et ſexto et ex tertio et
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quinto. </
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<
s
xml:id
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xml:space
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">et vniuerſaliter vbicū plures termini in
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numero impari arithmetice continuo proportio
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nantur ſemper aggregatum ex quibuſcū duo-
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bus equaliter diſtantibus a medio eſt due partes
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aliquote. </
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<
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">aggregati ex omnibus illis quarū par
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tium aliquotarum vtra denominatur a numero
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impari a quo denominantur illi termini. </
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<
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">vt ſi ter
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mini ſint vndeci3 denominabuntur due vndecime
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et ſi .13. due tridecime. </
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<
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">Probatur hoc correlarium /
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et ſigno tres terminos .a.b.c. / et arguo ſic / aggrega
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tum ex .a.c. eſt duplum ad .b. quia .b. eſt terminꝰ me
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dius inter .a.c. ſed aggregatum ex a.b.c. componi
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tur adeq̈te ex .b. et aggregato ex .a.c. duplo ad .b. /
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vt patet ex concluſione: ergo b. eſt vna tertia totiꝰ
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aggregati cum ter in illo contineatur adequate et
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per conſequens aggregatum ex .a.c. due tertie / qḋ
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fuit probandum. </
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<
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">Item poſitis quin trrminis .a
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b.c.d.e. aggregatum ex .a. et .e. eſt duplum ad ter-
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minum medium .c. et ſimiliter aggregatum ex .b. et
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d. / vt patet ex concluſioīe et totum aggregatum ex
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illis quin terminis componitur adequate ex c. et
<
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ex aggregato .a. et .e. et aggregato ex .b. et .d. et vtrū
<
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illorum aggregatorum eſt duplum ad .c. / vt pro
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batum eſt: ergo .c. eſt vna quinta totius aggrega-
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ti ex illis quin terminis: cum quīquies in illo ag
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gregato contineatur: et per conſequens aggrega
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tum ex .a. et .e. eſt due quinte: et ſimiliter aggrega-
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tum ex .b.d. cum ſit duplum ad .c </
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<
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">Et iſto modo pro
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babis capiendo quotcū alios terminos īpares
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continuo arithmetice ꝓportionabiles. </
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<
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xml:id
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">Et iſta ſit
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ſecunda proprietas medietatis arithmetice.</
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</
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<
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">Secunda concluſio </
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<
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">Si duo nume-
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ri a duobus numeris circum ſe poſitis equaliṫ di
<
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ſtent: illis coniunctis erunt equales. </
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<
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xml:id
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N122EF
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">Quod ſi eis
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equales fuerint: ab eis equidiſtare neceſſe eſt vt ca
<
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ptis his terminis .2.3.4.5. numerus quinarus et
<
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binarius circunſtantes quaternarium et ternariū
<
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equaliter ſimul iuncti equantur quaternario et ter
<
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nario ſimul iunctis et quia quinarius et binariꝰ
<
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ſimul iuncti equales ſunt quaternario et binario
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ſimul iuncti: ideo neceſſario ab illis equaliter di-
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ſtant. </
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<
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="
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">Probatur concluſio / et ſint .a.b.c.d.a.d. cir-
<
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cunſtantes reliqui vero intermedii: et diſtat .a. ab
<
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b.g. dnr̄a ita .a. ſit maior numerus et eadem .g
<
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dnr̄ia excedat .c. ipſum .d. / tunc dico / aggregatū
<
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ex .a.d. extremis numeris eſt equale aggregato ex
<
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b.c. intermediis a quibus alii equaliter diſtant.</
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</
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<
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="
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<
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xml:id
="
N12310
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xml:space
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">Quod probatur ſic / et volo / .a. perdat .g. dnr̄iaꝫ /
<
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ita fiat equale b. et .d. acquirat illam ita fiat
<
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equale .c. / et arguo ſic / facta tali variatione in a.d.
<
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aggregatū ex .a.d. ↄ̨ponit̄̄ adeq̈te ex duobꝰ eq̈libꝰ
<
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aliis duobus ex quibus adequate cõponitur ag-
<
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gretatum ex .b.c. / igitur facta tali variatiõe in .a.
<
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d. aggregatum ex .a.d. eſt equale aggregato ex .b
<
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c. et illud aggregatum ex .a.d. facta tali variatio
<
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ne eſt equale aggregato .a.d. ante talem variatio
<
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nem / vt patet ex vltima ſuppoſitione: igitur aggre
<
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gatum ex .a.c. ante talem variationem eſt equale </
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