Arithmetica

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Inc igitur ſi quis ad alias inequalitatꝭ ſpecies aīum tendat eādem conuenientiam intitubanter inuēiet. Quare pronūciādū eſtInec vlla trepidatione dubitāduꝫ qd̓ quēadmodum per ſe cōſtātis quantitatis vnitasprincipium ⁊ elementū eſt. ita ⁊ ad aliquid relate quātatis equalitas mater eſt. Demonſtrauimꝰ enim qd̓hic ⁊ eiª ꝓcreatio prīa foret: ⁊ ī ea rurſus poſtrēa ē.¶ De inueniendo in vnoquoqꝫ numero quot mueroseiuſdem proportionis poſſit procedere: eorūqꝫ deſcriptio: deſcriptioniſqꝫ expoſitio. Capitulum. ii.St āt quedam ⁊ hac re ꝓfunda ⁊ mirādaſpeculatō ⁊ vt ait nicomachus ennocophaton theorema proficiēs: ⁊ ad platonicā intimeo aīe gn̓ationē: ⁊ ad interualla armonice diſcipline. Ibi enī iubemur ꝓducereatqꝫ extēdere tres vel quatuor ſeſqualteros: vel quotlibet ſeſquitertias ꝓportiones: ⁊ ſeſꝗquartas cōparationes: eaſqꝫ ſecūdū propoſitū ordinē ſepe cōtinuasiubemur extendere. Ne āt hoc labore quodā ſemꝑ ꝙdem maxīo: frequentiꝰ inferaci fiat: hac nobis ratione in quot numeris quanti poſſint eſſe ſuperparticulares inueſtigandū eſt: Oēs enī multiplices tātarū ſimiliū ſibimet proportionū principes erunt: quoto ip̄loco ab vnitate diſceſſerunt. Quod āt dico ſibimet ſimilium: tale eſt: vt dupli ſēper multiplicitas vt ſuꝑiꝰdcm̄ ē ſequalteros creet: ⁊ triplex ſit dux ſeſquitertioruꝫ: qͣdruplꝰ ſeſquiquartꝭ. Primꝰ ergo duplex vnūſolū habebit ſequalteꝝ. ſcd̓s duos. tertiꝰ tres; quartus. 4. ⁊ ſcd̓m hūc ordinē eadē ſit in īfinitū ꝓgreſſioNeqꝫ vnquā fieri pōt: vt vel ſuꝑet proportionū nūerum vel ab eo ſit diminutior equabilis ab vnitate locatio. Primus ergo duplex ē binarius numerus: quivnū ſolū ſeſqualterū recipit .i. ternariū. Binarius .n.ꝯͣ ternariū cōparatꝰ ſeſqͣlterā efficit ꝓportinē. Ternarius v̓o qm̄ medietatē nō recipit: ſi eſt alter numeruſ ad quē in rōe ſeſqualtera cōparet̉. Quaternariv̓o nūerus ſcd̓us duplex eſt. hic ergo duos ſeſqualteros p̄cedit. Eſt .n. ad ipſū ꝗdē cōparatus ſenarius numerꝰ ad ſenariū v̓o qm̄ medietatē hꝫ: nouenariꝰ. ⁊ſunt duo ſeſqualteri. ad. 4. ſ. 6. ad ſex v̓o. 9. Nouenarius v̓o qm̄ medietate caret. ab hac ꝯparatiōe ſecluſus eſt. Tertiꝰ v̓o duplex eſt. 8. hic ergo 3. ſeſqualteros antecedit. Cōparatur .n. ad ip̄m duodenariꝰ nūerus. ad duodenariū. 18. ad. 18. rurſus. 27. At v̓o. 27.medio carēt. Idē quoqꝫ in ſequentibꝰ euenire neceſſe eſt. qd̓ nos cū propria ordinatione ſubdidimus. Sēꝑ .n. hoc diuina qͣdā nec hūana cōſtitutiōe ſpeculatōnibꝰ ocurit: vt quotienſcūqꝫ vltimus numerus inuenitur: qui loco duplicis ab vnitate ſit par: talis ſit vtin medietates diuidi ſecariqꝫ non poſſit. Latitudo12 4 5 16 326 12 24 48I 3an 9 18 39 7127 94 108la 81 162ris 243.Deꝫ cōtingit et in triplicibus. ex illis .n. ſeſuitercii procreant̉. Nā. qm̄ prmꝰ triplex ēternariꝰ nūerus: hꝫ vnū ſeſꝗteriū. i. 4 Cuius qͣternarii tertia ꝑs n̄ pōt īuēiri: atqꝫ iōIhic epitrito caret. Scd̓s v̓o ꝗ ē nouē hꝫ adſe duodenariū numeꝝ ſeſꝗtertiū. Duodenariꝰ āt qm̄

hꝫ tertiā ꝑtē. in ſeſquitertia ꝓportiōe ꝯparat̉ ad euꝫnūerꝰ ſedecī: ꝗ tertie ꝑtis ſectiōe ſolutꝰ eſt: Uigintiſeptē aūt qm̄ tertiꝰ ē triplex: hꝫ ad ſe ſeſꝗtertiū trigintaſex: ⁊ hic rurſus ad qͣdragintaocto eadē ꝓportionecōparat̉. Cui ſi ſexagintaqͣtuor appoſiti fuerīt: eandērurſus vī ꝓportōis explebūt. Quos ſexagintaqͣtuorad nullū ſeſquitertiū rurſuſ aptabis qm̄ ꝑte tertia nōtenent̉. Atqꝫ hoc in cūctis triplicibꝰ inuenit̉: vt extremus eiuſdē proportionis nūerus tātos ante ſe precedētes hꝫ: quāto pͥmꝰ eoꝝ ab vnitate diſceſſerit. Et ꝙtot ſuꝑ ſe eiuſdē proportiōis habuerit numeros quotus ab vnitate primꝰ eoꝝ iacet eiꝰ ꝑs quā illi cōparatus numerus poſſit eādē facere ꝓportionem īuenirineq̄at. Et triplicis ꝗdē hec ē deſcriptio. Latitudo.3 9 27 812434 36 108 324an 16 48 144 432gu 64 192 57 6la 156 768ris 1024T quadrupli ſecūdū hanc formā dereriptōaeſt: ad quā .ſ. ꝗ a prioribus inſtructꝰ acceſſerit: nulla ratiōe trepidabit: ⁊ de ceterꝭ ꝗdēmultiplicibus eandeꝫ conuenientiam pernotabit.Latitudo.I1 4 16 1 641 356 1 1024 120 80 320 1280an 25 100 400 1600gu 15 500 2000la 615 2509ris 3125Inc quoqꝫ ꝑſpicuū ē: iuꝑparticulariuꝫ queAadmodū priꝰ oſtēſuꝫ ē: primos eē multipliſcēs. Si ꝗdē duplices ſeſqualteros: tripliceſſeſꝗtertios: ⁊ cūcti multiplices cūctos in ordinē ſuꝑparticulares creāt. Eſt ēt ī his hocquoqꝫ mirabile. Nāqꝫ vbi prīa latitudo fuerit duplex⁊ ſub eiſdē ꝗ ſūt verſus cōtinui alternatī. poſiti ſcd̓mſeriē latitudinis dupliceſ erūt. Si v̓o fuerit triplices⁊ inferiores ordines tripla ſe ī ſuis terminis multiplicatōe ſuꝑabūt. At in quadrula qͣdrupli. atqꝫ hoc īfinita ductū ſpeculatiōe nō fallit. angulares āt oīuꝫ multiplices euēire necͣe ē. Erūt āt dupliciū ꝗdē triplices:tripliciū qͣdruplices qͣdruploꝝ v̓o ꝗncupli. ⁊ ẜꝫ eādēordīs īcōmutabilē rōeꝫ ſibimet cūcta ꝯſentiēt. ꝗbꝰ expoſitꝭ ad ſequētē oꝑiſ ſeriē ꝯpetēs diſputatō ꝯuertat̉¶ Qd̓ ml̓tiplex īteruallū ex ꝗbꝰ ſuꝑparticularibꝰ medietate poſita īteruall̓ fiat: eiuſqꝫ īuēiēdi regl̓a. ca. 3.I igit̉ due prīe ſuꝑparticulareſ ſpēſ ꝯiūgāt̉ pͥma ſpēſ ml̓tiplicitatꝭ exorit̉. oīſ .n. duplex ex ſeſqͣltero ſeſꝗtertioqꝫ ꝯpōit̉: ⁊ oīs ſeqͣlter ⁊ ſeſꝗtertiꝰ duplicē iūgūt. nā ternariꝰ ſeſqͣlter ē duoꝝ quatuor v̓o ſeſquitertiꝰ ternarii. ſed. 4. duplus duorum.

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