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<s><foreign lang="it"><emph type="center"/>OPERE <lb/>DI <lb/>EVANGELISTA TORRICELLI<emph.end type="center"/></foreign></s>

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<s><foreign lang="it"><emph type="center"/>VOLUME I: GEOMETRIA <lb/>PUBBLICATO PER CURA DI GINO LORIA<emph.end type="center"/></foreign></s>

<s><foreign lang="it"><emph type="center"/>PARTE I. <lb/>CON IL RITRATTO DI E. TORRICELLI E 373 FIGURE<emph.end type="center"/></foreign></s>

<s><foreign lang="it"><emph type="center"/><emph type="italics"/>FAENZA<emph.end type="italics"/><emph.end type="center"/></foreign></s>

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<s><foreign lang="it"><emph type="center"/>AMMINISTRATO DALL'ORFANOTROFIO MASCHI<emph.end type="center"/></foreign></s>

<pb/>

<s><foreign lang="it"><emph type="center"/><emph type="italics"/>Riservati tutti i diritti accordati dalla Legge <lb/>in Italia c all'Estero.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

<pb/>

<s><emph type="center"/>INDICE<emph.end type="center"/><lb/>

</s>

<pb/>

<table>

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<row><cell>DE AEQUALIT. PERIMETRORUM CYLINDRI CONI AC SPHAERAE</cell><cell>”</cell><cell>387—407</cell></row></table>

<s><foreign lang="it"><emph type="center"/>INTRODUZIONE.<emph.end type="center"/></foreign></s>

<pb/>

<s><foreign lang="it"><emph type="center"/><emph type="italics"/>INTRODUZIONE<emph.end type="italics"/><emph.end type="center"/></foreign></s>

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<s><foreign lang="it">Quantunque non siasi ancora stabilito perfetto accordo <lb/>fra i competenti intorno all'anno in cui, nella storia civile <lb/>e politica, debbasi collocare l'inizio dell'èra moderna, pure <lb/>l'evo medio si suole da tutti gli storici chiudere nell'ul<lb/>timo decennio del secolo XV o nel secondo del successivo. <lb/>L'analoga questione relativa alla storia delle matematiche, <lb/>per quanto ci consta, non venne sinora posta, almeno in <lb/>modo esplicito. Ma, ove lo fosse stata, non sarebbe certa<lb/>mente stato malagevole l'accordarsi nello scegliere la data <lb/>1650 come inizio dell'ultima delle grandi divisioni della <lb/>storia delle scienze esatte; chè allora appunto i germi fe<lb/>condi deposti da DESCARTES e FERMAT cominciarono a <lb/>produrre l'aritmetizzazione della geometria, cioè il grande <lb/>fenomeno che servì ad imprimerle una fisonomia del tutto <lb/>nuova ed era destinato a rinnovare tutta la scienza del<lb/>l'estensione figurata; inoltre, allora si trovavano in istato <lb/>d'imminente fioritura le idee ed i metodi chiamati ad as<lb/>sicurare sistematica unità alle indagini relative alla misura <lb/>della superficie e dei solidi a contorni arbitrari; allora <lb/>finalmente avevano intrapresa la loro gloriosa corsa nel <lb/>mondo i principi fondamentali posti alla dottrina dei moti <lb/>e delle forze dall'immortale autore dei <emph type="italics"/>Discorsi e dimo<lb/>strazioni matematiche intorno a due nuove scienze.<emph.end type="italics"/></foreign></s>

<s>Ora la matematica della rinascenza — al pari di lam<lb/>pada che, nell'istante in cui sta per spegnersi, diffonde

<pb xlink:href="090/01/006.jpg" pagenum="IV"/>all'intorno uno sprazzo di fulgidissima luce — presenta in <lb/>Italia un epilogo oltre ogni dire brillante in un perso<lb/>naggio di primo ordine, degno continuatore delle tradizioni <lb/>scientifiche che, nella patria nostra, sia pure con lunghe <lb/>deplorevoli lacune, si perpetuarono durante l'enorme pe<lb/>riodo storico che corre da ARCHIMEDE a GALILEO: è EVAN<lb/>GELISTA TORRICELLI, la cui altissima rinomanza presso i <lb/>contemporanei è attestata dall'anagramma </s>

<pb pagenum="IV"/>all'intorno uno sprazzo di fulgidissima luce — presenta in <lb/>Italia un epilogo oltre ogni dire brillante in un perso<lb/>naggio di primo ordine, degno continuatore delle tradizioni <lb/>scientifiche che, nella patria nostra, sia pure con lunghe <lb/>deplorevoli lacune, si perpetuarono durante l'enorme pe<lb/>riodo storico che corre da ARCHIMEDE a GALILEO: è EVAN<lb/>GELISTA TORRICELLI, la cui altissima rinomanza presso i <lb/>contemporanei è attestata dall'anagramma </s>

<s><foreign lang="it"><emph type="center"/><emph type="italics"/>En virescit Galileus alter<emph.end type="italics"/><emph.end type="center"/><lb/>che un suo anonimo ammiratore compose con le lettere che <lb/>ne formano il nome . </foreign></s>

<s><foreign lang="it"><emph type="center"/>II. — <emph type="italics"/>Alunnato e noviziato di E. Torricelli.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

<s>Colui a cui la sorte affidò il nobile còmpito di conso<lb/>lare gli ultimi giorni della travagliata esistenza di GALILEO <lb/>GALILEI, vide la luce il 15 ottobre 1608; sebbene nessun <lb/>documento lo dichiari esplicitamente, pure è pressochè <lb/>certo che egli nacque in Faenza , perchè suo padre GA<lb/>SPARE, apparteneva ad una famiglia, di modeste condizioni

<pb xlink:href="090/01/007.jpg" pagenum="V"/>di fortuna , la quale, a partire dalla metà del secolo XV <lb/>ebbe costante dimora in quella città ed ivi si spense sullo <lb/>scorcio del secolo XVII. </s>

<pb pagenum="V"/>di fortuna , la quale, a partire dalla metà del secolo XV <lb/>ebbe costante dimora in quella città ed ivi si spense sullo <lb/>scorcio del secolo XVII. </s>

<s><foreign lang="it">I suoi studi in umanità furono compiuti sotto la guida <lb/>amorosa e sapiente di uno zio paterno, ALESSANDRO (che <lb/>assunse il nome di JACOPO quando entrò nell'ordine ca<lb/>maldolese e che morì quasi novantenne priore del mona<lb/>stero di S. Giovanni della città natìa ); invece nelle <lb/>scienze, in particolare nelle matematiche, fu istruito dai <lb/>Padri Gesuiti. Ed in tali discipline diede prove lampanti <lb/>di tali spiccate attitudini che il suo ottimo zio persuase <lb/>la famiglia ad inviarlo a Roma allo scopo di perfezionarlo <lb/>sotto la direzione oculata di uno dei luminari del tempo, <lb/>BENEDETTO CASTELLI, il celebre discepolo di GALILEO che, <lb/>a partire dal marzo 1626, era lustro e decoro della Corte di <lb/>papa URBANO VIII. Ciò accadeva verso la metà dell'anno <lb/>1627. Subendo la benefica influenza di tanto istitutore, il <lb/>TORRICELLI fece progressi talmente rapidi e sorprendenti <lb/>che ben presto potè affermarsi pensatore originale con la <lb/>memoria, oggi notissima, <emph type="italics"/>Sul moto dei corpi naturalmente <lb/>discendenti<emph.end type="italics"/> , la quale d'un tratto lo collocò senza con<lb/>trasto in prima linea fra gli alunni del celebre matema<lb/>tico . </foreign></s>

<s>Nelle mani di P. BENEDETTO CASTELLI questa me<lb/>moria servì come mezzo per assicurare al diletto alunno <lb/>una situazione economica e sociale onorevole, lucrosa,

<pb xlink:href="090/01/008.jpg" pagenum="VI"/>stabile . Narra infatti VINCENZO VIVIANI per il tramite <lb/>di LODOVICO SERENAI quanto segue : </s>

<pb pagenum="VI"/>stabile . Narra infatti VINCENZO VIVIANI per il tramite <lb/>di LODOVICO SERENAI quanto segue : </s>

<s><foreign lang="it">“ Questo Padre (cioè il CASTELLI) nell'aprile del 1641 <lb/>venendo di Roma per Pisa a Firenze per passare a Ve<lb/>nezia al suo Capitolo generale, portò con sè manoscritto <lb/>il trattatello del Moto composto allora dal Torricelli sui <lb/>principi del medesimo Galileo, al quem il predetto Abate <lb/>fece sentire il contenuto e la diversità di maniera che in <lb/>varie cose aveva quegli tenuta per ampliare quella nuova <lb/>scienza meravigliosa del Galileo, di cui commiserando il <lb/>predetto Abate la cecità e scorgendo insieme il pericolo <lb/>in che, per la di lui grave età travagliata ancora da tante <lb/>indisposizioni, si stava di perdere per la di lui morte, il <lb/>residuo delle sue speculazioni non pubblicate che aveva <lb/>in sè e che gli rimanevano ancora da porre sulla carta, <lb/>glielo propose in ajuto; e il Galileo che dall'opera sopra <lb/>detta e dalle relazioni che di quella date gli aveva il Pa<lb/>dre, già ne aveva formato gran concetto, volontierissimo <lb/>lo accettò per ajuto e per compagno e restò col Padre <lb/>Castelli che al suo ritorno a Roma poteva trattar d'in<lb/>viarglielo, come seguì. </foreign></s>

<s>“ Giunse dunque il Torricelli alla Villa d'Arcetri (dove <lb/>abitava Galileo) verso la fine del settembre del medesimo <lb/>anno ed immantinente incominciò Galileo a comuni<lb/>cargli nei discorsi che quegli teneva tutto giorno, ciò che <lb/>gli rimaneva delle proprie fatiche e meditazioni, le quali <lb/>aveva stabilite di includere e distribuire in due giornate in

<pb xlink:href="090/01/009.jpg" pagenum="VII"/>dialogo da aggiungersi alle altre quattro dell'opera pochi <lb/>anni prima stampata sopra le sue due nuove scienze della <lb/>Meccanica e del Moto locale e la prima di quelle due con<lb/>tener doveva l'esplicazione d'alcune delle cose già dette <lb/>nelle prime quattro, e la soluzione di vari problemi natu<lb/>rali suoi e d'Aristotile, esaminati e purgati da alcune fal<lb/>lacie prese dal Filosofo e particolarmente nel Trattato <lb/>“ de incessu animalium ”. La seconda doveva comprendere <lb/>il racconto di varie esperienze antiche del Galileo da lui <lb/>fatte per l'investigazione della forza infinita della percossa <lb/>di cui stimava il medesimo Galileo di avere dopo lunghe <lb/>vigilie ed applicazioni arrivati i veri principi e fondamenti <lb/>da poter sopra di essi fabbricare una terza nuova scienza <lb/>e con progresso geometrico dimostrare proprietà stupende <lb/>e fuori della comune immaginazione. Ma iniqua sorte invi<lb/>diando agli uomini così grandi acquisti nelle scienze, volle <lb/>che (appena dato principio il Torricelli a distendere la <lb/>quinta giornata) in capo a poco più di tre mesi dopo la <lb/>congiunzione in terra di quei due grandi luminari, si estin<lb/>guesse il Maggiore conceduto alle vite umane da Dio, <lb/>sommo Sole, per dimostrar loro nei Cieli e nella Natura <lb/>novità ammirande e verità peregrine state occulte a tutta <lb/>l'Antichità ”. </s>

<pb pagenum="VII"/>dialogo da aggiungersi alle altre quattro dell'opera pochi <lb/>anni prima stampata sopra le sue due nuove scienze della <lb/>Meccanica e del Moto locale e la prima di quelle due con<lb/>tener doveva l'esplicazione d'alcune delle cose già dette <lb/>nelle prime quattro, e la soluzione di vari problemi natu<lb/>rali suoi e d'Aristotile, esaminati e purgati da alcune fal<lb/>lacie prese dal Filosofo e particolarmente nel Trattato <lb/>“ de incessu animalium ”. La seconda doveva comprendere <lb/>il racconto di varie esperienze antiche del Galileo da lui <lb/>fatte per l'investigazione della forza infinita della percossa <lb/>di cui stimava il medesimo Galileo di avere dopo lunghe <lb/>vigilie ed applicazioni arrivati i veri principi e fondamenti <lb/>da poter sopra di essi fabbricare una terza nuova scienza <lb/>e con progresso geometrico dimostrare proprietà stupende <lb/>e fuori della comune immaginazione. Ma iniqua sorte invi<lb/>diando agli uomini così grandi acquisti nelle scienze, volle <lb/>che (appena dato principio il Torricelli a distendere la <lb/>quinta giornata) in capo a poco più di tre mesi dopo la <lb/>congiunzione in terra di quei due grandi luminari, si estin<lb/>guesse il Maggiore conceduto alle vite umane da Dio, <lb/>sommo Sole, per dimostrar loro nei Cieli e nella Natura <lb/>novità ammirande e verità peregrine state occulte a tutta <lb/>l'Antichità ”. </s>

<s><foreign lang="it"><emph type="center"/>III. — <emph type="italics"/>Il periodo fiorentino.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

<s><foreign lang="it">Come è notorio la morte di GALILEO seguì il 6 gen<lb/>naio 1641. </foreign></s>

<s> “ Per sì funesto accidente non così presto aspetta<lb/>tosi dal Torricelli rimaneva qua egli come smarrito; ma <lb/>la gloriosa memoria del Ser. Gran Duca Ferdinando II sti<lb/>molata dalla sua nativa inclinazione a promuovere e pro<lb/>teggere le buone lettere e le matematiche in particolare, <lb/>prese subito a risarcire in parte così gran perdita di simil

<pb xlink:href="090/01/010.jpg" pagenum="VIII"/>soggetto statogli rappresentato dal Sen. Andrea Arrighetti <lb/>a relazione del medesimo Galileo di altissima aspettazione. <lb/>E mentre questi preparavasi a licenziarsi per tornarsene <lb/>a Roma fu fatto aspettare d'ordine del G. Duca che allora <lb/>trovavasi a Pisa e dichiarato successore ad un Galileo cioè <lb/>matematico di S. A. e per lui fu rinnovata l'antica ma per <lb/>lungo tempo dimessa lettura di matematiche in questo <lb/>Studio ” . </s>

<pb pagenum="VIII"/>soggetto statogli rappresentato dal Sen. Andrea Arrighetti <lb/>a relazione del medesimo Galileo di altissima aspettazione. <lb/>E mentre questi preparavasi a licenziarsi per tornarsene <lb/>a Roma fu fatto aspettare d'ordine del G. Duca che allora <lb/>trovavasi a Pisa e dichiarato successore ad un Galileo cioè <lb/>matematico di S. A. e per lui fu rinnovata l'antica ma per <lb/>lungo tempo dimessa lettura di matematiche in questo <lb/>Studio ” . </s>

<s><foreign lang="it">In conseguenza di questa provvida ed illuminata deli<lb/>berazione il Nostro, sorpassati di poco i trentadue anni, <lb/>vedeva assicurata a sè stessa una posizione pienamente <lb/>soddisfacente, perchè, in qualità di matematico dello studio <lb/>fiorentino, gli era assegnato lo stipendio annuo di scudi 200, <lb/>ed inoltre il governo toscano gli concesse gratuito alloggio <lb/>in un quartiere dell'antico palazzo dei Medici che divenne <lb/>poi palazzo Riccardi (oggi sede della prefettura); più <lb/>tardi, e precisamente in data 2 gennaio 1644, gli fu con<lb/>ferita anche l'ufficio di lettore di fortificazioni militari <lb/>nella fiorentina Accademia del Disegno con l'annuo emo<lb/>lumento di scudi 40 . </foreign></s>

<s>Non pago di avere liberato il grande faentino da ogni <lb/>sorta di preoccupazioni materiali, il Granduca di Toscana lo <lb/>colmò di altri benefici, il primo e forse più importante dei <lb/>quali (almeno dal punto di vista dei supremi interessi della <lb/>scienza), è quello di avere spontaneamente assunta tutte le <lb/>spese per la stampa e le figure dell'unica opera geometrica <lb/>che a lui fu concesso di presentare al pubblico , quel-

<pb xlink:href="090/01/011.jpg" pagenum="IX"/>l'opera che ebbe virtù di estendere ai lontani l'altissima <lb/>stima che il Torricelli si era acquistata da parte delle per<lb/>sone che ebbe la fortuna di avvicinarlo. </s>

<pb pagenum="IX"/>l'opera che ebbe virtù di estendere ai lontani l'altissima <lb/>stima che il Torricelli si era acquistata da parte delle per<lb/>sone che ebbe la fortuna di avvicinarlo. </s>

<s>Col suo trasferimento a Firenze comincia per il TOR<lb/>RICELLI il periodo più felice della sua esistenza, quello in <lb/>cui potè consacrarsi serenamente e con maggiore intensità <lb/>alla ricerca scientifica ed in cui si affollarono più nume<lb/>rose alla sua mente le idee originali . Libero da qual-

<pb xlink:href="090/01/012.jpg" pagenum="X"/>siasi preoccupazione materiale, essendo bello della persona, <lb/>gentile di maniera ed amante dell'onesto conversare, non <lb/>tardò a stringere rapporti di amicizia con le personalità <lb/>più spiccate viventi allora in Firenze (basti ricordare il <lb/>celebre pittore SALVATORE ROSA ed il dotto ellenista <lb/>CARLO DATI); anzi dai regolari convegni che egli tenne <lb/>con questi valentuomini trasse origine l'Accademia dei <lb/>“ Percossi ”, in seno alla quale forse egli fece conoscere <lb/>le <emph type="italics"/>Commedie<emph.end type="italics"/> che vuolsi scrivesse ed ove recitò, con <lb/>plauso generale, quell'<emph type="italics"/>Encomio del secol d'oro<emph.end type="italics"/> destinato a <lb/>prendere posto più tardi nella collezione delle sue <emph type="italics"/>Lezioni <lb/>accademiche<emph.end type="italics"/> . D'altronde l'Accademia della Crusca, onde <lb/>rendere solenne omaggio alla sua perizia nel maneggio <lb/>della patria lingua, volle annoverarlo fra i propri membri <lb/>ed egli, entrando a far parte di quel celebre sodalizio, ri<lb/>volse ai nuovi colleghi un ringraziamento che venne pure <lb/>accolto nella medesima raccolta . </s>

<pb pagenum="X"/>siasi preoccupazione materiale, essendo bello della persona, <lb/>gentile di maniera ed amante dell'onesto conversare, non <lb/>tardò a stringere rapporti di amicizia con le personalità <lb/>più spiccate viventi allora in Firenze (basti ricordare il <lb/>celebre pittore SALVATORE ROSA ed il dotto ellenista <lb/>CARLO DATI); anzi dai regolari convegni che egli tenne <lb/>con questi valentuomini trasse origine l'Accademia dei <lb/>“ Percossi ”, in seno alla quale forse egli fece conoscere <lb/>le <emph type="italics"/>Commedie<emph.end type="italics"/> che vuolsi scrivesse ed ove recitò, con <lb/>plauso generale, quell'<emph type="italics"/>Encomio del secol d'oro<emph.end type="italics"/> destinato a <lb/>prendere posto più tardi nella collezione delle sue <emph type="italics"/>Lezioni <lb/>accademiche<emph.end type="italics"/> . D'altronde l'Accademia della Crusca, onde <lb/>rendere solenne omaggio alla sua perizia nel maneggio <lb/>della patria lingua, volle annoverarlo fra i propri membri <lb/>ed egli, entrando a far parte di quel celebre sodalizio, ri<lb/>volse ai nuovi colleghi un ringraziamento che venne pure <lb/>accolto nella medesima raccolta . </s>

<s><foreign lang="it">Le più fruttifere indagini da lui condotte a termine nel <lb/>regno delle scienze matematiche e fisiche durante il pro<lb/>prio soggiorno alla Corte di Toscana diedero alcuni risul<lb/>tati a cui fu ben presto concesso un posto eminente nei <lb/>fasti della scienza; fra questi meritano uno speciale ri<lb/>cordo i seguenti: </foreign></s>

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<s><foreign lang="it">2. La classica esperienza col mercurio e la con<lb/>seguente invenzione del barometro ; </foreign></s>

<s><foreign lang="it">3. I procedimenti da lui ideati ed applicati per <lb/>costruire certi speciali microscopi e per ripulire le lenti <lb/>destinate ai telescopi . </foreign></s>

<s><foreign lang="it">Esse valsero a diffondere rapidamente la rinomanza del <lb/>Torricelli, non soltanto al di là delle mura di Firenze, ma <lb/>eziandio oltre i confini d'Italia. In pari tempo però fecero <lb/>pullulare detrattori invidiosi che vollero, alcuni diminuire <lb/>il valore dei suoi ritrovati, altri contestargli i diritti di prio<lb/>rità che egli con piena ragione accampava sopra di essi. <lb/>Da tale ingiustificata ostilità egli fu spinto a scrivere quel <lb/><emph type="italics"/>Racconto d'alcuni problemi ecc.<emph.end type="italics"/> che oggi rappresenta, dal <lb/>punto di vista umano, una delle più interessanti pagine, <lb/>delle sue <emph type="italics"/>Opere<emph.end type="italics"/> ; da ciò più tardi un suo amico, CARLO <lb/>DATI, consigliato e spalleggiato dal fedelissimo esecutore <lb/>testamentario LODOVICO SERENAI, fu indotto a pubblicare <lb/>nel 1663, sotto forma di lettera pseudonima, tutti i docu<lb/>menti atti ad illustrare e completamente chiarire alcune <lb/>controversie originate da invenzioni del Nostro e che sono <lb/>fra le più celebri che siano registrate nella storia delle <lb/>scienze positive . </foreign></s>

<s><foreign lang="it"><emph type="center"/>IV. — <emph type="italics"/>La morte.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

Line 148

<s><foreign lang="it">In data 14 ottobre 1647 scriveva LODOVICO SERENAI a <lb/>FRANCESCO fratello di EVANGELISTA TORRICELLI : </foreign></s>

<s>“ Si trova in letto malato il sig. Vangelista fratello di <lb/>V. S. con febbre che per otto giorni non è stata stimata <lb/>di gran pericolo, ma hiersera aggravò, e dopo essersi que<lb/>sta mattina confessato con grandissimo sentimento e haver <lb/>fatto testamento, e discorso di tutte le cose sue con gran<lb/>dissimo senno sino alle 21 ore incirca, ha poi sull'accen<lb/>sione della febbre dato in delirio, e delirò furioso a segno <lb/>che non si può ajutare con medicamenti senza gran diffi<lb/>cultà, e si teme d'incontrarla ancora nel cibarlo. Con la <lb/>commodità del primo riposo che conceda il delirio sarà <lb/>pronto il Parocc.<emph type="sup"/>no<emph.end type="sup"/> col Santissimo Viatico, e non si man<lb/>cherà di vigilanza per ogni rimedio spirituale; siccome

<pb xlink:href="090/01/016.jpg" pagenum="XIV"/>attorno al corpo s'è fatto, e si farà tutto il possibile: e <lb/>acciocchè V. S. possa crederlo sappia che oltre alla servitù <lb/>sua ordinaria ci assiste quasi del continuo il sig. Dottor <lb/>Buonajuti suo medico e amico carissimo, io me ne parto <lb/>soltanto tanto quanto vo à desinare, e a cena con mia <lb/>moglie habitando vicinissimo, e due astanti mandatici dal <lb/>Ser.<emph type="sup"/>mo<emph.end type="sup"/> Gran Duca non se ne partono punto. In compagnia <lb/>del sig.<emph type="sup"/>r<emph.end type="sup"/> Bonajuti viene alla cura il sig.<emph type="sup"/>r<emph.end type="sup"/> Dottore Scafucci <lb/>medico di S. A. S. la quale somministra regali di delizie e <lb/>di medicamenti preziosi della sua fonderia. Finalmente la <lb/>servitù e gli ajuti sono da principi e meritamente essendo <lb/>egli un Principe della sua professione ”. </s>

<pb pagenum="XIV"/>attorno al corpo s'è fatto, e si farà tutto il possibile: e <lb/>acciocchè V. S. possa crederlo sappia che oltre alla servitù <lb/>sua ordinaria ci assiste quasi del continuo il sig. Dottor <lb/>Buonajuti suo medico e amico carissimo, io me ne parto <lb/>soltanto tanto quanto vo à desinare, e a cena con mia <lb/>moglie habitando vicinissimo, e due astanti mandatici dal <lb/>Ser.<emph type="sup"/>mo<emph.end type="sup"/> Gran Duca non se ne partono punto. In compagnia <lb/>del sig.<emph type="sup"/>r<emph.end type="sup"/> Bonajuti viene alla cura il sig.<emph type="sup"/>r<emph.end type="sup"/> Dottore Scafucci <lb/>medico di S. A. S. la quale somministra regali di delizie e <lb/>di medicamenti preziosi della sua fonderia. Finalmente la <lb/>servitù e gli ajuti sono da principi e meritamente essendo <lb/>egli un Principe della sua professione ”. </s>

<s><foreign lang="it">Ma tutte queste cure, per quanto assidue, amorose, sa<lb/>pienti, non sortirono il desiderato effetto, chè nulla pote<lb/>rono contro il delirio che si rinnovava in modo sempre più <lb/>allarmante; ed il SERENAI addì 25 ottobre 1647 era co<lb/>stretto a riprendere la penna per informare FRANCESCO <lb/>TORRICELLI dell'avvenuta catastrofe : </foreign></s>

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<s><foreign lang="it">Tale epigrafe fu del seguente tenore : </foreign></s>

<s><emph type="center"/>EVANGELISTA TORRICELLIUS <lb/>FAVENTINUS <lb/>MAGNI DUCI ETRURIAE MATHEMATICUS <lb/>ET PHILOSOPHUS <lb/>OBIIT VIII KAL. NOVEMBRIS ANNO SALUTIS <lb/>M DC XLVII <lb/>AETATIS SUAE XXXIX;<emph.end type="center"/>

<pb xlink:href="090/01/017.jpg" pagenum="XV"/>disgraziatamente questa non fu sufficiente a far distinguere <lb/>per sempre le ossa del grande pensatore da quelle dei suoi <lb/>oscuri vicini ; infatti le ricerche dei suoi resti mortali, <lb/>eseguite per ordine della civica Amministrazione di Fi<lb/>renze in occasione del III Centenario della sua nascita, <lb/>non soltanto non condussero ad alcun risultato, ma gui<lb/>darono alla sconsolante conclusione di trovarsi di fronte <lb/>ad un problema che la scienza è nella impossibilità di scio<lb/>gliere . </s>

<pb pagenum="XV"/>disgraziatamente questa non fu sufficiente a far distinguere <lb/>per sempre le ossa del grande pensatore da quelle dei suoi <lb/>oscuri vicini ; infatti le ricerche dei suoi resti mortali, <lb/>eseguite per ordine della civica Amministrazione di Fi<lb/>renze in occasione del III Centenario della sua nascita, <lb/>non soltanto non condussero ad alcun risultato, ma gui<lb/>darono alla sconsolante conclusione di trovarsi di fronte <lb/>ad un problema che la scienza è nella impossibilità di scio<lb/>gliere . </s>

<s><foreign lang="it"><emph type="center"/>V. — <emph type="italics"/>Le disposizioni testamentarie.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

Line 170

<s><foreign lang="it">Nel testamento si legge quanto segue: </foreign></s>

<s>“ Item ordina al sopradetto Sig.<emph type="sup"/>r<emph.end type="sup"/> Lodovico suo esecu<lb/>tore che quanto prima, seguita sua morte, trasmetta e <lb/>mandi a spese della sua eredità al M. R. P. fra Bonaven<lb/>tura Cavalieri Matematico dello Studio di Bologna tutti <lb/>i suoi scritti, studii e fatiche di Geometria quali aveva <lb/>disegnato di pubblicare alla stampa, essendo già ordinate <lb/>con le dimostrazioni promesse, acciocchè detto Padre fra

<pb xlink:href="090/01/018.jpg" pagenum="XVI"/>Bonaventura ne pubblichi quella quantita che a esso libe<lb/>ramente parrà o piacerà, et il restante li mandi a Roma <lb/>al Sig.<emph type="sup"/>r<emph.end type="sup"/> Michelangelo Ricci gentiluomo splendidissimo et <lb/>amicissimo di detto Sig.<emph type="sup"/>r<emph.end type="sup"/> Testatore et intendentissimo di <lb/>queste scienze, acciò li metta insieme e li pubblichi, come <lb/>meglio ha significate et ordinate in vece al medesimo <lb/>Sig.<emph type="sup"/>r<emph.end type="sup"/> esecutore. Fra le quali scritture di Geometria detto <lb/>Sig.<emph type="sup"/>r<emph.end type="sup"/> Testatore intende che restino comprese lettere e ri<lb/>sposte passate fra lui e i Matematici di Francia ”. </s>

<pb pagenum="XVI"/>Bonaventura ne pubblichi quella quantita che a esso libe<lb/>ramente parrà o piacerà, et il restante li mandi a Roma <lb/>al Sig.<emph type="sup"/>r<emph.end type="sup"/> Michelangelo Ricci gentiluomo splendidissimo et <lb/>amicissimo di detto Sig.<emph type="sup"/>r<emph.end type="sup"/> Testatore et intendentissimo di <lb/>queste scienze, acciò li metta insieme e li pubblichi, come <lb/>meglio ha significate et ordinate in vece al medesimo <lb/>Sig.<emph type="sup"/>r<emph.end type="sup"/> esecutore. Fra le quali scritture di Geometria detto <lb/>Sig.<emph type="sup"/>r<emph.end type="sup"/> Testatore intende che restino comprese lettere e ri<lb/>sposte passate fra lui e i Matematici di Francia ”. </s>

<s><foreign lang="it">D'accordo ed a complemento di tali ordini stanno al<lb/>cuni <emph type="italics"/>Ricordi<emph.end type="italics"/> dettati al Serenai, ove fra l'altro è avver<lb/>tito che </foreign></s>

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<s><foreign lang="it">e riguardo al segreto per la fabbricazione dei vetri <lb/>è detto: </foreign></s>

<s><foreign lang="it">“ Dettogli: che vuol ella fare del suo segreto delli oc<lb/>chiali? Il negozio e segreto dei vetri non occorre neanche <lb/>mettercelo , perchè io farò che questa mattina sia in <lb/>mano al Gran Duca serrato: ma ha fatto male S. A. a <lb/>non mi far lavorare in sua presenza, perchè avrebbe ve<lb/>duto e imparato meglio; e non troverà chi lo faccia. Le <lb/>forme di vetri fatte da me con grandissima diligenza, che <lb/>S. A. non ne troverà, le lascio alla stessa Altezza S.; e <lb/>perchè mi costano molti denari, avendole fatte fare in <lb/>Galleria, dove sempre ho pagato e date mance larghissime, <lb/>desidero che S. A. se ne mostri benigno con i poveri miei <lb/>fratelli per quanto le parrà che elle vaglino ”. </foreign></s>

<s><foreign lang="it"><emph type="center"/>VI. — <emph type="italics"/>Alla ricerca di un editore.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

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<s><foreign lang="it">Le trattative col CAVALIERI si può dire che finirono <lb/>prima di venire iniziate; chè, avendo il SERENAI scritto <lb/>al geometra degli indivisibili sino dal 26 ottobre 1647 per <lb/>partecipargli la morte del TORRICELLI , ne ebbe risposta <lb/>di mano di un confratello, fra PLACIDO GHIRLANDI , <lb/>nella quale è detto che le condizioni di salute del CAVA<lb/>LIERI, da cattive che erano da molto tempo, si erano fatte <lb/>allarmanti: nè in tale apprezzamento vi era alcuna esage<lb/>razione, chè il giorno 30 del seguente novembre il CAVA<lb/>LIERI passava a miglior vita. </foreign></s>

<s>Il SERENAI volse allora il proprio pensiero a MICHE<lb/>LANGELO RICCI e, per assicurarsi la sua collaborazione, <lb/>pregò il MAGIOTTI (con lettere del 30 novembre e del 21 <lb/>dicembre 1647 ) di assumere la parte di intermediario; <lb/>il MAGIOTTI non rimase sordo alla preghiera dell'amico <lb/>(come risulta dalle risposte inviate il 15 dicembre 1647 <lb/>ed il 10 gennaio 1648 ), ma con esito ben poco soddi<lb/>sfacente. Tentò allora il SERENAI un assalto diretto alla

<pb xlink:href="090/01/020.jpg" pagenum="XVIII"/>troppo ben difesa fortezza , ma, con dolore dovette ri<lb/>conoscere ben presto che questa era inespugnabile: ciò è <lb/>attestato nel modo più chiaro dal seguente brano di let<lb/>tera inviata dal RICCI al SERENAI addì 11 aprile 1648 : </s>

<pb pagenum="XVIII"/>troppo ben difesa fortezza , ma, con dolore dovette ri<lb/>conoscere ben presto che questa era inespugnabile: ciò è <lb/>attestato nel modo più chiaro dal seguente brano di let<lb/>tera inviata dal RICCI al SERENAI addì 11 aprile 1648 : </s>

<s><foreign lang="it">“ Quel fervore, col quale gli anni passati intrapresi lo <lb/>studio delle Matematiche, incominciò ad intiepidirsi alcuni <lb/>mesi prima che morisse la buona memoria del Sig.<emph type="sup"/>r<emph.end type="sup"/> Tor<lb/>ricelli, e dopo la sua morte è di maniera diminuito che <lb/>sento più tosto repugnanza che diletto nell'applicarmivi. <lb/>Questo però non sarebbe sufficiente a ritardarmi da quel<lb/>l'impresa, alla quale mi aveva destinato il Sig.<emph type="sup"/>r<emph.end type="sup"/> Vangelista, <lb/>in riguardo forse di lasciare in sua morte un memorabile <lb/>onore nella persona d'un suo discepolo, e servitore affe<lb/>zionat.<emph type="sup"/>mo<emph.end type="sup"/>, cioè dichiarandomi abile alla revisione delle sue <lb/>degniss.<emph type="sup"/>me<emph.end type="sup"/> speculazioni: ma per il peso accresciutomi sù <lb/>le spalle per la morte di mio Zio, seguita sotto il 15 di <lb/>Gennaro, e per la grande età di mio Padre, restando sotto <lb/>la mia direzione quasi in tutto gli affari di mia casa; sono <lb/>talm.<emph type="sup"/>te<emph.end type="sup"/> occupato che nella varietà d'altri pensieri non <lb/>hanno luogo i concetti Geometrici, che richiedono per se <lb/>stessi tutto l'ingegno, e la fantasia. Dirò di vantaggio che <lb/>nei tre mesi decorsi dell'anno corrente sono stato quasi <lb/>sempre indisposto, et inetto alle fatiche della mente; per <lb/>le quali cose parmi di avere tanta ragione, e scusa che <lb/>V. S. possa rendersi persuasa della difficoltà grandiss.<emph type="sup"/>ma<emph.end type="sup"/>, <lb/>che forse potrebbesi e con altro titolo chiamare impossi<lb/>bilità, la quale mi fa ricusare la carica di ripulir le opere <lb/>del Sig. Torricelli ”. </foreign></s>

<s>Dolente per questo nuovo insuccesso, ma non scorag<lb/>giato, parve al SERENAI che il MAGIOTTI fosse persona <lb/>indicatissima per assumere l'ufficio a cui il TORRICELLI <lb/>aveva destinati il CAVALIERI ed il RICCI e si cullò nella <lb/>dolce illusione di potere giungere a persuaderlo ad addos<lb/>sarselo in occasione di una visita che quel valentuomo <lb/>aveva in animo di fare ad un proprio fratello residente in

<pb xlink:href="090/01/021.jpg" pagenum="XIX"/>Firenze ; ma, per ragioni che ignoriamo, anche il MA<lb/>GIOTTI declinò l'onorevole ma gravosissimo incarico. </s>

<pb pagenum="XIX"/>Firenze ; ma, per ragioni che ignoriamo, anche il MA<lb/>GIOTTI declinò l'onorevole ma gravosissimo incarico. </s>

<s><foreign lang="it">Il seguito delle spinose trattative è narrato dal SERENAI <lb/>con le seguenti parole : </foreign></s>

<s><foreign lang="it">“ Mancandomi pertanto così gran capitale di questi <lb/>due matematici e sapendo che tra i veri e buoni Amici <lb/>che si acquistasse quì il Torricelli il primo era stato Vin<lb/>cenzo Viviani, in quel tempo che vivevano amendue in<lb/>sieme ospiti e commensali del medesimo Galileo e avendo <lb/>veduto poi con quanto amore e con qual reciproca fami<lb/>gliarità si praticassero continuamente; ricorsi a questo <lb/>pregandolo contentarsi di faticare sopra le opere geome<lb/>triche lasciate in confuso e imperfette dall'Amico nostro, <lb/>ma egli per lungo tempo ricusò sempre, dicendo non co<lb/>noscersi abile a tanta impresa e quando ne fusse stato <lb/>non aver tempo da impiegarvelo stante le sue occupazioni <lb/>domestiche e negli affari pubblici e in servigio di S. A. che <lb/>già un tempo gli impedirono di proseguire i suoi propri <lb/>studi nonchè applicare agli altri, soggiungendomi altri <lb/>vari motivi che lo dissuadevano dall'intraprendere questo <lb/>lavoro. </foreign></s>

<s>“ Infine dopo reiterati assalti cedè alle istanze mie e <lb/>di altri Amici accettando il travagliare sopra tali mano<lb/>scritti in quei tempi che gli fossero restati liberi e quieti <lb/>come diceva richiedersi per lui in simili speculazioni; ma <lb/>però mi protestò apertamente che volentieri per far cosa <lb/>grata a me e servire alla memoria del comune amico ac<lb/>consentiva di faticarci, ma che siccome ciò faceva senza <lb/>alcun fine e speranza nè di guadagno nè di premio nè di <lb/>lode, così voleva almeno assicurarsi di non esporre abben<lb/>chè minimo sospetto la sua lealtà, e che però si dichiarava <lb/>di non voler mai nelle sue mani alcun benchè piccolo fo<lb/>gliuzzo degli Originali del Torricelli, con tutto che nume<lb/>rati, nè assai nè poco maneggiarli per tempo alcuno, ma

<pb xlink:href="090/01/022.jpg" pagenum="XX"/>che voleva solamente le copie puntuali con le figure foglio <lb/>per foglio come appresso di me si trovavano. A proposi<lb/>zione tanto rispettosa e discreta non seppi che replicare, <lb/>anzi questa m'insinuò di fare le copie domandate di mia <lb/>mano propria, come veramente con mia fatica incredibile <lb/>io lo feci di tutti gli originali matematici imitando giu<lb/>stamente, anzi dipingendo tutte le figure, non tanto le ben <lb/>fatte quanto le guaste e cassate, e ogni parola dello scritto <lb/>ancorchè cancellata ” . </s>

<pb pagenum="XX"/>che voleva solamente le copie puntuali con le figure foglio <lb/>per foglio come appresso di me si trovavano. A proposi<lb/>zione tanto rispettosa e discreta non seppi che replicare, <lb/>anzi questa m'insinuò di fare le copie domandate di mia <lb/>mano propria, come veramente con mia fatica incredibile <lb/>io lo feci di tutti gli originali matematici imitando giu<lb/>stamente, anzi dipingendo tutte le figure, non tanto le ben <lb/>fatte quanto le guaste e cassate, e ogni parola dello scritto <lb/>ancorchè cancellata ” . </s>

<s><foreign lang="it">Quando si tenga presente che molti manoscritti del <lb/>TORRICELLI consistevano di semplici appunti da lui rapida<lb/>mente presi nel corso delle sue ricerche, mentre altri rap<lb/>presentano prime stesure di lavori, alle quali erano indi<lb/>spensabili molteplici miglioramenti di sostanza e di forma, <lb/>e che il SERENAI era un giureconsulto di grande reputa<lb/>zione , ma affatto digiuno di studi matematici, si vedrà <lb/>chiaramente che la fatica a cui egli spontaneamente si <lb/>sobbarcava (e che occupò tutti i suoi ozi durante quattro <lb/>lunghi anni ) è la più eroica prova di amore per la <lb/>scienza e di disinteressata amicizia per un illustre defunto <lb/>che si trovi registrata nella storia. Tante pene avrebbero <lb/>ben meritato l'unico guiderdone che il SERENAI ne atten<lb/>deva, quello cioè di vedere esaudito l'ardente voto formu<lb/>lato dal TORRICELLI sul suo letto di morte; ma anche esso <lb/>fu negato da una sorte implacabilmente avversa! </foreign></s>

<s>Non è da credersi che il VIVIANI di deliberato proposito <lb/>sia venuto meno all'impegno che aveva assunto; la rac-

<pb xlink:href="090/01/023.jpg" pagenum="XXI"/>colta fiorentina consacrata ai “ <emph type="italics"/>Discepoli di Galileo<emph.end type="italics"/> ” sta a <lb/>provare quante ore di lavoro egli abbia speso a riordinare, <lb/>compilare, commentare, trascrivere le opere del diletto <lb/>commilitone ; ma sia che fosse distratto da indagini <lb/>sue proprie o quasi totalmente assorbito dalle altre cariche <lb/>affidategli, sia che le frequenti malattie gli vietassero con<lb/>tinuità di lavoro o che preferisse dedicare alla memoria del <lb/>suo venerato maestro GALILEO il meglio delle sue forze, sia <lb/>finalmente che, anche in questa contingenza, non gli riu<lb/>scisse di vincere la proverbiale incontentabilità che lo in<lb/>duceva a correggere, rifare, trascrivere un numero stermi<lb/>nato di volte tutto ciò che uscivagli dalla penna, fatto sta <lb/>che egli scese nella tomba prima che l'augurata edizione <lb/>si avviasse verso un lontano indizio di attuazione . </s>

<pb pagenum="XXI"/>colta fiorentina consacrata ai “ <emph type="italics"/>Discepoli di Galileo<emph.end type="italics"/> ” sta a <lb/>provare quante ore di lavoro egli abbia speso a riordinare, <lb/>compilare, commentare, trascrivere le opere del diletto <lb/>commilitone ; ma sia che fosse distratto da indagini <lb/>sue proprie o quasi totalmente assorbito dalle altre cariche <lb/>affidategli, sia che le frequenti malattie gli vietassero con<lb/>tinuità di lavoro o che preferisse dedicare alla memoria del <lb/>suo venerato maestro GALILEO il meglio delle sue forze, sia <lb/>finalmente che, anche in questa contingenza, non gli riu<lb/>scisse di vincere la proverbiale incontentabilità che lo in<lb/>duceva a correggere, rifare, trascrivere un numero stermi<lb/>nato di volte tutto ciò che uscivagli dalla penna, fatto sta <lb/>che egli scese nella tomba prima che l'augurata edizione <lb/>si avviasse verso un lontano indizio di attuazione . </s>

<s><foreign lang="it">Ed il SERENAI, probabilmente avvedendosi della cattiva <lb/>piega che stava prendendo l'impresa a cui aveva dedicata <lb/>la vita e sentendo approssimarsi la propria fine, nel testa<lb/>mento dettato il 26 settembre 1674 disponeva che tutti <lb/>i manoscritti torricelliani fossero consegnati ad AGOSTINO <lb/>NELLI e, in caso di morte di costui, a RIDOLFO PAGANELLI, <lb/>oppure, nell'ipotesi che anche questi premorisse al testa<lb/>tore, a CARLO DATI, in ogni caso a disposizione del VI<lb/>VIANI in servizio della progettata edizione: uscita questa <lb/>in luce tutte quelle carte dovevano essere (come da tempo <lb/>aveva consigliato il VIVIANI) consegnato al Gran Duca <lb/>regnante per venire depositato nella Libreria medicea di <lb/>S. Lorenzo. </foreign></s>

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<s><foreign lang="it"><emph type="center"/>VII. — <emph type="italics"/>Vicissitudini subite dai manoscritti torricelliani.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

<s>Morto AGOSTINO NELLI, il figliuol suo GIOVANNI BAT<lb/>TISTA, discepolo ed amico del VIVIANI, indusse questi ad <lb/>assumere la custodia di tutti i manoscritti che il SERENAI <lb/>aveva ricevuti in deposito fiduciario dal suo compianto <lb/>amico. Tale adesione non può non recare grande mera-

<pb xlink:href="090/01/025.jpg" pagenum="XXIII"/>viglia a chi ricordi l'invincibile riluttanza che il VIVIANI <lb/>aveva da giovane sentita e manifestata di assumere la <lb/>grave responsabilità di un siffatto deposito. Ma ancor più <lb/>stupefacente è il fatto che egli, sentendo approssimarsi la <lb/>grande ombra, mentre diede precise disposizioni a tutela <lb/>delle ricchissime collezioni di libri e di quadri dei quali <lb/>era legittimo proprietario, non fece alcun cenno del tesoro <lb/>di cui un capriccio della sorte (sempre nemica al TORRI<lb/>CELLI) avevalo fatto depositario. </s>

<pb pagenum="XXIII"/>viglia a chi ricordi l'invincibile riluttanza che il VIVIANI <lb/>aveva da giovane sentita e manifestata di assumere la <lb/>grave responsabilità di un siffatto deposito. Ma ancor più <lb/>stupefacente è il fatto che egli, sentendo approssimarsi la <lb/>grande ombra, mentre diede precise disposizioni a tutela <lb/>delle ricchissime collezioni di libri e di quadri dei quali <lb/>era legittimo proprietario, non fece alcun cenno del tesoro <lb/>di cui un capriccio della sorte (sempre nemica al TORRI<lb/>CELLI) avevalo fatto depositario. </s>

<s><foreign lang="it">Questa negligenza imperdonabile, e che non riusciamo <lb/>a non chiamare colpevole, visti i deplorevoli effetti che <lb/>ebbe, proietta sopra la figura morale del VIVIANI una <lb/>luce ancor più tetra di quanto sia il mancato impegno di <lb/>pubblicare gli scritti inediti del suo commensale di gio<lb/>ventù. In conseguenza di quella dimenticanza (che vo<lb/>gliamo giudicare pura e semplice conseguenza di senile <lb/>amnesia) i manoscritti torricelliani passarono, insieme ad <lb/>altre carte, in eredità, come mobili, all'abate JACOPO PAN<lb/>ZANINI (nipote del VIVIANI e lettore di matematica nello <lb/>studio fiorentino) e, morto costui (1733), ai suoi nipoti <lb/>CARLO ed ANGELO, i quali nell'incapacità di comprenderne <lb/>il valore, un brutto giorno, per fare spazio in armadi so<lb/>verchiamente ingombri di biancheria, ne vendettero una <lb/>parte ad un pizzicagnolo. “ Le vie di Dio son molte ” di<lb/>remo con ALESSANDRO MANZONI; giacchè fortuna volle che <lb/>quel negoziante, ignaro dei più elementari precetti del<lb/>l'igiene, si servisse di un autografo del Galilei per avvol<lb/>gere una piccola quantità di mortadella da lui venduta a <lb/>G. B. CLEMENTE NELLI; questi riconobbe subito la scrit<lb/>tura del celebre scienziato e, per il tramite di quel pizzi<lb/>cagnolo, si pose in relazione con i PANZANINI e riuscì ad <lb/>acquistare in blocco tutto il materiale scientifico del quale <lb/>essi a torto, benchè senza alcuna colpa, si consideravano <lb/>legittimi proprietari. </foreign></s>

<s>Ora dal paragone dell'inventario redatto dal NELLI di <lb/>tutti gli scritti del TORRICELLI da lui comperati con

<pb xlink:href="090/01/026.jpg" pagenum="XXIV"/>l'<emph type="italics"/>Elenco delle Opere inedite<emph.end type="italics"/> del TORRICELLI compilato dal <lb/>SERENAI con l'aiuto del VIVIANI risulta (fatto incredi<lb/>bile, ma pur vero) che, durante le traversie subìte da quei <lb/>manoscritti, se pure essi subirono qualche perdita, si tratta <lb/>di cosa pressochè insignificante. </s>

<pb pagenum="XXIV"/>l'<emph type="italics"/>Elenco delle Opere inedite<emph.end type="italics"/> del TORRICELLI compilato dal <lb/>SERENAI con l'aiuto del VIVIANI risulta (fatto incredi<lb/>bile, ma pur vero) che, durante le traversie subìte da quei <lb/>manoscritti, se pure essi subirono qualche perdita, si tratta <lb/>di cosa pressochè insignificante. </s>

<s><foreign lang="it">Giunto in possesso di tanto tesoro pensò il NELLI di <lb/>mostrarsi meritevole dell'insperata fortuna toccatagli col <lb/>portare a compimento la desideratissima edizione delle <lb/><emph type="italics"/>Opere inedite<emph.end type="italics"/> del TORRICELLI; mise anche mano ai lavori <lb/>preparatori, ma sgraziatamente non potè toccare l'ago<lb/>gnata mèta. Però, quando ebbe a perdere la speranza di <lb/>raggiungerla, provvide a che quel tesoro non fosse una <lb/>nuova volta esposto ad essere disperso; a tale scopo, nel <lb/>testamento da lui dettato il 14 dicembre 1793, impose ai <lb/>propri eredi che, qualora pensassero di alienarlo, prima di <lb/>trattare la vendita con privati, lo offrissero al Gran Duca <lb/>di Toscana; ora le meno floride condizioni finanziarie della <lb/>famiglia NELLI avendo consigliata tale vendita, nell'ot<lb/>tobre del 1818 Ferdinando III, che deteneva allora lo <lb/>scettro, esercitando il diritto di prelazione che eragli stato <lb/>fortunatamente conferito, entrò in possesso di quella ine<lb/>stimabile suppellettile scientifica, corrispondendo alla fa<lb/>miglia NELLI la somma di zecchini 1046, nella quale le <lb/>opere del TORRICELLI venivano valutate 80 zecchini. Così <lb/>finalmente gli scritti del sommo faentino toccavano un <lb/>porto sicuro! Essi, nel 1861, passarono dalla Biblioteca <lb/>Palatina alla Nazionale di Firenze, e, con gli altri scritti <lb/>dell'epoca, diedero origine alla collezione dei “ <emph type="italics"/>Discepoli di <lb/>Galileo<emph.end type="italics"/> ” da noi tante volte ricordata e che, per avventura, <lb/>è la più importante del genere che esista nel mondo . </foreign></s>

<s><foreign lang="it"><emph type="center"/>VIII. — <emph type="italics"/>Pubblicazioni parziali di lavori torricelliani.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

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<s><foreign lang="it">Anzitutto, sino dal 1674 il VIVIANI, col consenso del <lb/>SERENAI , pubblicava tre teoremi dell'opuscolo <emph type="italics"/>De pro<lb/>porlionibus<emph.end type="italics"/> nel corpo della nota sua opera <emph type="italics"/>Quinto Libro <lb/>degli Elementi di Euclide ovvero Scienza universale delle <lb/>Proporzioni spiegata con la dottrina del Galileo<emph.end type="italics"/> (Firenze). </foreign></s>

<s>Inoltre nel 1715 — come abbiamo già avuto occasione <lb/>di accennare — uscivano stampate le <emph type="italics"/>Lezioni accademiche,<emph.end type="italics"/><lb/>per merito di TOMMASO BUONAVENTURI, il benemerito eru<lb/>dito che, col concorso di GUIDO GRANDI, BENEDETTO BRE<lb/>SCIANI e GIUSEPPE AVERANI curò la prima edizione fio<lb/>rentina delle <emph type="italics"/>Opere<emph.end type="italics"/> di GALILEO: non è forse fuor di luogo <lb/>di rilevare, a scanso di equivoci, che per condurre a ter<lb/>mine questa importante pubblicazione il BUONAVENTURI <lb/>si giovò di materiali passati direttamente dalle mani del <lb/>SERENAI alla Libreria di palazzo Pitti. Gli stessi docu<lb/>menti resero possibile che le scritture del Nostro <emph type="italics"/>Sopra <lb/>la bonificazione della Valle di Chiana<emph.end type="italics"/> fossero nel 1768 in<lb/>seriti nel T. IV della celebre <emph type="italics"/>Raccolta d'autori che trattano <lb/>del moto delle acque<emph.end type="italics"/> e che dieci anni più tardi, giun<lb/>gesse in dominio del pubblico, come Appendice alla bio<lb/>grafia del Torricelli scritta da A. FABBRONI, il <emph type="italics"/>Racconto <lb/>d'alcune proposizioni proposte e passate scambievolmente tra <lb/>matematici di Francia e me dall'anno 1640, in qua<emph.end type="italics"/> . Nè

<pb xlink:href="090/01/028.jpg" pagenum="XXVI"/>è da credersi che alla pubblicazione integrale delle <emph type="italics"/>Opere<emph.end type="italics"/><lb/>del TORRICELLI si fosse in quell'epoca del tutto rinunziato; <lb/>giacchè da una lettera scritta da P. FRISI appunto al <lb/>FABBRONI il 3 settembre 1774 si apprende che allora <lb/>vi volgeva la mente un certo GIANNINI (forse il noto ma<lb/>tematico toscano PIETRO GIANNINI); ma, come ignoriamo <lb/>i particolari di questo progetto, ci sono ignote le ragioni <lb/>per le quali esso venne abbandonato. </s>

<pb pagenum="XXVI"/>è da credersi che alla pubblicazione integrale delle <emph type="italics"/>Opere<emph.end type="italics"/><lb/>del TORRICELLI si fosse in quell'epoca del tutto rinunziato; <lb/>giacchè da una lettera scritta da P. FRISI appunto al <lb/>FABBRONI il 3 settembre 1774 si apprende che allora <lb/>vi volgeva la mente un certo GIANNINI (forse il noto ma<lb/>tematico toscano PIETRO GIANNINI); ma, come ignoriamo <lb/>i particolari di questo progetto, ci sono ignote le ragioni <lb/>per le quali esso venne abbandonato. </s>

<s>Nessun nuovo piano dello stesso genere venne escogi<lb/>tato, per quanto ci consta, durante il secolo XIX. Però, <lb/>inaugurandosi nel 1864 un monumento marmoreo al TOR<lb/>RICELLI nella nativa Faenza, furono da G. GHINASSI pub<lb/>blicate, asssieme ad altri documenti, alcuni elementi inte<lb/>ressanti del suo carteggio scientifico . All'inesauribile <lb/>munificenza di BALDASSARRE BUONCOMPAGNI si deve la <lb/>pubblicazione, avvenuta undici anni dopo, di altre impor<lb/>tanti sezioni del medesimo carteggio . Ancora: una

<pb xlink:href="090/01/029.jpg" pagenum="XXVII"/>lettera del Torricelli si trova in una pubblicazione di <lb/>C. HENRY e nel corso degli anni 1891-98 moltissimi <lb/>squarci delle <emph type="italics"/>Opere inedite<emph.end type="italics"/> del TORRICELLI furono inseriti <lb/>dal CAVERNI nei primi cinque volumi della sua <emph type="italics"/>Storia del <lb/>metodo sperimentale in Italia,<emph.end type="italics"/> opera che (non è fuor di luogo <lb/>notarlo) conviene usare con somma cautela, chè troppo <lb/>spesso il desiderio di denigrare Galileo offusca nell'autore <lb/>la serenità del giudizio e l'onestà storica . Finalmente <lb/>allo spirare del secolo scorso giungevano in dominio del <lb/>pubblico gli studi del TORRICELLI sulla spirale logarit<lb/>mica , importante saggio delle sue ricerche sopra “ de <lb/>lineis novis ”, a cui egli attribuiva tanta importanza. </s>

<pb pagenum="XXVII"/>lettera del Torricelli si trova in una pubblicazione di <lb/>C. HENRY e nel corso degli anni 1891-98 moltissimi <lb/>squarci delle <emph type="italics"/>Opere inedite<emph.end type="italics"/> del TORRICELLI furono inseriti <lb/>dal CAVERNI nei primi cinque volumi della sua <emph type="italics"/>Storia del <lb/>metodo sperimentale in Italia,<emph.end type="italics"/> opera che (non è fuor di luogo <lb/>notarlo) conviene usare con somma cautela, chè troppo <lb/>spesso il desiderio di denigrare Galileo offusca nell'autore <lb/>la serenità del giudizio e l'onestà storica . Finalmente <lb/>allo spirare del secolo scorso giungevano in dominio del <lb/>pubblico gli studi del TORRICELLI sulla spirale logarit<lb/>mica , importante saggio delle sue ricerche sopra “ de <lb/>lineis novis ”, a cui egli attribuiva tanta importanza. </s>

<s><foreign lang="it">Ne va dimenticato e taciuto che le indagini amoro<lb/>samente condotte sopra le opere già edite guidarono a <lb/>rivendicare al TORRICELLI la scoperta del metodo delle <lb/>tangenti che porta il nome del ROBERVAL e la gloria <lb/>di avere scoperta la prima curva esattamente rettifica<lb/>bile . Da ultimo l'importanza dei risultati da lui otte<lb/>nuti studiando il celebre problema di FERMAT “ ricerca <lb/>del punto nel piano di un triangolo per cui è minima la <lb/>somma delle distanze dai vertici ” consigliarono a chia<lb/>mare <emph type="italics"/>circonferenze di Torricelli<emph.end type="italics"/> quelle che servono a risol<lb/>verlo e <emph type="italics"/>punto di Torricelli<emph.end type="italics"/> quello che ne rappresenta <lb/>la soluzione . </foreign></s>

<s><foreign lang="it"><emph type="center"/>IX. — <emph type="italics"/>Risorge il progetto d'un'edizione completa.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

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<s><foreign lang="it">Questa proposta riscossa l'unanime adesione dei pre<lb/>senti (fra i quali si trovava P. TANNERY, l'illustre storico <lb/>francese che così efficacemente contribuì al buon esito della <lb/>pubblicazione delle <emph type="italics"/>Opere<emph.end type="italics"/> di FERMAT e di DESCARTES), i <lb/>quali, nella seduta del 6 aprile 1903 concordi votarono il <lb/>seguente ordine del giorno: </foreign></s>

<s>“ La Sezione VIII del Congresso internazionale di <lb/>scienze storiche (Roma, 1903) fa voti che il governo di <lb/>S. M. il Re d'Italia affidi alla R. Accademia dei Lincei <lb/>il còmpito di esaminare le opere manoscritte di Evange<lb/>lista Torricelli nell'intento di determinare quali fra esse

<pb xlink:href="090/01/031.jpg" pagenum="XXIX"/>siano meritevoli di stampa; e di presiedere alla pubblica<lb/>zione completa di tutte le opere di lui già edite e di quelle <lb/>inedite, giudicatene degne, senza escludere il suo carteg<lb/>gio scientifico, completando così il lavoro intrapreso con <lb/>la edizione nazionale delle opere del Galilei ”. </s>

<pb pagenum="XXIX"/>siano meritevoli di stampa; e di presiedere alla pubblica<lb/>zione completa di tutte le opere di lui già edite e di quelle <lb/>inedite, giudicatene degne, senza escludere il suo carteg<lb/>gio scientifico, completando così il lavoro intrapreso con <lb/>la edizione nazionale delle opere del Galilei ”. </s>

<s><foreign lang="it">Nell'intento di rendere possibile l'esaudimento di tal <lb/>voto senza alcun momentaneo aggravio per il bilancio <lb/>dello Stato, il nostro Ministero della pubblica istruzione, <lb/>in principio dell'anno scolastico 1904-05, trasferì da Como <lb/>a Firenze GIOVANNI VAILATI, onde egli dedicasse tutte <lb/>le ore lasciategli libere dall'insegnamento della matema<lb/>tica in quell'Istituto tecnico all'esame preliminare dei <lb/>manoscritti torricelliani esistenti in quella Biblioteca Na<lb/>zionale. Ci è ignoto sino a quale punto il sempre rim<lb/>pianto studioso spingesse la sua opera investigatrice; ma <lb/>questa venne ben presto bruscamente interrotta quando, <lb/>instituita con R. Decreto 19 novembre 1905 la ben nota <lb/><emph type="italics"/>Commissione per l'ordinamento degli studi secondari in <lb/>Italia,<emph.end type="italics"/> il VAILATI fu chiamato a farne parte; onore da lui <lb/>ben meritato, ma che a ragione pareva racchiudere la mi<lb/>naccia di un rinvio “ sine die ” dell'esecuzione della de<lb/>siderata edizione. </foreign></s>

<s><foreign lang="it">Si approssimava intanto il III Centenario della nascita <lb/>del grande scienziato; e GIUSEPPE VASSURA, nel mentre <lb/>a nome del Municipio di Faenza invitava la Società Ita<lb/>liana di fisica a tenere nel 1908 il suo Congresso a Fa<lb/>enza, chiedeva venisse emesso un nuovo voto a favore <lb/>della pubblicazione delle <emph type="italics"/>Opere<emph.end type="italics"/> del TORRICELLI; tale lo<lb/>devole iniziativa trovò favorevole accoglienza e, nella se<lb/>duta del 27 aprile 1906, dopo elevata discussione, venne <lb/>ad unanimità presa la seguente deliberazione: </foreign></s>

<s><foreign lang="it">“ Il Congresso della Società Italiana di fisica tenutosi <lb/>in Roma nel 1906 sollecita il governo a dare appoggi ma<lb/>teriali e morali affinchè le opere di Evangelista Torricelli <lb/>vengano sollecitamente pubblicate ”. </foreign></s>

<s><foreign lang="it">Ma neppure questa nuova autorevole esortazione fu <lb/>sufficiente a convincere il nostro governo che l'invocata <lb/>pubblicazione costituiva un debito di gratitudine verso chi <lb/>aveva onorata la patria conservandole, per qualche tempo <lb/>dopo la morte di GALILEO, un primato che gli stranieri <lb/>avevano dovuto riconoscere all'Italia. </foreign></s>

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<s><foreign lang="it">A far ciò volse il pensiero il Comune di Faenza in <lb/>occasione del III Centenario della nascita dell'inventor <lb/>del barometro, affidando l'incarico di condurre a termine <lb/>la nobile e meritoria impresa a GIUSEPPE VASSURA . <lb/>Con quali criteri egli abbia deciso di adempiere il mandato <lb/>ricevuto venne da lui stesso esposto in due pubblicazioni <lb/>che da tempo si trovano a disposizione degli studiosi . <lb/>A noi basta rilevare quì che a lui appartiene la riparti<lb/>zione di tutto il materiale da pubblicarsi in tre volumi, <lb/>il I destinato ad accogliere tutte le <emph type="italics"/>Opere geometriche<emph.end type="italics"/> già <lb/>edite od in istato da potere venire utilmente pubblicate; <lb/>il II alle <emph type="italics"/>Lezioni accademiche,<emph.end type="italics"/> la <emph type="italics"/>Meccanica<emph.end type="italics"/> e <emph type="italics"/>Scritti vari;<emph.end type="italics"/><lb/>il III riserbato al <emph type="italics"/>Carteggio scientifico.<emph.end type="italics"/> E poichè in circa <lb/>quattro anni di assiduo lavoro egli portò a compimento i <lb/>volumi II e III così era generale la fiducia che si fosse <lb/>finalmente scoperta la via capace di porgere la sospirata <lb/>soluzione della secolare questione. </foreign></s>

<s>Se non che, allontanatosi il VASSURA dalla sua città <lb/>natale sullo scorcio dell'anno 1912, sorse inatteso e spa<lb/>ventoso ostacolo contro il compimento dell'iniziata edi<lb/>zione. Nell'intento di sormontarlo il Comune di Faenza — <lb/>dietro suggerimento dello stesso VASSURA — rivolse a

<pb xlink:href="090/01/033.jpg" pagenum="XXXI"/>me l'invito terribilmente onorevole di curare la pubblica<lb/>zione del Volume delle <emph type="italics"/>Opere<emph.end type="italics"/> del TORRICELLI dedicato <lb/>alla Geometria, cioè, in complesso, di tutti i suoi lavori <lb/>inediti. La gravità di tale missione e l'assoluta impossi<lb/>bilità da parte mia di allontanarmi per lungo tempo dalla <lb/>mia consueta residenza, ove mi trattengono sempre im<lb/>prescindibili doveri d'ufficio, mi lasciarono lungamente in <lb/>dubbio intorno alla deliberazione da prendere. Finalmente, <lb/>da un lato il desiderio di contribuire all'esaudimento di <lb/>un desiderio che era espressione di un grande interesse <lb/>scientifico e nazionale; e dall'altro l'avere il VASSURA <lb/>poste a mia disposizione le copie eseguite sotto la sua <lb/>direzione dei lavori torricelliani conservati a Firenze e <lb/>di altri importanti documenti relativi ed il fatto che io <lb/>trovai nel dott. C. MOCARINI, dell'Archivio di Stato di <lb/>Firenze, persona capace e disposta a collazionare ed even<lb/>tualmente completare le copie anzidette, finirono col vin<lb/>cere la mia troppo giustificata esitazione. Ed ora, superate <lb/>le difficoltà di ogni genere che intralciarono più e più <lb/>volte la regolarità del mio procedere (difficoltà che l'im<lb/>mane guerra delle nazioni in parte creò ed in parte acuì) <lb/>mi è dato chiudere la mia fatica presentando al pubblico, <lb/>in unione al VASSURA, le <emph type="italics"/>Opere<emph.end type="italics"/> di E. TORRICELLI, non <lb/>prima però di avere brevemente esposti i criteri da me <lb/>prescelti nella mia azione di editore . </s>

<pb pagenum="XXXI"/>me l'invito terribilmente onorevole di curare la pubblica<lb/>zione del Volume delle <emph type="italics"/>Opere<emph.end type="italics"/> del TORRICELLI dedicato <lb/>alla Geometria, cioè, in complesso, di tutti i suoi lavori <lb/>inediti. La gravità di tale missione e l'assoluta impossi<lb/>bilità da parte mia di allontanarmi per lungo tempo dalla <lb/>mia consueta residenza, ove mi trattengono sempre im<lb/>prescindibili doveri d'ufficio, mi lasciarono lungamente in <lb/>dubbio intorno alla deliberazione da prendere. Finalmente, <lb/>da un lato il desiderio di contribuire all'esaudimento di <lb/>un desiderio che era espressione di un grande interesse <lb/>scientifico e nazionale; e dall'altro l'avere il VASSURA <lb/>poste a mia disposizione le copie eseguite sotto la sua <lb/>direzione dei lavori torricelliani conservati a Firenze e <lb/>di altri importanti documenti relativi ed il fatto che io <lb/>trovai nel dott. C. MOCARINI, dell'Archivio di Stato di <lb/>Firenze, persona capace e disposta a collazionare ed even<lb/>tualmente completare le copie anzidette, finirono col vin<lb/>cere la mia troppo giustificata esitazione. Ed ora, superate <lb/>le difficoltà di ogni genere che intralciarono più e più <lb/>volte la regolarità del mio procedere (difficoltà che l'im<lb/>mane guerra delle nazioni in parte creò ed in parte acuì) <lb/>mi è dato chiudere la mia fatica presentando al pubblico, <lb/>in unione al VASSURA, le <emph type="italics"/>Opere<emph.end type="italics"/> di E. TORRICELLI, non <lb/>prima però di avere brevemente esposti i criteri da me <lb/>prescelti nella mia azione di editore . </s>

<s><foreign lang="it"><emph type="center"/>XI. — <emph type="italics"/>La presente edizione.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

<s>Le esigenze imposte ad una riproduzione per mezzo <lb/>della stampa di opere scientifiche sono di natura ben di<lb/>verse da quelle a cui deve soddisfare un lavoro analogo <lb/>di carattere letterario. Mentre in questo si richiede una <lb/>riproduzione diplomatica degli originali, che ne rispetti <lb/>persino la punteggiatura e l'ortografia, ad un'edizione di <lb/>scritti scientifici si domanda soltanto che vengano religio<lb/>samente conservati e fedelmente riprodotti le idee ed i <lb/>metodi. In conseguenza di ciò noi ci siamo astenuti dal <lb/>consegnare al tipografo quei frammenti torricelliani che <lb/>sono manifestazioni tangibili di pensieri che balenarono <lb/>dinnanzi alla mente dell'autore ed a cui egli non diede più <lb/>seguito, sia per averli poi ravvisati per “ fatica buttata <lb/>via ” , sia per mancanza di tempo. È il sistema che già <lb/>adottarono, ad esempio, gli editori di LAGRANGE, che di<lb/>chiararono di seguire coloro a cui fu affidato il gran<lb/>dioso compito di preparare la pubblicazione definitiva degli <lb/>scritti di LEIBNIZ e che, per ragioni ben note a tutti <lb/>i competenti, verrà abbandonato soltanto riguardo agli <lb/>scritti di LEONARDO DA VINCI. Perciò la presente edizione <lb/>è, nelle nostre intenzioni, <emph type="italics"/>completa<emph.end type="italics"/> ma non <emph type="italics"/>totale,<emph.end type="italics"/> confor<lb/>memente, d'altronde, ai voti formulati dal TORRICELLI nel <lb/>momento in cui preparavasi al viaggio senza ritorno, ed <lb/>alle intenzioni di tutti coloro che, prima di noi, si accin-

<pb xlink:href="090/01/035.jpg" pagenum="XXXIII"/>sero a soddisfarli. Ciò, naturalmente, non esclude in alcun <lb/>modo che altri più oculato, possa trovare nei manoscritti <lb/>che si salvarono dalla minacciata dispersione, materiali <lb/>per aggiunte ai volumi che noi oggi sottoponiamo al giu<lb/>dizio del pubblico; onde questi non hanno alcuna pretesa <lb/>di far cessare il commovente pellegrinaggio di cui da circa <lb/>un secolo è oggetto l'inesauribile raccolta dei “ <emph type="italics"/>Discepoli <lb/>di Galileo<emph.end type="italics"/> ”. </s>

<pb pagenum="XXXIII"/>sero a soddisfarli. Ciò, naturalmente, non esclude in alcun <lb/>modo che altri più oculato, possa trovare nei manoscritti <lb/>che si salvarono dalla minacciata dispersione, materiali <lb/>per aggiunte ai volumi che noi oggi sottoponiamo al giu<lb/>dizio del pubblico; onde questi non hanno alcuna pretesa <lb/>di far cessare il commovente pellegrinaggio di cui da circa <lb/>un secolo è oggetto l'inesauribile raccolta dei “ <emph type="italics"/>Discepoli <lb/>di Galileo<emph.end type="italics"/> ”. </s>

<s>I fogli relitti dal TORRICELLI furono investigati con <lb/>amorosa profondità — già lo abbiamo detto e più d'una <lb/>volta — dal SERENAI e dal VIVIANI, i quali li ordinarono, <lb/>onde fare di quelli che trattano argomenti affini un tutto <lb/>omogeneo e degli altri un artistico mosaico . Ora delle <lb/>loro fatiche altamente meritorie noi abbiamo tratto il <lb/>massimo profitto, non soltanto nell'egoistico intento di al<lb/>leviare il còmpito nostro, ma perchè quei due valentuo<lb/>mini vanno considerati come i più coscienziosi depositari <lb/>ed i più fedeli interpreti del pensiero torricelliano. Però, <lb/>anche dopo tale indiscutibile perfezionamento subìto da <lb/>tutti quei lavori, essi raggiunsero soltanto in piccolissima <lb/>parte l'esattezza di forma che si esige da qualsia scritto <lb/>scientifico ; doveva l'editore permettersi di correggere <lb/>di suo arbitrio le inesattezze riscontrate e di colmare le la-

<pb xlink:href="090/01/036.jpg" pagenum="XXXIV"/>cune da lui notate? A nostro avviso <emph type="italics"/>no;<emph.end type="italics"/> giacchè un siffatto <lb/>poco rispettoso ed arbitrario sistema avrebbe reso difficile, <lb/>e fors'anche impossibile, al lettore di avere dinnanzi una <lb/>fedele immagine del pensiero torricelliano; è nostra con<lb/>vinzione che il VIVIANI aveva vagheggiato di eseguire que<lb/>st'opera complementare, ma che poi l'abbandonò forse per <lb/>scrupoli ben giustificati; e probabilmente la vana ricerca <lb/>di un'altra procedura che consentisse di offrire al pubblico <lb/>le produzioni del suo venerato amico sotto aspetto del tutto <lb/>soddisfacente fu la cagione che spinse lui — che tanto <lb/>spesso e volontieri sacrificava alla Dea Procrastinazione — <lb/>a rinviare di giorno in giorno l'adempimento dell'impegno <lb/>assunto col SERENAI. Il procedimento indarno cercato dal<lb/>l'ultimo discepolo di GALILEO è forse quello a cui ai dì <lb/>nostri si appigliarono gli editori delle <emph type="italics"/>Opere<emph.end type="italics"/> dell'HUYGENS, <lb/>i quali, riguardo agli scritti inediti del sommo Olandese, <lb/>adottarono il sistema della riproduzione diplomatica, ac<lb/>compagnata da esaurienti commenti, sotto forma di note <lb/>a piè di pagina; è il sistema che noi pure avremmo pre<lb/>ferito ove l'edizione delle <emph type="italics"/>Opere<emph.end type="italics"/> di TORRICELLI, al pari di <lb/>quella di quel celebre scienziato, fosse stata assunta da un <lb/>sodalizio scientifico avente esistenza illimitata nel tempo; <lb/>ma, data invece l'enormità del lavoro consistente nel com<lb/>pletare e commentare tutti gli scritti inediti del TORRI<lb/>CELLI e data la brevità della vita umana, scegliendolo non <lb/>si sarebbe probabilmente ottenuto altro risultato che di <lb/>aggiungere un nuovo nome alla lunga teoria di persone <lb/>che tentarono indarno di porre in circolazione i frutti delle <lb/>elucubrazioni geometriche del celebre faentino. </s>

<pb pagenum="XXXIV"/>cune da lui notate? A nostro avviso <emph type="italics"/>no;<emph.end type="italics"/> giacchè un siffatto <lb/>poco rispettoso ed arbitrario sistema avrebbe reso difficile, <lb/>e fors'anche impossibile, al lettore di avere dinnanzi una <lb/>fedele immagine del pensiero torricelliano; è nostra con<lb/>vinzione che il VIVIANI aveva vagheggiato di eseguire que<lb/>st'opera complementare, ma che poi l'abbandonò forse per <lb/>scrupoli ben giustificati; e probabilmente la vana ricerca <lb/>di un'altra procedura che consentisse di offrire al pubblico <lb/>le produzioni del suo venerato amico sotto aspetto del tutto <lb/>soddisfacente fu la cagione che spinse lui — che tanto <lb/>spesso e volontieri sacrificava alla Dea Procrastinazione — <lb/>a rinviare di giorno in giorno l'adempimento dell'impegno <lb/>assunto col SERENAI. Il procedimento indarno cercato dal<lb/>l'ultimo discepolo di GALILEO è forse quello a cui ai dì <lb/>nostri si appigliarono gli editori delle <emph type="italics"/>Opere<emph.end type="italics"/> dell'HUYGENS, <lb/>i quali, riguardo agli scritti inediti del sommo Olandese, <lb/>adottarono il sistema della riproduzione diplomatica, ac<lb/>compagnata da esaurienti commenti, sotto forma di note <lb/>a piè di pagina; è il sistema che noi pure avremmo pre<lb/>ferito ove l'edizione delle <emph type="italics"/>Opere<emph.end type="italics"/> di TORRICELLI, al pari di <lb/>quella di quel celebre scienziato, fosse stata assunta da un <lb/>sodalizio scientifico avente esistenza illimitata nel tempo; <lb/>ma, data invece l'enormità del lavoro consistente nel com<lb/>pletare e commentare tutti gli scritti inediti del TORRI<lb/>CELLI e data la brevità della vita umana, scegliendolo non <lb/>si sarebbe probabilmente ottenuto altro risultato che di <lb/>aggiungere un nuovo nome alla lunga teoria di persone <lb/>che tentarono indarno di porre in circolazione i frutti delle <lb/>elucubrazioni geometriche del celebre faentino. </s>

<s><foreign lang="it">Per tali ragioni noi limitammo l'opera nostra ad insi<lb/>gnificanti ritocchi superficiali, a qualche sobria dilucida<lb/>zione a piè di pagina ed al sostituire gli schizzi nervosa<lb/>mente tracciati dall'autore con figure effettivamente capaci <lb/>di chiarire i ragionamenti esposti . </foreign></s>

<s><foreign lang="it">Coloro che prima di noi si accinsero a pubblicare gli <lb/>scritti di cui ci occupiamo si proposero di presentarli al <lb/>pubblico in modo da formare un tutto ben ordinato: pro<lb/>blema certo importante e bellissimo, ma che, secondo noi, <lb/>lo stesso autore non sarebbe stato in grado di risolvere. <lb/>Infatti si tratta, non di materiali destinati a costituire <lb/>un'opera unica, ma sibbene di svariatissime ricerche, rag<lb/>gruppantisi intorno ad alcuni centri; ond'è nostro con<lb/>vincimento che il TORRICELLI se ne sarebbe servito per <lb/>scrivere parecchie memorie staccate. “ Rebus sic stanti<lb/>bus ” per porre un po' d'ordine a quei materiali non si <lb/>poteva pensare che ad un ordinamento o cronologico, o <lb/>in base agli argomenti trattati, o tenendo conto dei me<lb/>todi di ricerca usati. Ora: </foreign></s>

Line 288

<s><foreign lang="it">3. Quanto al metodo d'indagine, pure essendo sempre <lb/>geometrico, talora è prettamente archimedeo, talora in<lb/>vece è ispirato alle idee del CAVALIERI; ora il comporre <lb/>una Sezione con i lavori scritti in istile antico ed una con <lb/>gli altri, avrebbe avuto come conseguenza un'evidente e <lb/>deplorevole violazione dell'ordine in cui si svolse il pen<lb/>siero dell'eminente scienziato. </foreign></s>

<s>Per tutte queste ragioni noi abbiamo rinunciato ad un <lb/>rigoroso ordinamento di tutta la materia. Dopo la ripro<lb/>duzione della parte non meccanica dell'<emph type="italics"/>Opera geometrica<emph.end type="italics"/> — <lb/>l'unica che egli potè presentare al pubblico — ponemmo un <lb/>brevissimo squarcio che ne costituisce un complemento, poi <lb/>gli scritti che, trattando nuovi problemi di contatti circo<lb/>lari, della teoria delle proporzioni e di svariate questioni di

<pb xlink:href="090/01/038.jpg" pagenum="XXXVI"/>planimetria e stereometria, porgono aggiunte alla geome<lb/>tria elementare degli antichi. Altrettanto può dirsi di una <lb/>ricca miscellanea di teoremi semplicemente enunciati ed in <lb/>gran parte desunti dalla precedente raccolta. Seguono ad <lb/>essa alcune pagine che rivelano i dubbi che, nel TORRICELLI <lb/>od in altri, sorsero contro la geometria dell'infinito, la <lb/>quale rigogliosamente fioriva intorno al 1650, e che si ritro<lb/>vano in altro suo scritto sugli indivisibili. S'incontrano poi <lb/>le ricerche baricentriche o stereometriche relative a por<lb/>zioni di quàdriche rotonde. Riunimmo finalmente le impor<lb/>tanti scritture relative a curve speciali le quali — secondo <lb/>gl'intendimenti manifestati dall'autore nell'esordio alla me<lb/>moria <emph type="italics"/>De proportionibus<emph.end type="italics"/> — dovevano essere ingredienti di <lb/>un'opera di lunga lena da intitolarsi <emph type="italics"/>De lineis novis.<emph.end type="italics"/></s>

<pb pagenum="XXXVI"/>planimetria e stereometria, porgono aggiunte alla geome<lb/>tria elementare degli antichi. Altrettanto può dirsi di una <lb/>ricca miscellanea di teoremi semplicemente enunciati ed in <lb/>gran parte desunti dalla precedente raccolta. Seguono ad <lb/>essa alcune pagine che rivelano i dubbi che, nel TORRICELLI <lb/>od in altri, sorsero contro la geometria dell'infinito, la <lb/>quale rigogliosamente fioriva intorno al 1650, e che si ritro<lb/>vano in altro suo scritto sugli indivisibili. S'incontrano poi <lb/>le ricerche baricentriche o stereometriche relative a por<lb/>zioni di quàdriche rotonde. Riunimmo finalmente le impor<lb/>tanti scritture relative a curve speciali le quali — secondo <lb/>gl'intendimenti manifestati dall'autore nell'esordio alla me<lb/>moria <emph type="italics"/>De proportionibus<emph.end type="italics"/> — dovevano essere ingredienti di <lb/>un'opera di lunga lena da intitolarsi <emph type="italics"/>De lineis novis.<emph.end type="italics"/></s>

<s><foreign lang="it">Con l'eleggere ed adottare siffatta distribuzione delle <lb/>materie noi non pretendiamo di avere divinate le inten<lb/>zioni del TORRICELLI (dato e non concesso che egli ne <lb/>avesse di definitive); ci lusinghiamo, però, di non avere <lb/>resa impossibile la ricostruzione della genesi del suo pen<lb/>siero scientifico. </foreign></s>

<s><foreign lang="it"><emph type="center"/>XII. — <emph type="italics"/>A che cosa miri la presente pubblicazione.<emph.end type="italics"/><emph.end type="center"/></foreign></s>

<s>Nel licenziare il frutto delle nostre lunghe fatiche ci <lb/>si affaccia spontaneamente la tormentosa domanda quale <lb/>sarà l'accoglienza che esso sarà per ricevere da parte del <lb/>pubblico matematico. Ora ci sembra fuor di questione che <lb/>la presente pubblicazione costituiva da parte dell'Italia un <lb/>preciso dovere verso uno dei più illustri suoi figli “ onde <lb/>assicurare contro i danni inevitabili del tempo quelle pa<lb/>gine venerande, cui troppo spesso invidiano gli inchiostri <lb/>seicenteschi, veramente edaci della loro carta ” . Ad <lb/>essa però non può certamente venir fatta l'accoglienza <lb/>festosa che il TORRICELLI giustamente sperava quando, <lb/>nella tregua del delirio, raccomandava agli amici i suoi <lb/>lavori tuttora inediti. Gli è che nei tre secoli ormai de<lb/>corsi dal giorno in cui egli scese nella tomba l'ambiente

<pb xlink:href="090/01/039.jpg" pagenum="XXXVII"/>matematico si è totalmente e radicalmente mutato. I pro<lb/>cedimenti di cui egli si serviva sono quelli foggiati da un <lb/>contemporaneo di PLATONE, EUDOSSO DA CNIDO, svolti ed <lb/>applicati da ARCHIMEDE e trasfigurati dal CAVALIERI; anzi <lb/>il TORRICELLI seppe servirsene con tanta abilità e disinvol<lb/>tura che ben a ragione questi valentuomini avrebbero po<lb/>tuto additarlo alla universale estimazione dicendo: “ Ecco <lb/>colui che </s>

<pb pagenum="XXXVII"/>matematico si è totalmente e radicalmente mutato. I pro<lb/>cedimenti di cui egli si serviva sono quelli foggiati da un <lb/>contemporaneo di PLATONE, EUDOSSO DA CNIDO, svolti ed <lb/>applicati da ARCHIMEDE e trasfigurati dal CAVALIERI; anzi <lb/>il TORRICELLI seppe servirsene con tanta abilità e disinvol<lb/>tura che ben a ragione questi valentuomini avrebbero po<lb/>tuto additarlo alla universale estimazione dicendo: “ Ecco <lb/>colui che </s>

<s><foreign lang="it"><emph type="center"/><emph type="italics"/>Mostrò ciò che potea la lingua nostra<emph.end type="italics"/> ”.<emph.end type="center"/></foreign></s>

<s>A quei vetusti procedimenti egli si attenne con fedeltà <lb/>ancora più rigorosa di quanto abbia fatto il suo contem<lb/>poraneo HUYGENS (matematico che giova citare pei nume<lb/>rosi punti di contatto che presenta col Nostro ); giacchè <lb/>mentre questi prestò di quando in quando facile orecchio <lb/>alla giovane algebra che allora presentavasi circonfusa di <lb/>promettenti lusinghe, il TORRICELLI austeramente respinse <lb/>ogni sorta d'inviti per quanto seducenti, onde nella storia <lb/>della matematica egli ci si presenta siccome l'ultimo dei <lb/>puristi . Per effetto di tali spiccate caratteristiche men<lb/>tali Egli era destinato ad annoverare in vita molti ammi<lb/>ratori e alcuni seguaci, ma fatalmente doveva essere ben <lb/>presto lasciato in completo abbandono. Perciò, se è indu<lb/>bitato che questo fenomeno si sarebbe manifestato pochi <lb/>decennii dopo la scomparsa del TORRICELLI — cioè dopo il <lb/>trionfo delle idee di DESCARTES e FERMAT, di LEIBNIZ e

<pb xlink:href="090/01/040.jpg" pagenum="XXXVIII"/>NEWTON — non è forse matematicamente certo che esso <lb/>apparirà sotto forma ancora più generale in un'epoca, come <lb/>l'attuale, che segue il secolo di LAGRANGE ed EULERO, non<lb/>chè quello in cui, per opera della pleiade di matematici <lb/>iniziatasi con ABEL e CAUCHY e chiusa con WEIERSTRASS e <lb/>POINCARÉ, l'analisi matematica raggiunse un'altezza, un'e<lb/>stensione, un'energia che sarebbe stato follia sperare?... </s>

<pb pagenum="XXXVIII"/>NEWTON — non è forse matematicamente certo che esso <lb/>apparirà sotto forma ancora più generale in un'epoca, come <lb/>l'attuale, che segue il secolo di LAGRANGE ed EULERO, non<lb/>chè quello in cui, per opera della pleiade di matematici <lb/>iniziatasi con ABEL e CAUCHY e chiusa con WEIERSTRASS e <lb/>POINCARÉ, l'analisi matematica raggiunse un'altezza, un'e<lb/>stensione, un'energia che sarebbe stato follia sperare?... </s>

<s><foreign lang="it">Perciò — sarebbe vano negarlo — la presente pub<lb/>blicazione, al pari delle analoghe che la precedettero in <lb/>Italia ed all'Estero, possiede un carattere, non pratico, <lb/>ma eminentemente storico; ad essa è affidata la nobile <lb/>missione di porgere al futuro storico della matematica gli <lb/>elementi, di cui sino ad oggi si lamentava l'assenza, per <lb/>lumeggiare in tutti i suoi più reposti meati il grande pe<lb/>riodo che prelude l'apparizione del calcolo infinitesimale e <lb/>per determinare il posto che spetta ai discepoli di GALILEO <lb/>fra i precursori dei sommi di cui vanno giustamente su<lb/>perbe l'Inghilterra e la Germania. </foreign></s>

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<s><foreign lang="it">GINO LORIA. </foreign></s>

<pb/>

<s><emph type="center"/>DE SPHAERA <lb/>ET SOLIDIS SPHAERALIBUS<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>LIBRI DUO.<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/>IN QUIBUS ARCHIMEDIS DOCTRINA <lb/>DE SPHAERA ET CYLINDRO DENUO COMPONITUR, <lb/>LATIÙS PROMOVETUR, <lb/>ET IN OMNI SPECIE SOLIDORUM, QUAE VEL CIRCA, <lb/>VEL INTRA SPHAERAM, <lb/>EX CONVERSIONE POLIGONORUM REGULARIUM <lb/>GIGNI POSSINT, UNIVERSALIUS PROPAGATUR.<emph.end type="center"/></s>

<pb/>

<s><emph type="center"/><emph type="italics"/>Serenissimo Magno Duci Etruriae<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>FERDINANDO II<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="italics"/>Erubescerem profectò, Serenissime Magne Dux, oblaturus <lb/>libellum hunc Serenissimae Celsitudini Tuae, nisi haberem <lb/>maxima Archimedis, et Galilei nomina, quae praetendere <lb/>possim audaciae meae: Exigua enim sunt opuscula haec, et <lb/>de rebus aetate nostra neglectis, nempe Geometricis. Attamen, <lb/>nisi fallor, duo maxima Geometriae opera promovebunt, cum <lb/>veterem De Sphaera, et Cylindro, novamque De Motu scien<lb/>tiam exequantur. Sed ego frustra Geometriam excuso apud <lb/>eum Principem, cui non solum haereditaria, sed etiam in<lb/>genita est Mathematicarum disciplinarum protectio. Serenis<lb/>simus enim Cosmus II Pater Tuus stipendijs celeberrimo <lb/>Galileo oblatis; deinde Ser. C. Tua, beneficijs maximis in <lb/>huiusmodi scientiae cultores collocatis, optime demonstravit <lb/>intelligere, quanti momenti sint Mathematicae scientiae, vel <lb/>in disponendis exercituum aciebus, vel in muniendis, exor<lb/>nandisque urbibus, utroque tempore belli, pacisque. Cum <lb/>enim (ut de Mechanica facultate sileam) totum penè civile <lb/>commercium pondere, numero, et mensura administretur, <lb/>quis non videat omne hominum negotium in Mathematicis <lb/>esse? quae tria quantitatis genera cum in Scholis nostris<emph.end type="italics"/>

<pb xlink:href="090/01/043.jpg"/><emph type="italics"/>quotidie agitentur, illi profectò maximè utiles Reip. habe<lb/>buntur, qui in huiusmodi studijs versati, exercitatique erunt. <lb/>Libellorum itaque non inutilium causa penitus mala non <lb/>erit quatenùs Geometrici sunt. Utinam mala non sit eo no<lb/>mine quòd sunt mei: Propterea humilitèr oro, ut illos <expan abbr="qua-lescumq;">qua<lb/>lescumque</expan> sint, Tibi tamen debitos, Tuaque munificentia editos, <lb/>S. C. Tua suscipere dignetur eo vultu, quo me quoque sup<lb/>plicem suscepit, atque ea humanitate, quae cum tanti Prin<lb/>cipis maiestate coniuncta, amorem elicit etiam ab ignotis. <lb/>Faveat Deus omnibus votis Tuis, et S. C. Tuam, <expan abbr="imperiumq;">imperiumque</expan> <lb/>diu tueatur, et augeat.<emph.end type="italics"/></s>

<pb/><emph type="italics"/>quotidie agitentur, illi profectò maximè utiles Reip. habe<lb/>buntur, qui in huiusmodi studijs versati, exercitatique erunt. <lb/>Libellorum itaque non inutilium causa penitus mala non <lb/>erit quatenùs Geometrici sunt. Utinam mala non sit eo no<lb/>mine quòd sunt mei: Propterea humilitèr oro, ut illos <expan abbr="qua-lescumq;">qua<lb/>lescumque</expan> sint, Tibi tamen debitos, Tuaque munificentia editos, <lb/>S. C. Tua suscipere dignetur eo vultu, quo me quoque sup<lb/>plicem suscepit, atque ea humanitate, quae cum tanti Prin<lb/>cipis maiestate coniuncta, amorem elicit etiam ab ignotis. <lb/>Faveat Deus omnibus votis Tuis, et S. C. Tuam, <expan abbr="imperiumq;">imperiumque</expan> <lb/>diu tueatur, et augeat.<emph.end type="italics"/></s>

<s><emph type="italics"/>Sereniss. C. Tuae<emph.end type="italics"/></s>

<s><emph type="italics"/>Humillimus servus<emph.end type="italics"/></s>

<s><emph type="italics"/>Evangelista Torricellius.<emph.end type="italics"/></s>

<pb/>

<s><emph type="center"/>PROEMIUM<emph.end type="center"/></s>

<s>Inter omnia opera ad Mathematicas disciplinas perti<lb/>nentia, iure optimo Principem sibi locum vindicare vi<lb/>dentur Archimedis inventa; quae quidem ipso subtilitatis <lb/>miraculo terrent animos. Verùm inter omnes libros egregij <lb/>Authoris longè eminet ille, qui De Sphaera, et Cylindro <lb/>inscribitur: neque enim posteritatis tantùm consensu, sed <lb/>etiam ipsius Scriptoris iudicio primas tenet. Certè hunc <lb/>ipse in titulum sepulcri elegit, <expan abbr="dignumq;">dignumque</expan> prae caeteris iu<lb/>dicavit, qui tanti viri tumulum exornaret, ostenderetque. <lb/>Hunc tamen si quis attentiùs considerare, et perpendere <lb/>velit, magnum quidem inventum fateatur necesse est, sed <lb/>fortasse non absolutum. Loquor equidem de primo tantùm <lb/>libro, in quo partem operis Theorematicam, et omnem <lb/>doctrinae inventionem exequitur: quo veluti iacto funda<lb/>mento, in secunda parte postea, quasi coronidis loco, pro<lb/>blemata quaedam tamquam corollaria ad eam rem spe<lb/>ctantia ipse subnectit. Titulus libri est De Sphaera, et <lb/>Cylindro; quae quidem verba apud nos idem sonant, ac <lb/>si dixisset De Sphaera, atque unico solido sphaerali; sed <lb/>sphaeralia solida, quorum unum est cylindrus, multitudine <lb/>sunt infinita, ut mox patebit. Ergo absolutior fortasse con<lb/>templatio videri potuisset, si eximius Author proportionem, <lb/>non tantùm eam, quae est inter sphaeram, unicumque <lb/>ex sphaeralibus solidis perquisisset, verumetiam omnem <lb/>aliam rationem, quae inter sphaeram ipsam, et <expan abbr="unumquodq;">unumquodque</expan> <lb/>ex infinitis sphaeralibus solidis inter cedit, ostendendam

<pb xlink:href="090/01/045.jpg" pagenum="6"/>sibi assumpsisset. Hoc itaque propositum erit, et institu<lb/>tum meum in praesenti libello. Doctrinam non solum de <lb/>Sphaera, et Cylindro, sed de sphaera, et sphaeralibus so<lb/>lidis omnibus prosequemur: <expan abbr="Mutatisq;">Mutatisque</expan> plerumque Archi<lb/>medaeis fundamentis, universaliori demonstratione illam <lb/>complecti conabimur, atque in omni specie solidorum, vel <lb/>intrà, vel circà sphaeram descriptorum, propagabimus. </s>

<pb pagenum="6"/>sibi assumpsisset. Hoc itaque propositum erit, et institu<lb/>tum meum in praesenti libello. Doctrinam non solum de <lb/>Sphaera, et Cylindro, sed de sphaera, et sphaeralibus so<lb/>lidis omnibus prosequemur: <expan abbr="Mutatisq;">Mutatisque</expan> plerumque Archi<lb/>medaeis fundamentis, universaliori demonstratione illam <lb/>complecti conabimur, atque in omni specie solidorum, vel <lb/>intrà, vel circà sphaeram descriptorum, propagabimus. </s>

<s>Ex libro Archimedis De Sphaera et Cylindro duo haec <lb/>colliguntur spectantia ad illa solida, quae nos sphaeralia <lb/>appellamus: Primum, quòd sphaera dupla est inscripti sibi <lb/>rombi solidi aequilateri; quod quidem unum est ex solidis <lb/>sphaeralibus, genitum ex revolutione quadrati inscripti, et <lb/>circa diagonalem conversi. Alterum; quòd cylindrus ad <lb/>inscriptam sibi sphaeram est sesquialter. quod quidem et <lb/>unum ex solidis sphaeralibus est, genitum ex conversione <lb/>quadrati circumscripti, et circa ipsius catetum revoluti. <lb/>Stantibus his, contemplatione dignum mihi videbatur uni<lb/>versalius aliquod problema huiusmodi. <lb/><gap desc="SM"/></s>

Line 348

<s>Si intrà circulum descriptum fuerit poligonum regulare <lb/>habens latera numero parià, et conver<lb/>

<arrow.to.target n="fig1"></arrow.to.target><lb/>tatur figura circa catetum B. Quaeri<lb/>tur ratio sphaerae ad factum soli<lb/>dum. </s>

<s>Continuetur ratio radij poligoni ad <lb/>catetum eiusdem, nempe A ad B in <lb/>quatuor terminis A, B, C, D. Erit que <lb/>sphaera ad solidum inscriptum, ut diameter sphaerae, hoc <lb/>est ut dupla ipsius A, ad <expan abbr="utramq;">utramque</expan> simul B, et D. </s>

<s><emph type="center"/><emph type="italics"/>Secunda species.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Si intra circulum descriptum fuerit po<lb/>

<arrow.to.target n="fig2"></arrow.to.target><lb/>ligonum regulare habens latera numero <lb/>paria, et cunvertatur figura circà diagona<lb/>lem AB. Quaeritur ratio sphaerae ad fa<lb/>ctum sphaerale solidum. </s>

<s>Ostenditur. Sphaeram esse ad solidum, </s>

Line 370

<s>Si intrà circulum describatur poligonum regulare ha<lb/>bens latera numero imparia, et con<lb/>

<arrow.to.target n="fig3"></arrow.to.target><lb/>vertatur figura circa catetum B. <lb/>Quaeritur ratio sphaerae ad factum <lb/>sphaerale solidum. </s>

<s>Continuetur ratio radij A ad ca<lb/>tetum B in quatuor terminis A, B, <lb/>C, D. <expan abbr="Eritq;">Eritque</expan> sphaera ad solidum, ut quadrupla ipsius A <lb/>

<arrow.to.target n="marg2"></arrow.to.target><lb/>ad B semel, C bis, et D semel <expan abbr="simulq;">simulque</expan> sumptas. </s>

Line 381

<s>Si circà circulum describatur poligo<lb/>

<arrow.to.target n="fig4"></arrow.to.target><lb/>num regulare, habens latera numero paria, <lb/>et convertatur figura circa catetum C. <lb/>Quaeritur ratio solidi ad sphaeram. </s>

<s>Ostenditur solidum esse ad inscriptam <lb/>sibi sphaeram, ut duo simul quadrata, <lb/>

<arrow.to.target n="marg3"></arrow.to.target><lb/>quorum unum fit ex radio D alterum ex cateto C, ad <lb/>duplum quadrati C. </s>

<s><margin.target id="marg3"></margin.target>Theor. 18. <lb/>Lib. 2.</s>

Line 396

<arrow.to.target n="marg4"></arrow.to.target></s>

<s><margin.target id="marg4"></margin.target>Theor. 6. <lb/>Lib. 2.</s>

<s>Ostenditur solidum ad inscriptam sibi <lb/>sphaeram esse ut radius A ad catetum B <lb/>hoc est ut axis solidi ad axem sphaerae. </s>

Line 404

<s>Si circa circulum describatur poligonum regulare ha<lb/>bens latera numero imparia, et con<lb/>

<arrow.to.target n="fig6"></arrow.to.target><lb/>vertatur figura circa B catetum. <lb/>Quaeritur ratio solidi ad sphaeram. </s>

<s>Continuetur ratio radij A ad ca</s>

Line 418

<s>Quibus demonstratis, varia pro Corollarijs Theoremata <lb/>statim emergebant; cuiusmodi sunt. Datis ex praedicta<lb/>rum sex specierum solidis duobus quibuscunque, alterius <lb/>ad alterum rationem notam facere. </s>

<s>Conum aequilaterum circa sphaeram descriptum, esse <lb/>ad ipsam sphaeram ut 9 ad 4. Nempe duplum sesqui quar<lb/>tum. Propterea si circa eandem sphaeram conus, <expan abbr="cylin-drusq;">cylin<lb/>drusque</expan> aequilateri descripti sint, tria solida, nempe conum, <lb/>cylindrum, et sphaeram fore inter se in continua propor<lb/>tione sesquialtera. </s>

<s>Sphaeram ad conum aequilaterum sibi inscriptum esse <lb/>ut 32 ad 9. </s>

Line 437

<s><emph type="center"/>DEFINITIONES.<emph.end type="center"/></s>

<s>1. Cuiuscunque poligoni regularis latera habentis nu<lb/>mero paria, <emph type="italics"/>Diagonalem<emph.end type="italics"/> voco lineam, quae per oppositos <lb/>flgurae angulos ducitur. <emph type="italics"/>Catetum<emph.end type="italics"/> verò voco lineam, quae <lb/>puncta media laterum oppositorum connectit: sive earum<lb/>dem semisses. Cuiuscunque verò poligoni regularis latera

<pb xlink:href="090/01/049.jpg" pagenum="10"/>habentis numero imparia, <emph type="italics"/>catetum<emph.end type="italics"/> voco lineam, quae ab <lb/>uno angulo per centrum figurae extenditur. </s>

<pb pagenum="10"/>habentis numero imparia, <emph type="italics"/>catetum<emph.end type="italics"/> voco lineam, quae ab <lb/>uno angulo per centrum figurae extenditur. </s>

<s>2. Si poligonum quodcunque regulare convertatur, <lb/>sivè circa diagonalem, sive circa catetum, donec ad eum <lb/>locum redeat unde caepit moveri, solidum illud quod ex <lb/>revolutione circumscribitur, <emph type="italics"/>sphaerale solidum<emph.end type="italics"/> appellare <lb/>visum est. Parilaterum quidem si poligonum habuerit la<lb/>tera numero paria, Imparilaterum verò, quando poligonum <lb/>latera numero imparia habebit. </s>

Line 448

<s>Supponimus. cuiuscunque prismatis circà cylindrum ae<lb/>quealtum descripti, superficiem maiorem esse cylindri <lb/>ipsius superficie. Cylindricam verò superficiem maiorem <lb/>esse superficie prismatis inscripti, basim habentis regula<lb/>rem. exceptis semper basibus. Item pyramidis circa conum <lb/>descriptae superficiem maiorem esse ipsius coni superficie; <lb/>Inscriptae verò pyramidis et regularem basim habentis, <lb/>supponimus superficiem minorem esse conica superficie. </s>

<s>Demonstrantur haec apud Archimedem propos. 9, 10, <lb/>11, 12 lib. I de Sph. et Cyl. Si quis verò ea tamquam <lb/>nota admittere velit, totum libellum nostrum percurrere <lb/>poterit. </s>

<pb/>

<s><emph type="center"/>DE SOLIDIS SPHAERALIBUS<emph.end type="center"/></s>

Line 460

<s>Esto cylindrus rectus ABCD, <expan abbr="seceturq;">seceturque</expan> plano <lb/>

<arrow.to.target n="fig7"></arrow.to.target><lb/>EF oppositis basibus parallelo; Dico cylindricam <lb/>superficiem AEFD, ad cylindricam EBCF, esse <lb/>ut axis ad axem, sive ut latus AE, ad latus EB. </s>

<s>Producatur utrimque in infinitum cylindrus, <lb/>et accipiatur recta EG multiplex ipsius EA, iuxtà <lb/>quamlibet multiplicitatem, sectaque EG in partes <lb/>ipsi EA aequales, agantur per puncta divisio<lb/>num H, I, G; plana oppositis basibus parallela. <lb/>Eritque tam multiplex recta GE ipsius EA: quàm <lb/>multiplex est cylindrica superficies EL, super<lb/>ficiei ED. </s>

<s>Sumatur etiam recta EM multiplex ipsius EB, iuxta <lb/>quamlibet multiplicationem; <expan abbr="similiq.">similique</expan> peracta constructione <lb/>ut supra; erit tam multiplex recta EM rectae EB, quàm <lb/>multiplex est cylindrica superficies EN, superficiei EC. </s>

<s>Manifestum ergo est, quod si recta EG maior fuerit, <lb/>sive minor, vel aequalis, rectae EM: tunc etiam cylindrica <lb/>superficies EL, maior erit, sive minor, vel aequalis super<lb/>ficiei EN: et hoc semper: Propterea erit, ut AE ad EB, <lb/>ita superficies AEFD, ad superficiem EBCF. Quod erat <lb/>demonstrandum. </s>

Line 475

<s>Esto poligonum regulare <lb/>

<arrow.to.target n="fig8"></arrow.to.target><lb/>ABCDEF, super quo conci<lb/>piatur prisma rectum, habens <lb/>pro altitudine AL quartam <lb/>partem cateti IH. Dico peri<lb/>metrum prismatis, constan<lb/>tem ex figuris rectangulis aequalibus quarum una sit LB, <lb/>aequalem esse poligono suae basis. </s>

<s>Ducantur enim diagonales AOD, BOE, et erectà per<lb/>pendiculari IM, iungantur AM, BM; </s>

Line 484

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Constat ergo, quod si altitudo prismatis maior, minorvè fuerit, quàm <lb/>quarta pars cateti suae basis, erit perimeter prismatis maior, minorvè <lb/>quàm poligonum suae basis.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO III.<emph.end type="center"/></s>

Line 492

<s>Esto cylindrus rectus, cu<lb/>

<arrow.to.target n="fig9"></arrow.to.target><lb/>ius basis circulus circa dia<lb/>metrum AB descriptus; alti<lb/>tudo verò AC, aequalis sit <lb/>quartae parti diametri AB. </s>

<s>Dico cylindricam superfi<lb/>ciem aequalem esse circulo <lb/>suae basis AB. </s>

Line 505

<arrow.to.target n="fig10"></arrow.to.target><lb/>drica superficie). Quod est <lb/>absurdum: est enim contra <lb/>praemissas suppositiones. </s>

<s><margin.target id="marg6"></margin.target>ex. Corollar. <lb/>praeced.</s>

<s>Ponatur deinde circulus <lb/>minor quàm cylindrica su<lb/>perficies: et supposità diffe<lb/>rentia G, describatur circa <lb/>circulum aliquod poligonum regulare DEFG, quod excedat

<pb xlink:href="090/01/053.jpg" pagenum="14"/>circulum spatio minori quàm sit C (quomodo hoc fiat con<lb/>stat apud Commentarios in Archim. et in XII Euclidis.) <lb/><expan abbr="eritq;">eritque</expan> etiam poligonum minus quàm cylindrica superficies. </s>

<pb pagenum="14"/>circulum spatio minori quàm sit C (quomodo hoc fiat con<lb/>stat apud Commentarios in Archim. et in XII Euclidis.) <lb/><expan abbr="eritq;">eritque</expan> etiam poligonum minus quàm cylindrica superficies. </s>

<s>Concipiatur suprà poligonum erigi prisma eiusdem al<lb/>titudinis cum cylindro; eritque altitudo prismatis quarta <lb/>pars cateti suae basis poligonae. (cum prismatis altitudo <lb/>eadem sit atq: cylindri; cylindri autem altitudo est quarta <lb/>pars rectae AB, quae aequalis est cateto poligoni, quod <lb/>est basis prismatis). <lb/>

<arrow.to.target n="marg7"></arrow.to.target></s>

Line 525

<s>Esto cylindrus rectus, cuius rectangulum <lb/>

<arrow.to.target n="fig11"></arrow.to.target><lb/>per axem sit ABCD; <expan abbr="sumptaq;">sumptaque</expan> BE, quae <lb/>quarta pars sit ipsius BC; Dico cylindricam <lb/>superficiem ABCD ad circulum suae basis <lb/>esse, ut AB ad BE. </s>

<s>Producatur cylindrus versus F, sectàque <lb/>BF aequali ipsi BE, erit per praecedentem, <lb/>cylindrica superficies FC aequalis circulo suae basis BC. </s>

Line 535

<s><margin.target id="marg8"></margin.target>per 1. huius.</s>

<s>Iam: cylindrica superficies BD, ad cylindricam super<lb/>ficiem FC est ut AB ad BF; superficies verò FC ad cir<lb/>culum BC (ob aequalitatem) est ut FB ad BE; Ergo ex <lb/>aequo erit cylindrica superficies BD ad circulum BC, ut <lb/>AB ad BE, nempe ut latus cylindri ad 1/4 diametri basis <lb/>eiusdem. Quod erat ostendendum. </s>

<s><emph type="center"/>PROPOSITIO V.<emph.end type="center"/></s>

Line 543

<s>Esto cylindrus rectus cuius re<lb/>

<arrow.to.target n="fig12"></arrow.to.target><lb/>ctangulum per axem sit AB, et <lb/>centrum basis H. Ponatur autem <lb/>circulus quilibet cuius semidia<lb/>meter CD. Dico cylindricam su<lb/>perficiem ad circulum ex CD, esse <lb/>ut rectangulum AB ad quadra<lb/>tum CD. </s>

<s>Fiat ex AE (quae quidem 4 pars sit rectae AL) qua<lb/>dratum FE, producaturque EG. </s>

Line 569

<s>Cylindrorum superficies inter se sunt ut eorumdem re<lb/>ctangula per axem homologè sumpta. </s>

<s>Sint cylindri recti quorum rectangula per axem sint

<pb xlink:href="090/01/055.jpg" pagenum="16"/>AB, CD. Dico cylindricam superficiem AB, ad cylindricam <lb/>CD esse, ut rectangulum AB ad rectangulum CD. </s>

<pb pagenum="16"/>AB, CD. Dico cylindricam superficiem AB, ad cylindricam <lb/>CD esse, ut rectangulum AB ad rectangulum CD. </s>

<s>Accipiatur pro circulo quolibet, <lb/>

<arrow.to.target n="fig13"></arrow.to.target><lb/>circulus circa diametrum AE. <lb/>

<arrow.to.target n="marg12"></arrow.to.target></s>

<s><margin.target id="marg12"></margin.target>per praeced.</s>

<s>Erit ergò cylindrica superficies <lb/>AB ad circulum quemlibet AE, ut <lb/>rectang. AB ad quadratum AF. Cir</s>

Line 591

<s>Esto pyramis recta, cuius ba<lb/>

<arrow.to.target n="fig14"></arrow.to.target><lb/>sis poligonum regulare AFED. <lb/>vertex verò G, et centrum basis <lb/>sit I. Secto deinde uno latere bi<lb/>fariam in H, <expan abbr="iunctisq;">iunctisque</expan> GH, IH, <lb/>erit GH catetus superficiei pyra<lb/>midis; IH vero semicatetus basis; <lb/>quandoquidem omnia triangula in superficie sunt aequi<lb/>cruria, et aequalia inter se; quod etiam verum est et <lb/>in basi. </s>

<s>Dico basim ad superficiem esse ut IH ad HG. </s>

<s>Triangulum enim AIF, ad triangulum AGF (cum sint <lb/>in eadem basi) est ut IH, ad HG, ergo etiam ipsorum <lb/>

<arrow.to.target n="marg14"></arrow.to.target><lb/>aequemultiplicia, nempe basis, et superficies pyramidis, in <lb/>eadem ratione erunt, nempe ut IH ad HG. Quod erat <lb/>ostendendum. </s>

<s><margin.target id="marg14"></margin.target>15. quinti.</s>

Line 607

<s>Esto conus rectus, cuius <lb/>

<arrow.to.target n="fig15"></arrow.to.target><lb/>basis AB, vertex verò C, axis <lb/>CD. </s>

<s>Dico circulum basis, ad re<lb/>liquam conicam superficiem, <lb/>esse ut DA, ad AC. </s>

Line 618

<s>Cum <expan abbr="itaq.">itaque</expan> poligonum ad conicam superficiem maiorem <lb/>habeat rationem quàm DA ad AC; multò maiorem ratio<lb/>nem habebit ad superficiem suae pyramidis, quàm DA ad <lb/>AC, vel DB ad BC. Sed poligonum ad superficiem pyra<lb/>midis, per praècedentem, est ut DH ad HC; habebit ergo <lb/>DH ad HC, sive DI ad IC, multò maiorem rationem quàm <lb/>DB ad BC, vel quàm DI ad IL. Et propterea IC minor <lb/>esset quam IL absurdum. </s>

<s>Nam quadratum IC aequale est duobus quadratis ID, <lb/>DC; cum quadratum IL aequale sit tantùm duobus ID, DL. <lb/>Ponatur deinde circulus basis AB minor quàm oportet esse <lb/>ut ad conicam superficiem sit quemadmodum recta DA <lb/>ad AC, sitque tantò minor quantum est spatium E. Cir<lb/>cumscribatur circulo AB poligonum aliquod excedens <lb/>circulum minori excessu quàm sit spatium E. <expan abbr="Habebitq.">Habebitque</expan>

<pb xlink:href="090/01/057.jpg" pagenum="18"/>poligonum ad conicam superficiem, adhuc minorem ratio<lb/>

<pb pagenum="18"/>poligonum ad conicam superficiem, adhuc minorem ratio<lb/>

<arrow.to.target n="marg15"></arrow.to.target><lb/>nem quàm DA ad AC; ergò poligonum ad perimetrum <lb/>suae pyramidis multò mino<lb/>

<arrow.to.target n="fig16"></arrow.to.target><lb/>rem rationem habebit quàm <lb/>DA ad AC. Sed poligonum <lb/>ad perimetrum suae pyra<lb/>midis est ut DF ad FC; <lb/>propterea DF ad FC, multò <lb/>minorem rationem habebit <lb/>quàm DA ad AC; quod est <lb/>impossibile. Aequales etenim sunt tam DF, DA, inter se, <lb/>quàm FC, AC, inter se. </s>

<s><margin.target id="marg15"></margin.target>per 7. huius.</s>

<s>Erit itaque basis coni recti àd reliquam superficiem, ut <lb/>DA ad AC. Quod erat demonstrandum. </s>

Line 643

<arrow.to.target n="marg17"></arrow.to.target></s>

<s><margin.target id="marg17"></margin.target>per 8. huius.</s>

<s>Nàm conica superficies ad circulum suae basis est ut <lb/>AB, àd AH, sive ut rectangulum BAH ad quadratum AH </s>

<s>

<arrow.to.target n="marg18"></arrow.to.target><lb/>circulus autem ex AH, ad cylindricam superficiem DE, <lb/>est ut quadratum AH, ad rectangulum DE. Propterea, ex

<pb xlink:href="090/01/058.jpg" pagenum="19"/>aequo, erit conica superficies ABC ad cylindricam DE, <lb/>ut rectangulum BAH ad rectangulum DE. Quod erat <lb/>ostendendum. </s>

<pb pagenum="19"/>aequo, erit conica superficies ABC ad cylindricam DE, <lb/>ut rectangulum BAH ad rectangulum DE. Quod erat <lb/>ostendendum. </s>

<s><margin.target id="marg18"></margin.target>per 5. huius.</s>

Line 659

<s>Sint duo coni recti ABC, DEF <lb/>

<arrow.to.target n="fig18"></arrow.to.target><lb/>quorum axes BG, EH. Dico curvam <lb/>coni ABC superficiem, ad curvam su<lb/>perficiem coni DEF esse ut rectan<lb/>gulum BAG, ad rectangulum EDH <lb/>quae nimirum sub lateribus conorum, <lb/>et semidiametris basium compraehen<lb/>duntur. </s>

<s>Conica enim superficies ABC, ad circulum AC, est ut <lb/>recta BA ad AG, sive ut rectangulum BAG; ad quadra<lb/>

<arrow.to.target n="marg19"></arrow.to.target><lb/>tum AG. Circulus verò AC ad DF circulum, est ut qua<lb/>dratum AG, ad DH; denique circulus DF ad conicam <lb/>superficiem DEF, est ut quadratum DH, ad rectangulum <lb/>

Line 673

<s>Si fuerit ABCD frustum coni recti, abscissum planis ad axem erectis <lb/>(hoc enim modo semper intelligemus frusta <lb/>

<arrow.to.target n="fig19"></arrow.to.target><lb/>conica) secenturque latera AB, DC bifariam in <lb/>punctis E, et H <expan abbr="iungaturq;">iungaturque</expan> EH. Dico rectam <lb/>EH componi ex utràque BL, AI, nempe ex <lb/>semidiametris basium oppositarum frusti <lb/>conici. </s>

<s>Iungantur BD, EI, LH; Et quoniam AI, ID aequales sunt; item AE, <lb/>EB, aequales: erunt parallelae EI, BD et ideo in parallelogrammo <lb/>

<arrow.to.target n="marg21"></arrow.to.target><lb/>aequalia erunt latera ID, EM. Ob eandem causam aequalia sunt BL, <lb/>MH. Ergo tota EH aequalis erit ipsis ID, BL simul sumptis. Quod <lb/>erat etc.<gap desc="/SM"/></s>

<s><margin.target id="marg21"></margin.target>per 2. sexti.</s>

Line 690

<s><emph type="center"/>PROPOSITIO XI.<emph.end type="center"/></s>

<s>Curva superficies frusti conici, planis ad axem erectis <lb/>abscissi, ad conicam quamlibet superficiem, est ut rectan<lb/>gulum proprium frusti, ad rectangulum sub latere, et se<lb/>midiametro basis ipsius coni. </s>

<s>Esto frustum conicum <lb/>ABCD abscissum planis ad <lb/>axem erectis, sitque conus <lb/>quilibet EFG, cuius axis FH. <lb/>Dico curvam frusti AC su<lb/>perficiem, ad curvam coni <lb/>EFG superficiem, esse, ut <lb/>rectangulum sub AB, et sub utraque AL, BI contentum, <lb/>ad rectangulum FEH. </s>

Line 700

<s><margin.target id="marg22"></margin.target>per 10. <lb/>huius.</s>

<s>Iam superficies curva coni AMD ad superficiem cur<lb/>vam coni BMC est ut rectangulum LAM ad rectangulum <lb/>IBM; nempe ut rectangulum AP ad <expan abbr="Bq;">Bque</expan> et dividendo, <lb/>erit curva frusti conici ABCD superficies, ad superficiem <lb/>coni BMC, ut gnomon AOP, ad rectangulum BQ hoc est <lb/>ut rectangulum sub AB; et utraque AN, BO, sive AL, BI, <lb/>ad rectangulum IBM. Curva verò superficies coni BMC <lb/>ad curvam coni EFG, est ut rectangul. IBM ad rect. FEH <lb/>ergò ex aequo curva frusti conici ABCD superficies ad <lb/>curvam coni EFG superficiem est ut rectan. contentum <lb/>sub AB, et utraque AL, BI ad rectangulum FEH. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

Line 712

<s>Esto frustum conicum ABCD, et cylindrus cuius rectan<lb/>gulum per axem sit EF. Secetur AB bifariam in H, et <lb/>

<arrow.to.target n="fig20"></arrow.to.target><lb/>agatur media Aritmetica HI aequidistanter ad BC. Dico <lb/>conicam frusti superficiem, ad cylindricam EF, esse ut <lb/>rectangulum sub HI, et AB, ad rectangulum EF. </s>

<s>Accipiatur conus quilibet LMN, cuius axis MO. <expan abbr="Eritq;">Eritque</expan> <lb/>curva frusti superficies ad conicam curvam LMN, ut re</s>

<s>

<arrow.to.target n="marg23"></arrow.to.target><lb/>ctangulum sub AB, HI, ad rectangulum MLO; sed curva <lb/>coni LMN ad curvam cylindri EF superficiem, est ut re<lb/>ctangulum MLO, ad rectangulum EF; ergo ex aequo curva <lb/>frusti conici superficies, ad curvam superficiem cylindri, <lb/>est ut rectangulum sub AB, et HI, nempe ut rectangulum <lb/>proprium frusti, ad rectangulum EF per axem cylindri. <lb/>Quod erat ostendendum. </s>

<s><margin.target id="marg23"></margin.target>per praeced.</s>

Line 726

<s>Curva superficies <expan abbr="cuiuscunq;">cuiuscunque</expan> frusti conici ABCD ae<lb/>qualis demonstratur circulo cuidam, cuius quidem circuli <lb/>semidiameter E media proportio<lb/>

<arrow.to.target n="fig21"></arrow.to.target><lb/>nalis sit inter latus AB frusti co<lb/>nici, et inter FH mediam Aritme<lb/>ticam eiusdem frusti. </s>

<s>Esto quadratum E aequale <lb/>rectangulo sub BA, FH sumatur<lb/>que cylindrus quilibet IL; et erit <lb/>

<arrow.to.target n="marg24"></arrow.to.target><lb/>curva frusti conici superficies ad <lb/>curvam cylindricam IL, ut rectan<lb/>

Line 742

<s>Esto circulus ADBC, quem duae <lb/>

<arrow.to.target n="fig22"></arrow.to.target><lb/>diametri AB, CD secent ad angulos <lb/>rectos. Duas insuper tangentes ha<lb/>beat alteram DG in extremitate dia<lb/>metri CD, alteram verò ubicunque <lb/>in I, et aequalitèr producantur hinc <lb/>inde ILIM; dumodo axem AB pro<lb/>ductum non secent. Agantur deinde <lb/>per L, et per M parallelae ad CD, <lb/>rectae LE, MF tum figura convertatur circa axem AB.

<pb xlink:href="090/01/062.jpg" pagenum="23"/>Tangens GH describet cylindricam quandam superficiem <lb/>cuius rectangulum per axem erit EFHG: Tangens verò <lb/>LM designabit frustum conicae superficiei; <expan abbr="deniq;">denique</expan> circulus <lb/>ipse sphaeram circumscribet. Dico cylindricam superficiem <lb/>à linea GH descriptam, et conicam superficiem à linea LM <lb/>factam aequales esse inter se. </s>

<pb pagenum="23"/>Tangens GH describet cylindricam quandam superficiem <lb/>cuius rectangulum per axem erit EFHG: Tangens verò <lb/>LM designabit frustum conicae superficiei; <expan abbr="deniq;">denique</expan> circulus <lb/>ipse sphaeram circumscribet. Dico cylindricam superficiem <lb/>à linea GH descriptam, et conicam superficiem à linea LM <lb/>factam aequales esse inter se. </s>

<s>Ducatur IP media Aritmetica conici frusti; et agatur <lb/>IR per centrum <expan abbr="q;">que</expan> <expan abbr="eritq;">eritque</expan> IR perpendicularis ad LM: Du<lb/>catur etiam MT perpendicularis ad EG. </s>

Line 753

<s>Si circulum tetigerit recta linea aequalitèr <expan abbr="utrinq;">utrinque</expan> pro<lb/>ducta, et convertatur circulus circa axem, qui cum tangente <lb/>conveniat in extremitate ipsius tangentis, erit superficies <lb/>coni, quae à tangente describitur, aequalis superficiei cy<lb/>lindri, eandem cum cono altitudinem <lb/>

<arrow.to.target n="fig23"></arrow.to.target><lb/>habentis, et circà eandem sphaeram <lb/>descriptibilis. </s>

<s>Positis ijsdem ut in praecedentis <lb/>propositionis constructione; si linea <lb/>ML incidat in axem BL productum, <lb/><expan abbr="sintq;">sintque</expan> aequales utrinque IL, IM, tunc <lb/>describet ipsa ML conicam superfi<lb/>ciem, Dico conicam huiusmodi su<lb/>perflciem aequalem esse superficiei cylindri EFHG eandem

<pb xlink:href="090/01/063.jpg" pagenum="24"/>altitudinem habentis cum ipso cono, et circa eandem <lb/>sphaeram descriptibilis. </s>

<pb pagenum="24"/>altitudinem habentis cum ipso cono, et circa eandem <lb/>sphaeram descriptibilis. </s>

<s>Fiat enim angulus LMT rectus, et cum LM dupla po<lb/>natur ipsius LI, erit MT dupla ipsius IR, hoc est aequalis <lb/>diametro sphaerae, sive ipsi FH cum autem, per quartam <lb/>sexti, sit ut ML ad LN, ita TM ad MN erit rectangulum <lb/>LMN aequale rectangulo sub TM, LN, hoc est rectangulo <lb/>sub FH, LN, quod quidem per axem est cylindri EFHG. <lb/>

<arrow.to.target n="marg26"></arrow.to.target><lb/>Aequalis ergo est superficies coni OLM, superficiei cy<lb/>lindri EFHG. Quod etc. </s>

Line 772

<arrow.to.target n="marg27"></arrow.to.target></s>

<s><margin.target id="marg27"></margin.target>per praeced.</s>

<s>Superficies enim coni BAF aequalis est superficiei cy<lb/>lindri ML; Superficies autem frusti conici, quae inter plana <lb/>BF, CE intercipitur, aequalis est superficiei cylindri inter </s>

<s>

<arrow.to.target n="marg28"></arrow.to.target><lb/>eadem plana intercepti: et sic de singulis partibus super<lb/>ficierum, quae solidum sphaerale circumsepiunt; Ergò

<pb xlink:href="090/01/064.jpg" pagenum="25"/>omnes simul superficies ambientes sphaerale solidum ae<lb/>quales erunt superficiei cylindri GHIL. Quod erat osten<lb/>dendum. </s>

<pb pagenum="25"/>omnes simul superficies ambientes sphaerale solidum ae<lb/>quales erunt superficiei cylindri GHIL. Quod erat osten<lb/>dendum. </s>

<s><margin.target id="marg28"></margin.target>3. huius.</s>

Line 786

<s>Si circulum duae diametri AB, CD, ad angulos rectos secuerint, <lb/><expan abbr="eundemq;">eundemque</expan> circulum duae aequales rectae lineae <lb/>

<arrow.to.target n="fig25"></arrow.to.target><lb/>AF, BG tetigerint in extremitatibus axis AB. Tum <lb/>figura circà axem AB convertatur, describent AF, <lb/>BG duos circulos aequales, cum ipsae aequales <lb/>sint. Oportet segmentum cylindri circà eandem <lb/>sphaeram descriptibilis reperire, cuius superficies <lb/>aequalis sit duobus simul circulis ex AF, BG dc<lb/>scriptis. </s>

<s>Fiat angulus HGI rectus, <expan abbr="eritq;">eritque</expan> BI altitudo <lb/>quaesiti cylindri. Nam propter angulum rectum <lb/>HGI, erit rectangulum HBI aequale quadrato BG; <lb/>et rectangulnm ABI hoc est rectangulum LM duplum erit quadrati BG. <lb/>Propterea superficies cylindri LM dupla erit circuli ex BG descripti, et <lb/>

<arrow.to.target n="marg29"></arrow.to.target><lb/>ideo aequalis ambobus circulis ex BG, AF simul sumptis. Quod etc.<gap desc="/SM"/></s>

Line 799

<s>Esto circulus ABCD, quem secent duae diametri AC, <lb/>

<arrow.to.target n="fig26"></arrow.to.target><lb/>BD ad angul. rectos, et circa ipsum sit poligona figura

<pb xlink:href="090/01/065.jpg" pagenum="26"/>habens latera numero paria, sivè à quaternario mensu<lb/>rentur, ut in prima figura; sive tantùm à binario, ut in <lb/>secunda: Tum convertatur figura circa catetum AC, hoc <lb/>est circa lineam connectentem bisectiones laterum oppo<lb/>sitorum; Ex revolutione poligoni solidum sphaerale descri<lb/>betur contentum sub circularibus, conicisque superficiebus, <lb/>et una cylindrica, ut in prima figura, sive circularibus, et <lb/>conicis tantùm, ut in secunda. Fiat deinde ut IC ad CL, <lb/>ita CL ad CM, quod facile erit si fiat angulus ILM rectus; <lb/>et per M agatur planum NO erectum ad axem. Dico uni<lb/>versam superficiem solidi sphaeralis aequalem esse super<lb/>ficiei cylindri ENOH. </s>

<pb pagenum="26"/>habens latera numero paria, sivè à quaternario mensu<lb/>rentur, ut in prima figura; sive tantùm à binario, ut in <lb/>secunda: Tum convertatur figura circa catetum AC, hoc <lb/>est circa lineam connectentem bisectiones laterum oppo<lb/>sitorum; Ex revolutione poligoni solidum sphaerale descri<lb/>betur contentum sub circularibus, conicisque superficiebus, <lb/>et una cylindrica, ut in prima figura, sive circularibus, et <lb/>conicis tantùm, ut in secunda. Fiat deinde ut IC ad CL, <lb/>ita CL ad CM, quod facile erit si fiat angulus ILM rectus; <lb/>et per M agatur planum NO erectum ad axem. Dico uni<lb/>versam superficiem solidi sphaeralis aequalem esse super<lb/>ficiei cylindri ENOH. </s>

<s>Hoc autem patet ex praemissis; Nam tota sphaeralis <lb/>solidi superficies, demptis circulis oppositis, aequalis est <lb/>

<arrow.to.target n="marg30"></arrow.to.target><lb/>superficiei cylindricae inter plana EH, FG compraehensae. <lb/>Duo verò circuli oppositi quorum centra A, et C aequales <lb/>sunt (per praecedens lemma) superficiei cylindricae inter <lb/>duo plana FG, NO contentae. Propterea universa simul <lb/>sphaeralis solidi superficies aequalis erit superficiei cylindri <lb/>ENOH circa eandem sphaeram descripti, et altitudinem <lb/>habentis AM, quae componitur ex diametro sphaerae AC, <lb/>et ex recta CM, quae quidem tertia proportionalis est ad <lb/>semidiametrum IC, et semilatus, CL. Quod etc. </s>

Line 811

<s>Si circulum ABCD duae diametri AC, BD secent ad angulos rectos; <lb/>recta autem linea CE eundem contingat in extre<lb/>

<arrow.to.target n="fig27"></arrow.to.target><lb/>mitate axis AC et convertatur figura circa AC; <lb/>ipsa CE circulum describet. Oportet segmentum <lb/>cylindri circa eandem sphaeram descripti reperire, <lb/>cuius superficies aequalis sit circulo ex CE de<lb/>scripto. </s>

<s>Fiat angulus AEH rectus, ductoque plano per H <lb/>ad axem erecto. Dico cylindricam superficiem MILN <lb/>aequari circulo ex CE. Est enim ob angulum <lb/>rectum AEH, rectangulum ACH, hoc est rectan<lb/>

<arrow.to.target n="marg31"></arrow.to.target><lb/>gulum ML, aequale quadrato CE. Proptereà superficies cylindri MILN <lb/>aequalis erit circulo ex CE. Quod etc.<gap desc="/SM"/></s>

<s><margin.target id="marg31"></margin.target>per 5. huius.</s>

Line 825

<s>Esto circulus ABCD, circa quem <lb/>

<arrow.to.target n="fig28"></arrow.to.target><lb/>sit poligonum EFGHI habens latera <lb/>numero imparia; et convertatur figura <lb/>circa catetum EC, nempe circa lineam, <lb/>quae ab uno angulo E perducitur ad <lb/>bisectionem lateris oppositi; <expan abbr="orieturq;">orieturque</expan> <lb/>solidum sphaerale contentum sub co<lb/>nicis superficiebus, unicoque circulo. </s>

<s>Facto deinde angulo recto AHL, <lb/><expan abbr="ductoq;">ductoque</expan> per L plano MN ad axem erecto. Dico universam <lb/>solidi superficiem aequalem esse superficiei cylindri OMNP. <lb/>

<arrow.to.target n="marg32"></arrow.to.target></s>

Line 839

<s>Hemisphaerij superficies aequalis est superficiei curvae <lb/>cylindri eandem ipsi basim, et eandem altitudinem habentis. </s>

<s>Esto hemisphaerium ABC, et circa ipsum cylindrus <lb/>eiusdem altitudinis, ADEC. </s>

<s>Dico superficiem hemisphaerij aequalem esse super<lb/>ficiei cylindri ADEC. </s>

<s>Si enim non est aequalis, vel maior erit, vel minor. <lb/>Ponatur primùm sphaerica super<lb/>

<arrow.to.target n="fig29"></arrow.to.target><lb/>ficies maior: fiatque ut cylindri <lb/>superficies ad superficiem hemi<lb/>spherij, quae maior ponitur, ita <lb/>recta AD ad AG: <expan abbr="intelligaturq;">intelligaturque</expan> <lb/>cylindrus productus usque ad GF. <lb/>Secetur deinde arcus AB bifa<lb/>riam, <expan abbr="iterumq;">iterumque</expan> portiones eius bifariam, et hoc semper, <lb/>donec poligoni circà semicirculum ABC descripti semilatus <lb/>VL minus sit quam recta DG (quod fieri posse constat ex <lb/>prima Decimi; semilatera enim poligonorum circulo cir<lb/>cumscriptorum ex continua arcuum bisectione semper mi<lb/>nuuntur plusquam pro medietate, ut ab alijs ostensum <lb/>est). Factum ergò sit; et esto poligonum HILMN, conver<lb/>sàque figura circa axem LO, fiat ex poligono, semisolidum <lb/>sphaerale sub conicis superficiebus compraehensum. Cum <lb/>itaque recta DG maior sit quam semilatus LV, multo <lb/>maior eadem erit quàm LB, et propterea planum PQ pro<lb/>ductum per L intra puncta D et G cadet. </s>

<s>Iam quia superficies cylindri AE ad superficiem hemis<lb/>phaerij est ut AD ad AG, hoc est ut cylindrica super</s>

Line 860

<s>Assumpsimus conicam quae describitur à linea HS maiorem esse <lb/>quàm illa superficies, quae describitur à linea AS quod patet ex 12 <lb/>huius. Rectangulum enim proprium conicae superficiei multò maius est <lb/>quam rectangulum per axem cylindricae, quando quidem sub maioribus <lb/>lateribus continetur.<gap desc="/SM"/></s>

<s>Ponatur iam sphaerica ABC minor quam cylindrica <lb/>ADEC. Fiat ut superficies cylindrica ADEC ad sphaeri<lb/>cam, quae ponitur minor; ita recta AF ad FL. Fiatque

<pb xlink:href="090/01/068.jpg" pagenum="29"/>ex FL semidiametro aliud hemisphaerium LNI, priori <lb/>concentricum, et circà ipsum intelligatur cylindrus LHMI: <lb/>Inscribatur etiam intrà se micirculum ABC figura laterum <lb/>

<pb pagenum="29"/>ex FL semidiametro aliud hemisphaerium LNI, priori <lb/>concentricum, et circà ipsum intelligatur cylindrus LHMI: <lb/>Inscribatur etiam intrà se micirculum ABC figura laterum <lb/>

<arrow.to.target n="fig30"></arrow.to.target><lb/>aequalium, ita ut latera ipsius non tangant semicirculum <lb/>LNI (quod fieri posse constat ex Euclide). <expan abbr="Describaturq;">Describaturque</expan> <lb/>alius semicirculus semidiametro FO, qui contingat singula <lb/>latera factae figurae, et convertatur universa figura circa <lb/>FB ita ut fiat semisolidum sphaerale AVBTC conicis <lb/>superficiebus circumseptum; ex semicirculo autem FO <lb/>fiat aliud hemisphaerium, circà quod concipiatur cylin<lb/>drus RQSP. </s>

<s>Iam sic; superficies cylindri ADEC ad superficiem he<lb/>misphaerij est, per constructionem, ut AF ad FL, hoc est <lb/>ut AC ad LI, hoc est ut rectangulum AE ad rectangulum <lb/>

<arrow.to.target n="marg35"></arrow.to.target><lb/>LM, hoc est ut cylindrica AE ad cylindricam LM. Quare <lb/>sphaerica superficies aequalis erit cylindricae LM, et pro<lb/>pterea minor quàm cylindrica RS, hoc est quàm omnes <lb/>

Line 877

<s><emph type="center"/>PROPOSITIO XIX.<emph.end type="center"/></s>

<s>Cuiuscunque minoris portionis Sphaerae superficies ae<lb/>qualis est curvae superficiei cylindri circà integram sphae<lb/>ram descripti, et eandem altitudinem cum ipsa portione <lb/>habentis.

<arrow.to.target n="marg37"></arrow.to.target></s>

<s><margin.target id="marg37"></margin.target>15. quinti.</s>

<s>Esto minor sphaerae <lb/>

<arrow.to.target n="fig31"></arrow.to.target><lb/>portio ABC, et portio cy<lb/>lindri FDEG; circa inte<lb/>gram sphaeram descripti, <lb/>eandem tamen altitudi<lb/>nem HB cum ipsa por<lb/>tione sphaerica habentis. <lb/>Dico sphaericam superfi<lb/>ciem ABC aequalem esse superficiei cylindri FDEG. </s>

<s>Si enim non est aequalis, vel maior erit vel minor. </s>

Line 895

<s>Assumpsimus cylindricam superficiem FSTG maiorem esse omnibus <lb/>conicis <expan abbr="INOPq.">INOPque</expan> Hoc enim patet. Nam ex 13, 14 et 15 huius colligi <lb/>potest, conicas INOPQ aequales esse superficiei cylindricae contentae <lb/>inter planum ST, et planum quod duceretur per puncta <expan abbr="Iq.">Ique</expan> </s>

<s>Assumpsimus etiam, ductà tangente AV conicam superficiem, quae <lb/>fit à linea IV, maiorem esse quàm illa quae fit linea AV. Quod quidem <lb/>demonstratur apud Archimedem ad Propositionem 37 de Sphaera et <lb/>Cylindro. Sed et ex nostris deduci potest. Nam rectangulum proprium <lb/>superficiei, quae fit à linea IV, maius est quàm rectangulum proprium <lb/>illius quae fit à linea AV. Continetur enim sub lineis maioribus.<gap desc="/SM"/></s>

<s>Ponatur deinde sphaerica superficies portionis ABC <lb/>min. quam cylindrica FDEG. </s>

<s>Fiat ut cylindrica FDEG ad sphaericam superficiem <lb/>ABC, quae minor ponitur, ita FH ad HM et centro T <lb/>semidiametro autem HM fiat hemisphaerium OQP, circa <lb/>quod intelligatur cylindrus OLNP. Intra arcum autem <lb/>ABC figura inscribatur multorum laterum AVBXC per <lb/>continuam bisectionem arcuum ita ut latera ipsius non <lb/>tangant semicirculum OQP, et convertatur universa figura <lb/>circa axem BT. Intelligatur autem radio TZ (quae recta <lb/>perpendicularis sit ad unum latus figurae inscriptae) de<lb/>scribi sphaeram, quae tangat singula figurae AVBXC la<lb/>tera, et circa huiusmodi sphaeram descriptus concipiatur <lb/>suus cylindrus <foreign lang="greek">gbed. </foreign></s>

Line 921

<s><margin.target id="marg41"></margin.target>15. huius.</s>

<s>Constat ergò superficiem ABC aequalem esse cylin<lb/>dricae DFGE cum demonstratum sit neque maiorem esse, <lb/>neque minorem. Quod etc. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium I.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Ex prima duarum praemissarum Propositionum pa<lb/>

<arrow.to.target n="fig32"></arrow.to.target><lb/>tet superficiem integram sphaerae, aequalem esse su<lb/>perficiei cylindri sibi circumscripti, et eiusdem cum <lb/>ipsa sphaera altitudinis. </s>

<s>Cum enim haemisphaerium ABC superficiem habeat <lb/>aequalem superficiei cylindri AEHC, et item hemispae<lb/>rium alterum ADC, superficiem habeat aequalem super<lb/>ficiei cylindri AFGC, erit coniunctim tota sphaerae superficies aequalis <lb/>superficiei cylindri FEHG; exceptis semper basibus.<gap desc="/SM"/></s>

Line 938

<arrow.to.target n="marg42"></arrow.to.target></s>

<s><margin.target id="marg42"></margin.target>Corollar. <lb/>praeced.</s>

<s>Cum enim integra sphaerae superficies aequalis sit <lb/>superficiei cylindri IDGL, et demonstratum sit super<lb/>ficiem segmenti minoris ABC aequalem esse superficiei <lb/>cylindri EDGF, erit reliqua superficies sphaerae AHC, aequalis reliquae <lb/>superficiei EILF. Quod oportebat etc.<gap desc="/SM"/></s>

Line 948

<s>Sit sphaera ABCD cuius diameter AC; et <lb/>

<arrow.to.target n="fig34"></arrow.to.target><lb/>circà ipsam intelligatur cylindrus eiusdem <lb/>altitudinis FEHG. </s>

<s>Dico superficiem sphaerae quadruplam <lb/>esse maximi circuli in ea descriptibilis. </s>

Line 957

<s>

<arrow.to.target n="marg43"></arrow.to.target><lb/>descriptum, ut EF ad quar. partem ipsius FG, hoc est ut <lb/>FG ad quar. partem ipsius FG; hoc est quadrupla. Pro<lb/>

<arrow.to.target n="marg44"></arrow.to.target><lb/>pterea etiam superficies sphaerae, quae cylindricae est <lb/>aequalis, quadrupla erit circuli circa AC descripti, qui in <lb/>sphaera maximus est. Quod etc. </s>

<s><margin.target id="marg43"></margin.target>4. huius.</s>

Line 976

<s>Esto sphaerae portio sive minor sive maior ABC. cuius <lb/>ex polo ducta sit recta AB. Dico superficiem portionis <lb/>

<arrow.to.target n="fig35"></arrow.to.target><lb/>aequalem esse circulo qui fit ex AB tamquam semidia<lb/>metro. </s>

<s>Cum enim quadratum AB aequale sit rectangulo DBE <lb/>ob circulum, aequale erit et rectangulo GFIH, quod idem <lb/>est ac rectangulum DBE. Propterea circulus ex AB ae<lb/>qualis erit superficiei cylindri, cui per axem sit rectang. <lb/>

<arrow.to.target n="marg46"></arrow.to.target><lb/>GFIH, et ideo aequalis etiam superficiei sphaericae por<lb/>tionis ABC. Quod etc. <lb/><gap desc="SM"/></s>

Line 984

<s><margin.target id="marg46"></margin.target>5. huius.</s>

<s>Tria haec Theoremata, quae sequuntur, ex Archimede desumpta <lb/>sunt; quod quidem fecimus ne lector Archimedem adire cogeretur, sed <lb/>universam hanc doctrinam in hoc libello haberet.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXII.<emph.end type="center"/></s>

<s>Sint duo coni recti ABC, DEF. <expan abbr="Sitq;">Sitque</expan> curvae coni ABC <lb/>superficiei aequalis circulus DF; nempe basis alterius <lb/>coni DEF; rectae verò IH, quae <lb/>

<arrow.to.target n="fig36"></arrow.to.target><lb/>ex centro I ducitur perpendicu<lb/>lariter ad latus AB, aequalis sit <lb/>altitudo EL: Dico conos ABC, <lb/>DEF esse aequales. </s>

<s>Nam altitudo BI ad altitudi<lb/>

<arrow.to.target n="marg47"></arrow.to.target><lb/>nem EL est ut BI ad IH (ob aequalitem) sive ut BA, <lb/>ad AI, nempe ut curva superficies ABC ad basim AC; <lb/>sive ut basis DF ad basim AC reciprocè. Quare aequales <lb/>erunt coni ABC, DEF. Quod erat etc. </s>

Line 1005

<s>Si fuerit rombus solidus ABCD, ex duobus conis rectis <lb/>compositus; <expan abbr="Sitq;">Sitque</expan> conus EFG habens basim EG aequalem <lb/>superficiei curvae alterius conorum rombi, puta, BAD; al<lb/>titudinem verò FH aequalem <lb/>

<arrow.to.target n="fig37"></arrow.to.target><lb/>rectae CL, quae quidem ex ver<lb/>tice reliqui coni BCD ducitur <lb/>perpendiculariter in latus AB <lb/>productum alterius coni BAD. <lb/>Dico rombum solidum ABCD <lb/>aequalem esse cono EFG. </s>

<s>Ducatur IN perpendicularis ad AB. Iam, conus BCD, <lb/>ad conum BAD, est ut CI ad IA; et componendo, rombus

<pb xlink:href="090/01/074.jpg" pagenum="35"/>ABCD ad conum BAD est ut CA ad AI; sive ut CL, <lb/>ad IN. Conus verò BAD ad conum EFG est ut IN ad FH: <lb/>ergo ex aequo rombus ABCD ad conum EFG est ut CL <lb/>

<pb pagenum="35"/>ABCD ad conum BAD est ut CA ad AI; sive ut CL, <lb/>ad IN. Conus verò BAD ad conum EFG est ut IN ad FH: <lb/>ergo ex aequo rombus ABCD ad conum EFG est ut CL <lb/>

<arrow.to.target n="marg48"></arrow.to.target><lb/>ad FH. Ergo aequalis. Quod erat, etc. </s>

<s><margin.target id="marg48"></margin.target>per Cor. <lb/>praeced.</s>

Line 1017

<s>Si fuerit conus sive rombus solidus ABCD sectus plano <lb/>EF ad basim parallelo. Intelligaturque ex integro solido <lb/>ABCD ablatus rombus solidus EBFD. Dico reliquum so<lb/>

<arrow.to.target n="fig38"></arrow.to.target><lb/>lidum ex cavatum AEDFC quod superest, equale esse <lb/>cono cuidam M, cuius basis M sit aequalis frusto curvae <lb/>superficiei conicae AEFC inter plana EF, AC, interceptae, <lb/>altitudo verò M sit aequalis perpendiculari DI, quae à <lb/>vertice ablati rombi D ducitur in latus BA. </s>

<s>Intelligantur tres coni aequealti L, M, N quorum uni<lb/>cuique altitudo sit aequalis rectae DI; basis verò coni L <lb/>sit aequalis curvae superficiei coni EBF at basis M ae<lb/>qualis sit segmento conicae superficiei inter plana EF, AC <lb/>intercepto: coni tandem N basis aequalis sit utrisque simul <lb/>praedictis basibus; sive (quod idem est) integrae superficiei <lb/>curvae coni ABC. </s>

<s>Manifestum est quod integrum solidum ABCD aequale <lb/>erit cono N (per alterutram praecedentium duarum Pro<lb/>pos.) sed etiam duo coni L et M simul sumpti aequales <lb/>

<arrow.to.target n="marg49"></arrow.to.target><lb/>sunt eidem cono N ergo integrum solidum ABCD aequale <lb/>erit duobus conis L et M simul sumptis. Demptis itaque,

<pb xlink:href="090/01/075.jpg" pagenum="36"/>rombo EBFD, et cono L, qui per praecedentem sunt ae<lb/>quales, reliquum solidum excavatum AEDFC aequale erit <lb/>reliquo cono M. Quod erat etc. </s>

<pb pagenum="36"/>rombo EBFD, et cono L, qui per praecedentem sunt ae<lb/>quales, reliquum solidum excavatum AEDFC aequale erit <lb/>reliquo cono M. Quod erat etc. </s>

<s><margin.target id="marg49"></margin.target>ex 24. <lb/>Quinti.</s>

Line 1033

<s>Esto cylindrus, cuius rectangu<lb/>

<arrow.to.target n="fig39"></arrow.to.target><lb/>lum per axem sit ABCD et ex ipso <lb/>auferatur conus BEC, ut dictum est. <lb/>Sumatur autem alius conus FIL, <lb/>cuius basis FL aequalis sit superficiei curvae cylindri, al<lb/>titudo aequalis rectae NB hoc est semidiametro basis cy<lb/>lindri. Dico reliquum ex cylindro solidum, dempto cono <lb/>BEC, aequale esse cono FIL. </s>

<s>Secetur BN bifariam in O. Conus ergo FIL ad conum <lb/>BEC, rationem habet compositam ex ratione altitudinum <lb/>HI ad BA, hoc est NB ad BA, et ex ratione basium, hoc <lb/>

<arrow.to.target n="marg50"></arrow.to.target><lb/>est basis quae circa FL ad basim quae circa BC, sive quod <lb/>idem est, superficiei cylindricae ad basim propriam quae <lb/>circa BC, hoc est, lineae AB ad BO. Erit ergò conus FIL <lb/>ad conum BEC, ut NB ad BO, nempe duplus: solidum <lb/>etiam cylindricum excavatum, dempto cono BEC, duplum <lb/>est eiusdem coni BEC. Propterea solidum cylindricum <lb/>excavatum aequale erit cono FIL, cuius basis aequatur <lb/>superficiei cylindri, altitudo verò aequalis est semidiametro <lb/>basis cylindri. Quod etc. </s>

Line 1043

<s><emph type="center"/>PROPOSITIO XXVI.<emph.end type="center"/></s>

<s>Si ex cono conus auferatur eandem habens basim alti<lb/>tudinem verò minorem, erit excavatum solidum conicum, <lb/>quod relinquitur, aequale cono cuidam, cuius quidem basis <lb/>aequalis sit curvae superficiei totius prioris coni, alti-

<pb xlink:href="090/01/076.jpg" pagenum="37"/>tudo verò aequalis perpendiculari, quae ex vertice ablati <lb/>coni demittitur in latus maioris coni. </s>

<pb pagenum="37"/>tudo verò aequalis perpendiculari, quae ex vertice ablati <lb/>coni demittitur in latus maioris coni. </s>

<s>Esto conus rectus ABC ex quo auferatur conus ADC, <lb/>uti dictum est. Ponatur autem <lb/>

<arrow.to.target n="fig40"></arrow.to.target><lb/>conus EFG, habens basim EG, <lb/>aequalem curvae superficiei coni <lb/>ABC; altitudinem verò HF ae<lb/>qualem rectae DI, quae per<lb/>pendicularitèr à vertice ablati <lb/>coni cadit in latus AB. Dico solidum conicum excavatum <lb/>ADCB, dempto cono ADC, aequale esse cono EFG. </s>

<s>Nam cum triangula BLA, BID, rectangula sint, ha<lb/>beantque angulum communem ABL, similia erunt. Sed <lb/>conus EFG ad conum ADC rationem habet compositam <lb/>ex ratione basium, nempe circuli circa EG, sive superficiei <lb/>curvae coni ABC, ad circulum circa AC, hoc est rectae <lb/>

<arrow.to.target n="marg51"></arrow.to.target><lb/>BA ad AL; sive BD ad DI, et ex ratione altitudinum, <lb/>

Line 1061

<s>Si ab eadem magnitudine AB duae magnitudines inae<lb/>

<arrow.to.target n="fig41"></arrow.to.target><lb/>quales auferantur AC, maior, et AD minor, <expan abbr="fueritq;">fueritque</expan> DC, nempe <lb/>differentia inter ablatas, aequalis differentiae sive excessui, <lb/>quo maius residuum BD superat quandam magnitudinem E. <lb/>Dico ipsam E minori residuo CB aequalem esse. </s>

<s>Patet hoc. Cum enim maius residuum DB superet magni<lb/>tudinem E excessu DC; si excessu abijciatur, erit reliqua CB <lb/>aequalis magnitudini E. Propterea magnitudo E aequalis est <lb/>minori residuo. Quod etc.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXVII.<emph.end type="center"/></s>

<s>Si ex conico frusto conus auferatur, qui pro basi ha<lb/>beat maiorem frusti basim, altitudinem verò eandem cum <lb/>frusto; Erit reliquum excavatum solidum aequale cono

<pb xlink:href="090/01/077.jpg" pagenum="38"/>cuidam, qui basim habeat aequalem superficiei curvae <lb/>frusti, altitudinem verò aequalem perpendiculari quae du<lb/>citur ex vertice ablati coni in latus alterum conici frusti. </s>

<pb pagenum="38"/>cuidam, qui basim habeat aequalem superficiei curvae <lb/>frusti, altitudinem verò aequalem perpendiculari quae du<lb/>citur ex vertice ablati coni in latus alterum conici frusti. </s>

<s>Esto conicum frustum ABCD, <lb/>

<arrow.to.target n="fig42"></arrow.to.target><lb/>cuius maior basis sit circulus <lb/>circa BC. Et ex ipso auferatur <lb/>conus BEC, cuius basis sit idem <lb/>circulus circa BC; altitudo verò <lb/>FE eadem cum frusto. Dico re<lb/>liquum solidum excavatum dem<lb/>pto cono BEC, aequale esse cono <lb/>cuidam, cuius basis aequalis sit <lb/>curvae superficiei conici frusti ABCD altitudo vero sit <lb/>linea EH quae nimirùm ex E vertice ablati coni cadit <lb/>perpendicularitèr in AB latus conici frusti. </s>

<s>Inscribatur alius conus AFD habens basim circà AD, <lb/>et verticem in F. Ducaturque AI parallela ad EF, eritque <lb/>tota IC aequalis utrique simul semidiametro basium, nempe <lb/>ipsi EA, <expan abbr="ipsiq;">ipsique</expan> FB. Fiat deinde circa FB semicirculus FOB, <lb/>in quo applicetur BO aequalis ipsi FI, sive ipsi EA; <expan abbr="eritq;">eritque</expan> <lb/>circulus ex semidiametro FO differentia inter duos circu<lb/>los, quorum semidiametri sint, FB, BQ, sive FB, et EA; <lb/>nempe differentia inter bases oppositas conici frusti, hoc <lb/>est inter bases conorum BEC, AFD, et propterea conus <lb/>cuius basis sit circulus ex FO semidiametro, altitudo verò <lb/>FE, differentia erit, sive excessus, quo maior conus BEC <lb/>superat minorem AFD. </s>

Line 1087

<s><margin.target id="marg54"></margin.target>5. secundi.</s>

<s>Cum autem rectangulum BIC simul cum quadrato FI <lb/>aequale sit quadrato FB, vel quadratis FO, OB, demptis

<pb xlink:href="090/01/078.jpg" pagenum="39"/>aequalibus BO, FI erit reliquum rectangulum BIC qua<lb/>drato FO aequale. </s>

<pb pagenum="39"/>aequalibus BO, FI erit reliquum rectangulum BIC qua<lb/>drato FO aequale. </s>

<s>Concipiatur iam conus AFD detrahi ex conico frusto <lb/>ABCD, <expan abbr="eritq;">eritque</expan> reliquum excavatum solidum dempto prae</s>

Line 1108

<s>Sit conicum frustum ABCD cuius <lb/>

<arrow.to.target n="fig43"></arrow.to.target><lb/>maior basis BC, et ex ipso auferatur <lb/>conus BEC, altitudinem habens eandem <lb/>cum frusto, et pro basi, maiorem ipsius <lb/>frusti basim. Compleatur conus BGC, <lb/>cuius datum erat frustum, ductaque EH <lb/>

<arrow.to.target n="marg57"></arrow.to.target><lb/>ad angulos rectos ipsi BG, ponatur IL <lb/>media proportionalis inter GB, BF, <expan abbr="eritq;">eritque</expan> <lb/>circulus ex IL semidiametro descriptus, <lb/>aequalis superficiei coni BGC fiat circa IL semicirculus IML, in quo <lb/>aptetur IM media proportionalis inter GA, AE, <expan abbr="eritq;">eritque</expan> circulus ex semi-

<arrow.to.target n="marg58"></arrow.to.target><lb/>diametro IM factus aequalis superficiei coni AGD; Reliquus circulus <lb/>ex semidiametro ML factus, aequalis erit superficiei conicae frusti <lb/>ABCD. (si enim ab aequalibus aequalia demas reliqua sunt aequalia). </s>

<s><margin.target id="marg57"></margin.target>per Cor. 8. <lb/>huius.</s>

<s><margin.target id="marg58"></margin.target>per Cor. 8. <lb/>huius.</s>

<s>Dico reliquum solidum frusti conici ABCD, ablato cono BEC, aequale <lb/>esse cono cuidam, cuius altitudo sit EH; basis verò aequalis super<lb/>ficiei conicae ispsius frusti; hoc est circulus ex semidiametro ML de<lb/>scriptus. </s>

Line 1133

<s>Esto cylindrus AB cuius axis <lb/>

<arrow.to.target n="fig44"></arrow.to.target><lb/>CD. Tubus verò cylindricus EF <lb/>(dempto nimirum cylindro GH) <lb/>aequealtus sit cum cylindro AB. <lb/>Dico cylindrum AB a tubum EF esse ut quadratum AC <lb/>semidiametri basis cylindri, ad rectangulnm EGI, nempe <lb/>ad rectangulum basis tubi, hoc est quod fit à differentia <lb/>EG et ab aggregato GI semidiametrorum basis ipsius tubi. </s>

<s>Nam cylindrus integer EF ad cylindrum GH, est ut <lb/>quadratum EL ad LG quadratum. Et dividendo, Tubus <lb/>cylindricus EF ad cylindrum GH est ut rectangulum EGI <lb/>ad quadratum GL. Sed cylindrus GH ad AB cylindrum <lb/>est ut quadratum GL ad quadratum BC. Ergo ex aequo

<pb xlink:href="090/01/080.jpg" pagenum="41"/>erit tubus cylindricus EF ad cylindrum AB ut rectan<lb/>gulum EGI ad quadratum AC. Convertendo igitur patet <lb/>quod propositum erat. </s>

<pb pagenum="41"/>erit tubus cylindricus EF ad cylindrum AB ut rectan<lb/>gulum EGI ad quadratum AC. Convertendo igitur patet <lb/>quod propositum erat. </s>

<s><emph type="center"/>PROPOSITIO XXIX.<emph.end type="center"/></s>

Line 1144

<s>Esto cylindrus ABCD, cuius <lb/>

<arrow.to.target n="fig45"></arrow.to.target><lb/>axis EF: <expan abbr="datoq;">datoque</expan> intra cylindrum <lb/>solido AED circa eundem axem <lb/>EF revoluto, sive hemisphaerium, <lb/>sive conus, vel conoides sit, oportet <lb/>ipsi solido AED duas figuras ex <lb/>cylindris aequealtis compositas, al<lb/>teram quidem inscribere, alteram verò circumscribere ita <lb/>ut circumscripta superet inscriptam minori excessu quam <lb/>sit quodlibet datum solidum K. </s>

<s>Secetur bifariam cylindrus AC plano HG ad axem EF <lb/>erecto; <expan abbr="iterumq;">iterumque</expan> cylindrus HD bifariam secetur plano IL; <lb/>et hoc fiat semper donec cylindrus aliquis puta AL minor <lb/>remaneat quàm solidum K. Tunc diviso toto cylindro AC <lb/>in cylindros aequealtos ac ipse AL, oriantur in solido AED <lb/>sectiones MN, OP, QR. Concipiamus super <expan abbr="unoquoq;">unoquoque</expan> cir<lb/>culorum MN, OP, QR, duos cylindros, alterum quidem <lb/>versus E, alterum autem versus partes F conversum. <lb/><expan abbr="Eruntq;">Eruntque</expan> omnes simul cylindri qui verticem habent versus <lb/>F, aequales omnibus simul cylindris verticem versus E <lb/>habentibus (cum singuli singulis aequales sint). Ergo si <lb/>omnibus cylindris qui verticem habent versus E, addas <lb/>cylindrum AL, superabit iam figura circa solidum AED <lb/>descripta, figuram eidem inscriptam, differentia AL; Nempe <lb/>minori excessu quàm sit solidum K. Quod erat etc. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Hinc patet quòd data figura solida, sive hemispherium sit, sive <lb/>conus, sive conoides etc. ipsi duae figurae solidae ex cylindris aeque<lb/>altis compositae altera inscribi potest, altera vero circumscribi; ità ut

<pb xlink:href="090/01/081.jpg" pagenum="42"/>differentia inter datam solidam figuram, et descriptarum alterutram, <lb/>minor sit quolibet dato solido K. </s>

<pb pagenum="42"/>differentia inter datam solidam figuram, et descriptarum alterutram, <lb/>minor sit quolibet dato solido K. </s>

<s>Differentia enim inter figuram datam et alteram descriptarum minor <lb/><expan abbr="utiq;">utique</expan> erit quam differentia inter descriptas (est enim pars eiusdem) ergo <lb/>multò minor quàm solidum K.<gap desc="/SM"/></s>

Line 1161

<s>Esto circulus cuius centrum A; quadratum ipsi cir<lb/>cumscriptum sit BCDE; iunctisque EA, AD. convertatur <lb/>

<arrow.to.target n="fig46"></arrow.to.target><lb/>figura circa axem FG ita ut à quadrato fiat cylindrus, à <lb/>sphaera circulus; à triangulo EAD, conus EAD. </s>

<s>Dico sphaeram quadruplam esse coni EAD. Nisi enim <lb/>quadrupla sit, non erit haemisphaerium aequale solido, <lb/>quod describitur à triangulo EHA circa axem FG converso <lb/>(cum hoc solidum duplum sit coni EAD). Erit <expan abbr="itaq;">itaque</expan> he<lb/>misphaerium vel maius, vel minus solido trianguli EHA. <lb/>

<arrow.to.target n="marg60"></arrow.to.target></s>

Line 1169

<s><margin.target id="marg60"></margin.target>per 29. <lb/>huius.</s>

<s>Esto primùm maius, si potest esse; sitque excessus <lb/>aequalis solido K. Inscribatur in hemisphaerio figura ex <lb/>cylindris aequealtis constans ita ut ab hemisphaerio de<lb/>ficiat minori defectu quam sit solidum K. Et erit flgura <lb/>inscripta adhuc maior quàm solidum trianguli EHA. Se<lb/>cetur etiam axis AG in tot partes aequales in quot sectus <lb/>erit AF. <expan abbr="Ductisq;">Ductisque</expan> per puncta sectionum planis ad axem <lb/>erectis, intelligatur in solido trianguli EHA inscripta figura <lb/>ex tubis cylindricis aequealtis constans, quorum unus sit, <lb/>cuius sectio est rectangulum HO. </s>

<s>Iam cylindrus IL ad tubum cylindricum HO, est ut <lb/>quadratum IP ad rectangulum MON. Sed quadratum IP </s>

Line 1179

<s><margin.target id="marg61"></margin.target>per 28. <lb/>huius</s>

<s>Esto deinde, si fieri potest, hemisphaerium minus solido <lb/>trianguli EHA; <expan abbr="sitq;">sitque</expan> defectus aequalis solido K. </s>

<s>Circumscribatur ipsi hemisphaerio figura solida ex cy<lb/>

<arrow.to.target n="marg62"></arrow.to.target><lb/>lindris aequealtis constans, ita ut excessus figurae super <lb/>hemisphaerium minus sit solido K. Tunc enim circum<lb/>scripta figura adhuc minor erit solido trianguli EHA. Con<lb/>cipiamus deinde solido trianguli EHA aliquam figuram <lb/>esse circumscriptam constantem ex tubis cylindricis aeque<lb/>altis ac cylindri ex quibus constat figura haemisphaerio <lb/>circumscripta. </s>

Line 1190

<s>Secundus cylindrus GI ad secundum tubum LM, est <lb/>

<arrow.to.target n="marg63"></arrow.to.target><lb/>ut quadratum GN ad rectangulum LTF, nempe aequalis

<pb xlink:href="090/01/083.jpg" pagenum="44"/>(quadratum enim GN, aequale est rectangulo ONP, sive <lb/>LTF, nam recta ON rectae BQ, sive LE, sive LT, aequalis <lb/>est, et reliqua NP reliquae TF). </s>

<pb pagenum="44"/>(quadratum enim GN, aequale est rectangulo ONP, sive <lb/>LTF, nam recta ON rectae BQ, sive LE, sive LT, aequalis <lb/>est, et reliqua NP reliquae TF). </s>

<s><margin.target id="marg63"></margin.target>per 28. <lb/>huius.</s>

Line 1203

<s>Hinc patet sphaeram subsesquialteram esse cylindri, cuius basis <lb/>aequalis sit maximo sphaera circulo, altitudo verò diametro sphaerae <lb/>aequalis. </s>

<s>Nam sph. ostenditur esse ad conum EAD ut 4, ad unum, conus <lb/>vero EAD ad cylindrum EBCD est ut unum ad 6 ergo ex aequo sphaera <lb/>ad cylindrum EBCD erit ut 4 ad 6. Nempe subsesquialtera.<gap desc="/SM"/></s>

<pb/>

<s><emph type="center"/>DE SOLIDIS SPHAERALIBUS<emph.end type="center"/></s>

Line 1215

<s>Esto circa sphaeram, cuius cen<lb/>

<arrow.to.target n="fig47"></arrow.to.target><lb/>trum A, descriptus conus BCD, <lb/>(qui videlicet sphaeram tangat et <lb/>lateribus, et basi) <expan abbr="Ponaturq;">Ponaturque</expan> alius <lb/>conus EFG, qui basim habeat EG <lb/>aequalem tum curvae superficiei, <lb/>tum etiam basi coni BCD, altitudinem verò HF habeat <lb/>aequalem radio sphaerae AL. </s>

<s>Dico conos BCD, EFG aequales esse. </s>

<s>Solidum enim conicum excavatum quod fit ex revolu<lb/>tione trianguli CBA circa axem IC, aequale est cono <lb/>

<arrow.to.target n="marg64"></arrow.to.target><lb/>cuidam, qui basim habeat aequalem curvae superficiei <lb/>conicae BCD, altitudinem verò aequalem perpendiculari <lb/>AL, nempe radio sphaerae: Talis ergò conus unà cum <lb/>cono BAD (cum habeant eandem altitudinem) aequales <lb/>erunt cono EFG; Quandoquidem conus EFG basim habet

<pb xlink:href="090/01/085.jpg" pagenum="46"/><expan abbr="utriq;">utrique</expan> simul basi aequalem, altitudinem verò alterutrae <lb/>aequalem. Proptereà et conus BCD, qui duobus praedictis <lb/>conis aequatur, aequalis erit cono EFG. Quod etc. </s>

<pb pagenum="46"/><expan abbr="utriq;">utrique</expan> simul basi aequalem, altitudinem verò alterutrae <lb/>aequalem. Proptereà et conus BCD, qui duobus praedictis <lb/>conis aequatur, aequalis erit cono EFG. Quod etc. </s>

<s><margin.target id="marg64"></margin.target>26. p. partis</s>

Line 1237

<s>Superficies ergò coni BCD sine basi, ad circulum suae basis est <lb/>ut CB ad BI, nempe ut CA ad AI, et componendo, et <lb/>

<arrow.to.target n="fig48"></arrow.to.target><lb/>per conversionem rationis, erit universa superficies coni <lb/>BCD cum basi, ad superficiem eiusdem coni sine basi, <lb/>ut IC ad CA, hoc est ut IM ad AL. </s>

<s>Propterea si reciprocè adhibeantur bases, et altitu<lb/>dines, erit conus cuius altitudo AL, basis verò aequalis <lb/>universae superficiei coni BCD cum basi, aequalis cono <lb/>cuius altitudo sit IM, basis verò curva tantum superficies <lb/>conica BCD, hoc est cono BCD (aequales enim sunt, conus cuius altitudo <lb/>IM, basis verò conica superficies BCD; et conus BCD per 22 huius).<gap desc="/SM"/></s>

Line 1247

<s>Esto circa sphaeram ABC de<lb/>

<arrow.to.target n="fig49"></arrow.to.target><lb/>scriptus conus DEF; Dico huius<lb/>modi conum esse ad sphaeram, <lb/>ut coni superficies una cum basi, <lb/>ad superficiem sphaerae. </s>

<s>Ponatur conus HIL ut in prae<lb/>cedenti, cuius basis aequalis sit integro perimetro coni <lb/>DEF una cum basi, altitudo verò PI aequalis radio <lb/>sphaerae OC, <expan abbr="eritq;">eritque</expan> conus HIL aequalis cono DEF. </s>

Line 1262

<s>

<arrow.to.target n="marg68"></arrow.to.target><lb/>est ut basis HL ad basim MN, conus denique MCN ad <lb/>sphaeram, est ut basis MN ad superficiem sphaerae (nempe

<pb xlink:href="090/01/086.jpg" pagenum="47"/>in ratione sub quadrupla) quare ex aequo erit conus DEF <lb/>ad sphaeram, ut universus perimeter coni DEF ad super<lb/>ficiem sphaerae. Quod etc. </s>

<pb pagenum="47"/>in ratione sub quadrupla) quare ex aequo erit conus DEF <lb/>ad sphaeram, ut universus perimeter coni DEF ad super<lb/>ficiem sphaerae. Quod etc. </s>

<s><margin.target id="marg68"></margin.target>20. et 30. p. <lb/>partis.</s>

Line 1272

<s>Esto circa sphaeram, cuius diameter <lb/>

<arrow.to.target n="fig50"></arrow.to.target><lb/>DE, descriptus conus quilibet ABC. Dico <lb/>conum ad sphaeram esse ut rectangulum <lb/>sub BAD tamquàm unà linea, et sub AD <lb/>compraehensum, ad quadratum DE. </s>

<s>Curva enim superficies coni ABC ad <lb/>

<arrow.to.target n="marg69"></arrow.to.target><lb/>circulum suae basis est ut BA ad AD, <lb/>et componendo erit totus coni perimeter ad eundem cir<lb/>culum basis ut BA, AD simul ad AD; hoc est ut rectan<lb/>gulum sub linea BAD, et sub AD ad quadratum AD; <lb/>circulus verò basis coni, ad circulum circa DE, est ut qua<lb/>dratum AD ad quadratum DF, circulus denique circa DE <lb/>ad sphaerae superficiem, est ut quadratum DF ad quadra<lb/>tum DE, ergò ex aequo universus coni ABCA perimeter <lb/>ad superficiem sphaerae (hoc est conus ipse ad sphaeram <lb/>per praecedentem) erit ut rectangulum sub recta linea <lb/>BAD, et sub AD, ad quadratum DE. Quod etc. </s>

Line 1288

<s>Possent hic Theoremata non pauca proponi circa solidorum circum<lb/>scriptionem, et inscriptionem: qualia sunt. </s>

<s>Si circa sphaeram prisma concipiatur, quod singulis suis parallelo<lb/>grammis sphaeram contingat; sitque eiusdem altitudinis; Erit prisma

<pb xlink:href="090/01/087.jpg" pagenum="48"/>ad sphaeram, ut perimeter basis prismatis ad duas tertias periphaeriae <lb/>maximi circuli sphaerae. </s>

<pb pagenum="48"/>ad sphaeram, ut perimeter basis prismatis ad duas tertias periphaeriae <lb/>maximi circuli sphaerae. </s>

<s>Si verò non eiusdem sit altitudinis; ratio prismatis ad sphaeram <lb/>componetur ex praedicta, et ex ratione altitudinum; altitudo autem <lb/>sphaerae diameter est. </s>

Line 1305

<s>Si circà circulum describatur poligonum habens latera <lb/>numero paria, sive à quaternario, sive à binario mensurata, <lb/>et revolvatur figura circa diagonalem, erit factum sphae<lb/>rale solidum aequale cono cuidam qui basim habeat ae<lb/>qualem superficiei solidi, altitudinem verò semidiametro <lb/>sphaerae aequalem. <lb/><gap desc="SM"/></s>

<s>Hoc autem quandò numerus laterum mensuratur à quaternario <lb/>demonstratum fuit ab Archimede Prop. 29 sive mavis 25 lib. p. de <lb/>Sph. et Cylin. Quando verò laterum numerus etiam à binario tantnm <lb/>mensuratur, ostendemus sic, eritque demonstratio (exceptis quae de <lb/>ultimo solido cylindrico dicentur) eadem cum ea quam affert Archi<lb/>medes.<gap desc="/SM"/></s>

<s>Esto poligonum ABCDEFG habens latera à binario <lb/>tantum mensurata, ut in prima figura. Ergò semipoligo<lb/>

<arrow.to.target n="fig51"></arrow.to.target><lb/>num ABCDEF habebit latera numero imparia, latusque <lb/>unum tanget circulum in puncto T, <expan abbr="atq;">atque</expan> ideo cylindricam <lb/>superficiem in conversione describet. Intelligatur conus <lb/>MNO, cuius basis sit circulus MO aequalis universae su<lb/>perficiei solidi sphaeralis, altitudo verò PN, aequalis sit <lb/>radio sphaerae. Dico sphaerale solidum aequale esse cono <lb/>MNO. </s>

<s>Rombus enim solidus factus in conversione figurae à <lb/>triangulo ABQ, aequalis est cono cuidam cuius basis ae<lb/>

<arrow.to.target n="marg70"></arrow.to.target><lb/>qualis sit conicae superficiei descriptae à linea AB, alti<lb/>tudo verò sit radius QR. Solidum autem excavatum factum <lb/>in conversione à triangulo BCQ, aequatur cono cuidam <lb/>

<arrow.to.target n="marg71"></arrow.to.target><lb/>cuius basis aequalis sit conicae superficiei descriptae à <lb/>linea BC altitudo verò aequalis radio sphaerae QS et sic <lb/>semper procedatur. Ultimum denique solidum cylindricum <lb/>

<arrow.to.target n="marg72"></arrow.to.target><lb/>excavatum descriptum à triangulo CTQ, aequale est cono <lb/>cuidam, cuius basis aequalis sit superflciei cylindricae à <lb/>linea CT factae, altitudo verò aequalis sit semidiametro <lb/>cylindri, QT; Et sic de solidis circa alterum hemisphae<lb/>rium TFV descriptis. Ergo universum sphaerale solidum, <lb/>aequale erit omnibus praedictis conis simul sumptic: ijsdem <lb/>autem omnibus praedictis conis aequalis est conus MNO <lb/>(cum basim habeat omnibus simul illorum basibus aequa<lb/>lem, nempe superficiei solidi sphaeralis, altitudinem verò <lb/>unicuique illorum aequalem, nempe radio sphaerae). Pro<lb/>pterea praedictum solidum sphaerale aequale erit cono <lb/>MNO. Quod etc. </s>

<s><margin.target id="marg70"></margin.target>23. p. partis.</s>

Line 1329

<s>Manente praecedentis Propositionis constructione; Esto <lb/>sphaerale solidum cuius diagonalis, atque axis sit AB, cen<lb/>trum autem sphaerae sit C. Dico sphaerale solidum ad <lb/>

<arrow.to.target n="fig52"></arrow.to.target><lb/>inscriptam sibi sphaeram esse, ut superficies solidi ad su<lb/>perficiem sphaerae. </s>

<s>Inscribatur n. in hemisphaerio conus DEF, et ponatur <lb/>conus GIH cuius basis GH aequalis sit universae super<lb/>ficiei solidi sphaeralis ut in praecedenti altitudo verò LI <lb/>aequalis radio sphaerae, et erit per praecedentem sphaerale <lb/>solidum aequale cono GIH. </s>

<s>Propter aequalitatem ergò, erit sphaerale solidum ad <lb/>conum GIH ut superficies universa sphaeralis solidi ad ba<lb/>sim coni GIH; conus autem GIH ad conum DEF (ob <lb/>aequalem altitudinem) est ut basis circa GH ad basim <lb/>circa DF; conus denique DEF ad sphaeram, est ut ba<lb/>sis circa DF ad superficiem sphaerae (nempe in ratione <lb/>subquadrupla). Propterea erit ex aequo sphaerale solidum <lb/>ad inscriptam sibi sphaeram ut universa sphaeralis solidi <lb/>superficies ad superficiem sphaerae. Quod etc. </s>

<s><emph type="center"/>PROPOSITIO VI.<emph.end type="center"/></s>

<s>Si circa circulum describatur poligonum habens latera <lb/>numero paria, et convertatur figura circa diagonalem, erit <lb/>factum sphaerale solidum ad inscriptam sibi sphaeram ut <lb/>axis solidi ad axem sphaerae. </s>

<s>Manente praecedentium constructione; esto sphaerale <lb/>solidum, cuius diagonalis, atque axis sit AB centrum verò <lb/>sphaerae sit C, et diameter HI. </s>

<s>Dico sphaerale solidum ad inscriptam sibi sphaeram <lb/>esse ut AB ad HI. </s>

Line 1363

<s>Si intra circulum describatur poligonum habens latera <lb/>numero paria, et convertatur figura circa diagonalem, erit <lb/>sphaera ad inscriptum sibi sphaerale solidum, ut quadra<lb/>tum diametri sphaerae, ad quadratum cateti poligoni. </s>

<s>Sit n. circ. cuius cent. A, et diamet. BC poligonum <lb/>regulare, cuius diagonalis sit linea BC, et convertatur

<pb xlink:href="090/01/091.jpg" pagenum="52"/>figura circa BC. Dico sphaeram circumscriptam ad in<lb/>clusum sphaerale solidum, esse ut qua<lb/>

<pb pagenum="52"/>figura circa BC. Dico sphaeram circumscriptam ad in<lb/>clusum sphaerale solidum, esse ut qua<lb/>

<arrow.to.target n="fig53"></arrow.to.target><lb/>dratum AC, ad quadratum cateti po<lb/>ligoni AD. Ducatur enim DE ex D <lb/>perpendicularis ad BC, et EF perpen<lb/>

<arrow.to.target n="marg76"></arrow.to.target><lb/>dicularis ad AD, <expan abbr="eruntq;">eruntque</expan> in continua <lb/>proportione quatuor rectae AC, AD, <lb/>AE, AF. Concipiatur etiam radio AD <lb/>aliam sphaeram describi, quae singulas <lb/>conicas superficies solidi sphaeralis <lb/>continget; necnon cylindricam, si quam huiusmodi sphae<lb/>rale solidum habebit. <lb/>

<arrow.to.target n="marg77"></arrow.to.target></s>

Line 1371

<s><margin.target id="marg76"></margin.target>4. sexti.</s>

<s><margin.target id="marg77"></margin.target>ultima duo<lb/>decimi.</s>

<s>Erit itaque sphaera maior ad sphaeram minorem, ut <lb/>CA ad AF; minor verò sphaera ad sphaerale solidum, <lb/>quod sibi circumscribitur (per praecedentem) est ut DA <lb/>ad AC, hoc est, ut AF ad AE; Proptereà ex aequo erit <lb/>circumscripta sphaera maior, ad inscriptum solidum sphae<lb/>rale, ut CA ad AE; nempe ut quadratum CA ad quadra<lb/>tum AD. Quod erat etc. </s>

Line 1389

<s><emph type="center"/>PROPOSITIO IX.<emph.end type="center"/></s>

<s>Si circa sphaerale solidum, natum ex revolutione <lb/>poligoni circà diagonalem revoluti, sphaera circumscri<lb/>batur, et altera inscribatur: Erit superficies solidi sphae-

<pb xlink:href="090/01/092.jpg" pagenum="53"/>ralis media proportionalis inter superficies duarum sphae<lb/>rarum. </s>

<pb pagenum="53"/>ralis media proportionalis inter superficies duarum sphae<lb/>rarum. </s>

<s>Manente figura, et constructione <lb/>

<arrow.to.target n="fig54"></arrow.to.target><lb/>praecedentium propositionum. Dico tres <lb/>superficies, nempe maioris sphaerae, so<lb/>lidi sphaeralis, <expan abbr="minorisq;">minorisque</expan> inscriptae <lb/>sphaerae, esse inter se in continua pro<lb/>portione. </s>

<s>Superficies enim circumscriptae <lb/>sphaerae est ad superficiem inscriptae, <lb/>ut quadratum CA ad quadratum AD; <lb/>superficies autem solidi ad superficiem eiusdem inscriptae <lb/>sphaerae, est ut recta CA ad rectam AD: Ergò tres su<lb/>

<arrow.to.target n="marg79"></arrow.to.target><lb/>perficies praedictae sunt in continua proportione; et qui<lb/>dem perimeter sphaeralis solidi medius proportionalis est. <lb/>inter superficies duarum sphaerarum. Quod etc. </s>

Line 1406

<s>Esto circa circulum figura poligona aequilatera ABC<lb/>DEH habens latera numero paria, et convertatur figura <lb/>

<arrow.to.target n="fig55"></arrow.to.target><lb/>circa catetum IL, <expan abbr="orieturq;">orieturque</expan> solidum contentum sub co<lb/>nicis, circularibus, et una cylindrica superficie, quando

<pb xlink:href="090/01/093.jpg" pagenum="54"/>numerus laterum à quaternario mensuratur; quandò verò <lb/>à binario tantum, tunc erit solidum sphaerale sub co<lb/>nicis, et circularibus tantum superficiebus compraehensum. <lb/>Dico <expan abbr="utrumq;">utrumque</expan> sphaerale solidum aequale esse cono cuidam <lb/>MNO, qui basim habeat aequalem universae solidi sphae<lb/>ralis superficiei, altitudinem verò PN aequalem radio <lb/>sphaerae. </s>

<pb pagenum="54"/>numerus laterum à quaternario mensuratur; quandò verò <lb/>à binario tantum, tunc erit solidum sphaerale sub co<lb/>nicis, et circularibus tantum superficiebus compraehensum. <lb/>Dico <expan abbr="utrumq;">utrumque</expan> sphaerale solidum aequale esse cono cuidam <lb/>MNO, qui basim habeat aequalem universae solidi sphae<lb/>ralis superficiei, altitudinem verò PN aequalem radio <lb/>sphaerae. </s>

<s>Hoc ostendetur similiter ut propositione 4 factum est. <lb/>Nam conus qui fit à triangulo IAQ in conversione circa <lb/>axem IL, aequatur cono qui basim habeat aequalem cir<lb/>culo qui fit ex radio IA, altitudinem verò aequalem radio <lb/>sphaerae QI, quia idem prorsus est. Solidum autem exca<lb/>

<arrow.to.target n="marg80"></arrow.to.target><lb/>vatum, quod fit à triangulo ABQ, aequale probatur cono <lb/>cuidam, cuius basis aequalis sit conicae superficiei factae <lb/>à linea AB, altitudo verò sit QR radius sphaerae. Ultimum <lb/>denique cylindricum solidum excavatum, factum à trian<lb/>gulo BQS (quando poligoni latera à quaternario mensu<lb/>

Line 1422

<s>Si circa circulum describatur poligonum habens latera <lb/>numero pario, et convertatur figura circa catetum, habebit <lb/>factum sphaerale solidum ad inscriptam sibi sphaeram <lb/>eam rationem, quam universa solidi sphaeralis superficies <lb/>habet ad superficiem sphaerae. </s>

<s>Manente praecedentis propositionis constructione, esto <lb/>sphaerale solidum cuius catetus, et axis sit AB; centrum <lb/>autem sphaerae sit C. Dico sphaerale solidum ad inscri-

<pb xlink:href="090/01/094.jpg" pagenum="55"/>ptam sibi sphaeram esse ut universa ipsius solidi super<lb/>ficies ad superficiem sphaerae. </s>

<pb pagenum="55"/>ptam sibi sphaeram esse ut universa ipsius solidi super<lb/>ficies ad superficiem sphaerae. </s>

<s>Concipiatur enim in hemisphaerio conus DAE, et in<lb/>telligatur alius conus FGH, cuius basis FH aequalis sit <lb/>

<arrow.to.target n="fig56"></arrow.to.target><lb/>universae superficiei solidi sphaeralis, altitudo verò IG <lb/>aequalis radio sphaerae; et erit per praecedentem sphae<lb/>rale solidum aequale cono FGH. </s>

<s>Propter aequalitatem ergo, erit sphaerale solidum ad <lb/>conum FGH, ut superficies universa sphaeralis solidi, <lb/>ad basim coni FGH, conus autem FGH, ad conum DAE <lb/>(ob aequalem altitudinem) est ut basis circa FH, ad basim <lb/>circa DE; denique conus DAE, ad sphaeram, est ut <lb/>basis circa DE ad superficiem sphaerae (nempe in ratione <lb/>subquadrupla): Propterea erit ex aequo sphaerale solidum <lb/>ad inscriptam sibi sphaeram, ut universa sphaeralis solidi <lb/>superficies ad superficiem sphaerae. Quod etc. </s>

Line 1435

<s>Si circa circulum describatur poligonum habens latera <lb/>numero paria, et convertatur figura circa catetum; Ha<lb/>bebit factum sphaerale solidum ad inscriptam sibi sphae<lb/>ram, eam rationem, quam habet composita recta linea ex <lb/>diametro sphaerae, et ex tertia proportionali (si fiat ut <lb/>semidiameter sphaerae ad semilatus poligoni, ita semilatus <lb/>ad aliam), ad diametrum sphaerae. </s>

<s>Manente praecedentium propositionum constructione, <lb/>esto sphaerale solidum cuius catetus, et axis sit AB; cen<lb/>trum autem sphaerae sit C. Fiat angulus CDE rectus, <expan abbr="eritq;">eritque</expan>

<pb xlink:href="090/01/095.jpg" pagenum="56"/>BE tertia proportionalis ad semidiametrum CB, et semi<lb/>latus poligoni BD. Dico sphaerale solidum ad inscriptam <lb/>

<pb pagenum="56"/>BE tertia proportionalis ad semidiametrum CB, et semi<lb/>latus poligoni BD. Dico sphaerale solidum ad inscriptam <lb/>

<arrow.to.target n="fig57"></arrow.to.target><lb/>sibi sphaeram esse ut EA ad AB; nempe ut composita <lb/>ex diametro sphaerae AB, et tertia proportionali BE, ad <lb/>diametrum sphaerae AB. Concipiatur circa sphaeram de<lb/>scriptus cylindrus FLMI, et per puncta A; B; E produ<lb/>cantur plana FI, LM, GH, ad axem erecta. </s>

<s>Erit ergo, per praecedentem, sphaerale solidum ad in<lb/>scriptam sibi sphaeram, ut superficies solidi ad superficiem <lb/>

<arrow.to.target n="marg82"></arrow.to.target><lb/>sphaerae; hoc est, sumptis aequalibus, ut superficies cy<lb/>lindri FGHI ad superficiem cylindri FLMI; hoc est ut <lb/>

Line 1453

<s>Esto circa circulum, cuius centrum A, descriptum po<lb/>

<arrow.to.target n="fig58"></arrow.to.target><lb/>ligonum habens latera numero paria, et convertatur figura <lb/>circa catetum BC: <expan abbr="factoq;">factoque</expan> angulo recto ADE, erit (per

<pb xlink:href="090/01/096.jpg" pagenum="57"/>praecedentem) solidum sphaerale ad suam sphaeram ut <lb/>EB ad BC. Dico insuper solidum sphaerale ad suam sphae<lb/>ram esse, ut quadratum ex AD, simul cum quadrato ex <lb/>AC, ad duplum quadrati ex AC. </s>

<pb pagenum="57"/>praecedentem) solidum sphaerale ad suam sphaeram ut <lb/>EB ad BC. Dico insuper solidum sphaerale ad suam sphae<lb/>ram esse, ut quadratum ex AD, simul cum quadrato ex <lb/>AC, ad duplum quadrati ex AC. </s>

<s>Nam EA ad AC est ut quadratum DA ad quadratum <lb/>AC; et componendo, erunt EA, et AC simul, hoc est <lb/>tota EB, ad AC, ut duo quadrata DA, AC simul ad qua<lb/>dratum AC; sumptisque consequentium duplis, erit EB <lb/>

<arrow.to.target n="marg84"></arrow.to.target><lb/>ad BC (hoc est solidum sphaerale ad sphaeram) ut duo <lb/>quadrata DA, AC simul, ad duplum quadrati ex AC. <lb/>Quod etc. </s>

Line 1467

<s>Sit in circulo cuius diameter AB <lb/>

<arrow.to.target n="fig59"></arrow.to.target><lb/>poligonum habens latera numero pa<lb/>ria, et convertatur figura circa cate<lb/>tum CD: Ducanturque perpendicu<lb/>lares DF ad rectam HE, et FI ad <lb/>HD; et erunt quatuor lineae EH, HD, <lb/>HF, HI, in continua ratione semidia<lb/>metri HE ad semicatetum HD. Dico <lb/>sphaeram ad inscriptum solidum esse, <lb/>ut dupla HE ad <expan abbr="utramq;">utramque</expan> simul DH, HI. Vel ut integra <lb/>diameter sphaerae ad CI. </s>

<s>Intelligatur alia sphaera intra solidum inscripta. Erit <lb/>ergo exterior sphaera ad interiorem, ut EH ad HI, sive <lb/>

<arrow.to.target n="marg85"></arrow.to.target><lb/>ut dupla EH ad duplam HI; interior verò sphaera ad <lb/>solidum sphaerale est, ut duo quadrata ex HD, ad duo <lb/>quadrata HD, HE, hoc est ut duo quadrata ex HI, ad <lb/>

<arrow.to.target n="marg86"></arrow.to.target><lb/>duo quadrata ex HI, HF, hoc est (ut infrà ostendemus) <lb/>ut dupla HI ad HI, HD; Propterea erit ex aequo sphaera <lb/>exterior ad inscriptum sibi sphaerale solidum, ut dupla <lb/>HE, hoc est integra diameter sphaerae, ad HI, et HD

<pb xlink:href="090/01/097.jpg" pagenum="58"/>simul; quae quidem sunt secunda, et quarta in ratione <lb/>semidiam. sphaerae ad semicatetum poligoni. Quod etc. </s>

<pb pagenum="58"/>simul; quae quidem sunt secunda, et quarta in ratione <lb/>semidiam. sphaerae ad semicatetum poligoni. Quod etc. </s>

<s><margin.target id="marg85"></margin.target>Ultima duo<lb/>decimi.</s>

Line 1488

<s>Esto circuli centrum A, polig. <lb/>

<arrow.to.target n="fig60"></arrow.to.target><lb/>verò perimeter BCDEFGH. Et sint <lb/>latera eius numero imparia; conver<lb/>taturque figura circa catetum BI, ut <lb/>oriatur solidum sphaerale contentum <lb/>sub conicis superficiebus unicoque <lb/>circulo circa diametrum EF descri<lb/>pto. Ponatur iam conus LMN, qui <lb/>basim habeat aequalem universae su<lb/>perficiei sphaeralis solidi, altitudinem <lb/>verò OM aequalem radio sphaerae <lb/>AI. Dico solidum sphaerale aequale esse cono LMN. </s>

<s>Agatur per centrum sphaerae planum PQ ad axem <lb/>erectum, quod transuersè, secabit aliquod latus poligoni, <lb/>puta CD. </s>

<s>Erit iam rombus solidus, factus à conversione triang. <lb/>

<arrow.to.target n="marg87"></arrow.to.target><lb/>BCA, aequalis cono, qui basim habeat aequalem conicae <lb/>superficiei factae à linea BC; altitudinem autem aequalem <lb/>radio sphaerae AR. Solidum verò conicum excavatum

<pb xlink:href="090/01/098.jpg" pagenum="59"/>quod fit ex giro trianguli CPA, aequale erit cono, qui <lb/>

<pb pagenum="59"/>quod fit ex giro trianguli CPA, aequale erit cono, qui <lb/>

<arrow.to.target n="marg88"></arrow.to.target><lb/>basim habeat aequalem superficiei, quae fit à linea CP <lb/>altitudinem verò aequalem radio sphaerae AS. Solidum <lb/>quoque excavatum, factum ex revolutione trianguli PDA, <lb/>

<arrow.to.target n="marg89"></arrow.to.target><lb/>aequatur cono, qui basim habeat aequalem superficiei co<lb/>nicae quae fit à motu lineae PD, altitudinem autem ae<lb/>qualem radio shpaerae AS. Eadem prorsum eodem modo <lb/>dicuntur de solido conico excavato, facto à triangulo DAE; <lb/>et de ultimo cono facto à revolutione trianguli EIA. Pro<lb/>pterea totum sphaerale solidum aequale erit omnibus prae<lb/>dictis conis simul sumptis, vel cono LMN, qui omnibus <lb/>illis praedictis aequivalet: (habet enim basim omnibus si<lb/>mul illorum basibus aequalem, altitudinem verò aequalem <lb/><expan abbr="unicuiq;">unicuique</expan> illorum). Quod etc. </s>

Line 1512

<s>Si circa circulum describatur poligonum habens latera <lb/>numero imparia, et convertatur <lb/>

<arrow.to.target n="fig61"></arrow.to.target><lb/>figura circa catetum; habebit fa<lb/>ctum sphaerale solidum ad inscri<lb/>ptam sibi sphaeram, eam rationem <lb/>quam universa sphaeralis solidi su<lb/>perficies habet, ad superficiem <lb/>sphaerae. </s>

<s>Manente praecedentis proposi<lb/>tionis constructione, sit sphaerale <lb/>solidum cuius catetus, sive axis sit <lb/>AB, centrum verò sphaerae sit C. Dico sphaerale solidum

<pb xlink:href="090/01/099.jpg" pagenum="60"/>ad inscriptam sibi sphaeram esse, ut ipsius solidi integra <lb/>superficies ad superficiem sphaerae. </s>

<pb pagenum="60"/>ad inscriptam sibi sphaeram esse, ut ipsius solidi integra <lb/>superficies ad superficiem sphaerae. </s>

<s>Concipiatur in hemisphaerio conus DEF; et intelligatur <lb/>conus GHI cuius basis GI aequalis sit universae superfi<lb/>ciei solidi sphaeralis, altitudo verò LH aequalis sit radio <lb/>sphaerae, et erit per praecedentem, sphaerale solidum <lb/>aequale cono GHI. </s>

Line 1527

<s>Manente praecedentium constructione, <lb/>

<arrow.to.target n="fig62"></arrow.to.target><lb/>sit sphaerale solidum cuius catetus, at<lb/>que axis sit AB, centrum verò sphaerae <lb/>C, et diameter DB. Fiat angulus rectus <lb/>DEF, <expan abbr="eritq;">eritque</expan> BF tertia proportionalium, <lb/>posita diametro DB pro prima, et semi<lb/>latere poligoni BE pro secunda. Dico <lb/>sphaerale solidum ad inscriptam sibi <lb/>sphaeram esse ut tota AF ad DB. </s>

<s>Concipiatur circa sphaeram cylindrus MNOP, et per <lb/>puncta A, D, B, F, plana agantur ad axem erecta. </s>

<s>Erit ergo, per praecedentem, sphaerale solidum ad in<lb/>scriptam sibi sphaeram, ut superficies sphaeralis solidi

<pb xlink:href="090/01/100.jpg" pagenum="61"/>ad superficiem sphaerae; hoc est, sumptis aequalibus, <lb/>

<pb pagenum="61"/>ad superficiem sphaerae; hoc est, sumptis aequalibus, <lb/>

<arrow.to.target n="marg90"></arrow.to.target><lb/>ut superficies cylindri GHIL ad superficiem cylindri <lb/>MNOP; hoc est ut recta AF ad BD per primam p. par<lb/>tis. Quod etc. </s>

<s><margin.target id="marg90"></margin.target>per 17. et 18. <lb/>p. partis.</s>

Line 1543

<s>Esto circulus cuius diameter AB, cen<lb/>

<arrow.to.target n="fig63"></arrow.to.target><lb/>trum verò G, <expan abbr="ipsiq;">ipsique</expan> circumscribatur poli<lb/>gonum habens latera numero imparia, <lb/>cuius catetus sit GB, et convertatur fi<lb/>gura circa CB; Factoque angulo GDF <lb/>recto, erit ratio rectae GB ad GD con<lb/>tinuata in tribus terminis GB, GD, GF; <lb/>uti propositum est. Dico solidum ad <lb/>sphaeram esse, ut GF, GB, simul cum GD bis sumpta, ad <lb/>ipsam GB quater sumptam. </s>

<s>Fiat alius angulus ADE rectus; <expan abbr="eritq;">eritque</expan> solidum ad <lb/>sphaeram per praecedentem, ut CE ad diametrum sphaerae <lb/>AB, hoc est ut EG, GD simul, ad diametrum sphaerae (sunt <lb/>enim aequales GC, GD) hoc est ut dupla EG, et dupla GD <lb/>

<arrow.to.target n="marg91"></arrow.to.target><lb/>ad duas diametros, hoc est ut FG, GB cum dupla GD, ad <lb/>quatuor semidiametros GB. Quod erat demon. etc. </s>

Line 1553

<s><emph type="italics"/>Quod autem assumptum fuit, ostendemus sic.<emph.end type="italics"/> Dico ipsam <lb/>EG bis sumptam, aequalem esse duabus FG, GB. </s>

<s>Quoniam ob angulum rectum, rectangula ABE, GBF, <lb/>aequalia sunt eidem quadrato BD, aequalia erunt et inter <lb/>se; ideoque latera eorum reciproca, nempe ut AB ad BG <lb/>subduplam, ita erit FB ad BE subduplam; aequales ergo <lb/>sunt FE, EB et tres rectae GF, GE, GB sunt in propor<lb/>tione Aritmetica; ideo EG bis sumpta aequalis erit duabus <lb/>FG, GB. Quod etc. </s>

<s><emph type="center"/>PROPOSITIO XIX.<emph.end type="center"/></s>

Line 1561

<s>Sit circulus cuius diameter AI, cen<lb/>

<arrow.to.target n="fig64"></arrow.to.target><lb/>trum verò C, et inscribatur poligonum <lb/>habens latera numero imparia; tum con<lb/>vertatur figura circa catetum AD. Fiant<lb/>que anguli CEB et CBF recti, eritque <lb/>ratio CD ad CE continuata in quatuor <lb/>terminis CD, CE, CB, CF. Dico sphaeram <lb/>ad inscriptum sibi solidum sphaerale <lb/>esse, ut CF quater sumpta, ad CB semel, CE bis, et CD <lb/>semel, simulque sumptas. </s>

<s>Intelligatur alia sphaera cuius semidiameter CD: in<lb/>

<arrow.to.target n="marg92"></arrow.to.target><lb/>scripta in solido. Erit ergo maior sphaera ad minorem ut <lb/>cubus EC ad cubum CD vel recta FC ad CD, vel ut FC <lb/>quater, ad CD quater; sphaera verò minor inscripta, est <lb/>ad solidum sphaerale, (per praecedentem) ut CD quater <lb/>sumpta, ad CB semel, CE bis, et CD semel: Propterea <lb/>erit ex aequo, maior, sive circumscripta sphaera, ad suum <lb/>sphaerale solidum, ut CF quater sumpta, ad CB semel, CE <lb/>bis, et CD semel sumptas. Quod etc. </s>

Line 1571

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Hactenus sex praecipua Theoremata de solidis sphaeralibus demon<lb/>strata sunt. Sequuntur nunc quaedam scitu non iniucunda, et ad do<lb/>ctrinam spectantia.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XX.<emph.end type="center"/></s>

<s>Si intra sphaeram descriptum sit sphaerale solidum pa<lb/>rilaterum, <expan abbr="circaq;">circaque</expan> diagonalem revolutum: erit sphaera ad <lb/>excessum, quo ipsa solidum superat, in duplicata ratione <lb/>diametri sphaerae ad latus poligoni. </s>

<s>Sit in circulo cuius centrum A de<lb/>

<arrow.to.target n="fig65"></arrow.to.target><l

<arrow.to.target n="fig65"></arrow.to.target><lb/>scriptum poligonum habens latera numero <lb/>paria, et convertatur circa diagonalem BC. <lb/>Dico sphaeram ad excessum, quo ipsa so<lb/>lidum superat, esse ut quadratum BC ap <lb/>quadratum CD. </s>

<s>Ducatur AE ex centro perpendicularis <lb/>ad latus CD, et producatur. </s>

<s>Quoniam per demonstrata, est ut sphaera ad solidum <lb/>

<arrow.to.target n="marg93"></arrow.to.target><lb/>sphaerale ita quadratum FA ad quadratum AE, erit per <lb/>conversionem rationis sphaera ad excessum, ut quadratum <lb/>FA ad differentiam quadratorum FA, AE, hoc est ad re<lb/>ctangulum FEG, sive ad quadratum EC. Constat ergò <lb/>sphaeram ad excessum quo ipsa superat inscriptum sphae<lb/>rale solidum esse ut quadratum FA ad quadratum EC, <lb/>sive ut quadratum BC ad quadratum CD. Quod etc. </s>

<s><margin.target id="marg93"></margin.target>7 huius.</s>

<s><emph type="center"/>PROPOSITIO XXI.<emph.end type="center"/></s>

<s>Si in eadem sphaera duo solida parilatera, et circa dia<lb/>gonalem revoluta, concipiantur, erit differentia unius à <lb/>sphaera, ad differentiam alterius a sphaera, homologè in <lb/>duplicata ratione laterum. </s>

<s>Sint in circulo cuius diameter AB duo <lb/>

<arrow.to.target n="fig66"></arrow.to.target><lb/>semipoligona ACB, ADB; et convertatur <lb/>figura circà diagonalem AB. Dico diffe<lb/>rentiam inter sphaeram, et sphaerale soli<lb/>dum ACB, ad differentiam inter sphaeram <lb/>et sphaerale solidum ADB, esse ut qua<lb/>dratum CB ad quadratum BD. </s>

<s>Demonstratum enim est differentiam ACB, esse ad <lb/>

<arrow.to.target n="marg94"></arrow.to.target><lb/>sphaeram, ut quadratum BC, ad quadratum AB, sed sphaera

<arrow.to.target n="marg95"></arrow.to.target><lb/>ad differentiam ADB, est ut quadratum AB ad quadratum <lb/>BD, ergo ex aequo erit differentia ACB ad differentiam <lb/>ADB ut quadratum BC ad quadratum BD. Quod etc. </s>

<s><margin.target id="marg94"></margin.target>per praeced.</s>

<s><margin.target id="marg95"></margin.target>per eandem.</s>

<s><emph type="center"/>PROPOSITIO XXII.<emph.end type="center"/></s>

<s>Si eidem sphaerae duo solida parilatera, et similia, <lb/>circaque diagonalem revoluta, alterum circumscribatur, <lb/>alterum verò inscribatur; superficies sphaerae media pro<lb/>portionalis erit inter superficies <lb/>

<arrow.to.target n="fig67"></arrow.to.target><lb/>duorum solidorum. </s>

<s>Sit circulus, cuius diameter AB <lb/>atque ipsi duo poligona, alterum <lb/>circumscribatur, alterum verò in<lb/>scribatur, <expan abbr="habeatq;">habeatque</expan> <expan abbr="utrumq;">utrumque</expan> latera <lb/>numero paria, et sit numerus late<lb/>rum unius aequalis numero laterum <lb/>alterius, ut sphaeralia solida similia <lb/>evadant. Tum convertatur figura <lb/>circa diagonalem CD. </s>

<s>Dico superficiem factae sphaerae mediam proportio<lb/>nalem esse inter superficies factorum solidorum. </s>

<s>Ducatur ex centro G recta GL ad contactus M et L, <lb/>et radio GM fiat sphaera IMH. </s>

<s>Iam superficies solidi AF ad superficiem sphaerae IM <lb/>intra ipsum inscriptae est ut solidum AF ad sphaeram <lb/>IM, per 5. huius, nempe ut axis AG ad GM, per 6. huius, <lb/>hoc est ut rectangulum AGM ad quadratum GM; Super<lb/>ficies verò sphaerae IM ad superficiem sphaerae ALF est <lb/>ut quadratum GM ad quadratum GA. Ergò ex aequo su<lb/>perficies solidi AF ad superficiem sphaerae AL erit ut <lb/>rectangulum AGM ad quadratum GA, nempe ut recta MG <lb/>ad GA vel ut recta LG ad GC. Sed superficies sphaerae <lb/>ALF ad superficiem solidi CE est ut LG ad GC (quod <lb/>probatur eodem modo ut factum fuit supra) ergo in con<lb/>tinua proportione sunt superficies universa solidi AMF, <lb/>superficies sphaerae ALF et superficies solidi CE. Quod <lb/>erat etc. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Hinc patet etiam quod si eidem solido sphaerali parilatero circa <lb/>diagonalem revoluto duae sphaerae, altera circumscribatur, altera verò <lb/>inscribatur, tres superficies in continua proportione erunt inter se.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXIII.<emph.end type="center"/></s>

<s>Sphaeralia solida parilatera circa diagonalem revoluta, <lb/>et eidem sphaerae, vel aequalibus sphaeris circumscripta, <lb/>inter se sunt ut axes. </s>

<s>Sint circa circulum cuius centrum A <lb/>

<arrow.to.target n="fig68"></arrow.to.target><lb/>duo poligona dissimilia, quorum latera nu<lb/>mero paria sint, et convertantur circa dia<lb/>gonalem. <expan abbr="Sitq;">Sitque</expan> alterius factorum solidorum <lb/>BFC, axis BC; alterius verò nempe DGE, <lb/>esto axis DE. Dico solidum BFC, ad so<lb/>lidum DGE esse ut BC ad DE. </s>

<s>Hoc autem patet. Quoniam solidum <lb/>BFC ad sphaeram est ut BC ad diametrum HI; sphaera <lb/>

<arrow.to.target n="marg96"></arrow.to.target><lb/>vero ad alterum solidum DGE est ut diameter HI ad <lb/>

<arrow.to.target n="marg97"></arrow.to.target><lb/>axem DE, erit ex aequo, solidum BFC ad solidum DGE, <lb/>ut BC ad DE. Quod erat etc. <lb/><gap desc="SM"/></s>

<s><margin.target id="marg96"></margin.target>5. huius.</s>

<s>Hinc facile ostendi potest excessum, quo solidum BFC superat <lb/>sphaeram, ad excessum quo solidum DGE superat eandem sphaeram, <lb/>esse ut BH, ad HD. </s>

<s>Cum enim solidum BFC ad sphaeram sit ut BA ad AH, erit divi<lb/>

<arrow.to.target n="marg98"></arrow.to.target><lb/>dendo excessus BFC ad sphaeram, ut BH ad HA. Eadem ratione <lb/>sphaera ad excessum DGE erit, ut HA ad HD; ergo ex aequo, excessus <lb/>BFC ad excessum DCE, supra sphaeram erit ut BH, ad HD. Quod etc.<gap desc="/SM"/></s>

<s><margin.target id="marg98"></margin.target>6. huius.</s>

<s><emph type="center"/>PROPOSITIO XXIV.<emph.end type="center"/></s>

<s>Solida sphaeralia parilatera, eidem, vel aequalibus <lb/>sphaeris inscripta, et circa diagonalem revoluta, sunt inter <lb/>se in duplicata ratione catetorum. </s>

<s>Inscribantur in circulo cuius diameter AC duo semi<lb/>poligona ABC, ADC, et convertatur figura circa diago-

<pb pagenum="66"/>nalem AC, ut describantur duo solida sphaeralia ut im<lb/>peratum est. </s>

<s>Dico solidum sphaerale factum ex <lb/>

<arrow.to.target n="fig69"></arrow.to.target><lb/>poligono ABC, ad solidum sphaerale <lb/>factum ex poligono ADC, esse ut qua<lb/>dratum cateti IE, ad quadratum ca<lb/>teti IH. <lb/>

<arrow.to.target n="marg99"></arrow.to.target></s>

<s><margin.target id="marg99"></margin.target>7. huius.</s>

<s>Solidum enim ex ABC ad sphae<lb/>ram, est ut quadratum IE ad quadra<lb/>tum IC; sphaera autem ad solidum ADC, est ut qua</s>

<s>

<arrow.to.target n="marg100"></arrow.to.target><lb/>dratum IC ad quadratum IH; ergo ex aequo solidum <lb/>ABC ad solidum ADC erit, ut quadratum IE ad quadra<lb/>tum IH. Quod erat etc. </s>

<s><margin.target id="marg100"></margin.target>per eandem.</s>

<s><emph type="center"/>PROPOSITIO XXV.<emph.end type="center"/></s>

<s>Si intra aequales, vel eandem sphaeram, cuius dia<lb/>meter AB, descripta fuerint duo solida sphaeralia parila<lb/>tera, quorum duo latera sint BC, BD; demittanturque ex <lb/>punctis C, D, perpendiculares CE, DF ad diametrum; erit <lb/>solidum cuius latus BC, ad solidum cuius latus BD, ut <lb/>AE ad AF. </s>

<s>Ducantur enim ex centro I ad latera <lb/>

<arrow.to.target n="fig70"></arrow.to.target><lb/>BC, BD perpendiculares IG, IH. </s>

<s>Recta EA ad rectam AB, est ut qua<lb/>dratum AC ad quadratum AB (ob angu<lb/>lum in semicirculo rectum ACB) recta <lb/>autem BA ad AF, est ut quadratum AB, <lb/>ad quadratum AD, ergo ex aequo recta <lb/>EA ad rectam AF, est ut quadratum AC <lb/>ad quadratum AD, hoc est ut quadratum IG ad quadra<lb/>

<arrow.to.target n="marg101"></arrow.to.target><lb/>tum IH, hoc est ut solidum cuius latus est BC ad solidum <lb/>cuius latus est BD. Quod erat etc. </s>

<s><margin.target id="marg101"></margin.target>per praeced.</s>

<s><emph type="center"/>PROPOSITIO XXVI.<emph.end type="center"/></s>

<s>Si intra sphaeram cuius diameter AB descriptum sit <lb/>solidum sphaerale parilaterum, et circa diagonalem revo<lb/>lutum; demittaturque ab extremitate lateris BC quod dia-

<pb pagenum="67"/>metrum contingit, recta CD perpendicularis ad diametrum <lb/>circuli AB, erit conus cuius basis circulus AFCBE al<lb/>

<arrow.to.target n="fig71"></arrow.to.target><lb/>titudo verò sit AD, subduplus solidi <lb/>sphaeralis; conus verò, cuius eadem sit <lb/>basis, et altitudo DB, erit subduplus <lb/>differentiae, quae inter sphaeram, et so<lb/>lidum sphaerale est. </s>

<s>Sphaera enim ad inscriptum solidum <lb/>est ut quadratum diametri ad quadra<lb/>

<arrow.to.target n="marg102"></arrow.to.target><lb/>tum cateti AC (est enim AC ob angu<lb/>lum rectum ACB, aequalis cateto poligoni), hoc est ut BA <lb/>recta ad rectam AD. </s>

<s><margin.target id="marg102"></margin.target>7. huius.</s>

<s>Iam quia conus, cuius basis AFCBE altitudo verò sit <lb/>AB; aequalis est haemisphaerio in eadem basi constituto; <lb/>

<arrow.to.target n="marg103"></arrow.to.target><lb/>erit dictus conus, hoc est hemisphaerium, ad conum cuius <lb/>basis eadem AFCBE, altitudo verò AD, ut AB ad AD. <lb/>Sed hemisphaerium etiam ad semisolidum est ut AB ad <lb/>AD; ut ostendimus supra. Propterea conus cuius basis <lb/>circulus AFCBE, altitudo autem AD, erit aequalis semi<lb/>solido sphaerali, sive subduplus solidi sphaeralis. Quod etc. <lb/><gap desc="SM"/></s>

<s><margin.target id="marg103"></margin.target>ex 30. <lb/>p. partis.</s>

<s>Similiter inferetur, conum cuius basis eadem AFCBE, altitudo verò <lb/>DB, subduplum esse excessus illius, quo sphaera solidum superat.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Demonstramus etiam singula illa solida rotunda annularia, quae <lb/>describuntur in revolutione figurae à bilineis mixtis, quale unum est <lb/>FC, et solidum sphaerale circundant; aequalia esse singulis sphaeroidi<lb/>bus, quarum <expan abbr="uniuscuiusq;">uniuscuiusque</expan> maximus circulus sit circa diametrum FC. <lb/>Axis verò aequalis sit portioni rectae ex AB quae intercipitur inter <lb/>duas perpendiculares ad ipsam AB ductas ex punctis F et C et sic de <lb/>reliquis. Sed hoc alibi.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXVII.<emph.end type="center"/></s>

<s>Si eidem circulo duo poligona parilatera alterum cir<lb/>cumscribatur, alterum verò inscribatur; et convertatur <lb/>circumscriptum quidem circa catetum, inscriptum verò <lb/>circa diagonalem: erit differentia inter circumscriptum et <lb/>sphaeram, ad differentiam inter sphaeram et inscriptum,

<pb pagenum="68"/>ut quadratum lateris circumscripti ad duplum quadrati <lb/>lateris inscripti. </s>

<s>Esto circuli diameter AB, latus verò <lb/>

<arrow.to.target n="fig72"></arrow.to.target><lb/>poligoni circumscripti CD et inscripti AE. <lb/>Dico excessum, quo maius solidum sphae<lb/>ram superat, ad excessum, quo sphaera <lb/>superat minus esse ut quadratum CD ad <lb/>duo quadrata ex AE. <lb/>

<arrow.to.target n="marg104"></arrow.to.target></s>

<s><margin.target id="marg104"></margin.target>13. huius.</s>

<s>Solidum enim circumscriptum est ad <lb/>sphaeram ut duo quadrata CI, IA ad duplum quadrati ex <lb/>IA; ergo dividendo, erit excessus solidi supra sphaeram, <lb/>ad ipsam sphaeram, ut quadratum CA ad duplum quadrati <lb/>ex IA, sive ut quadr. CD, ad duplum quadr. ex AB. <lb/>Sphaera autem ad excessum, quo ipsa superat minus so<lb/>lidum, est ut quadratum AB ad quadratum AE, vel ut duo </s>

<s>

<arrow.to.target n="marg105"></arrow.to.target><lb/>quadrata ex AB ad duo quadrata ex AE. Propterea ex <lb/>aequo excessus solidi maioris supra sphaeram, ad excessum <lb/>sphaerae supra minus solidum, erit ut quadratum ex CD <lb/>ad duo quadrata ex AE. Quod etc. </s>

<s><margin.target id="marg105"></margin.target>20. huius.</s>

<s><emph type="center"/>PROPOSITIO XXVIII.<emph.end type="center"/></s>

<s>Quodlibet sphaerale solidum circa diagonalem revolu<lb/>tum (cuius latera numero quidem paria sint, sed nullo <lb/>modo à quaternario mensurentur, ut sunt 6, 10, 14, 18, <lb/>22 etc.) inscripti sibi rombi solidi duplum est. </s>

<s>Sit solidum quale dictum est ABC <lb/>

<arrow.to.target n="fig73"></arrow.to.target><lb/>DEFG circa axem sive diagonalem DI <lb/>revolutum. Manifestum est quod duo <lb/>latera opposita BL, FM contingent <lb/>sphaeram in extremitatibus A, G, dia<lb/>metri AG, quae quidem perpendicu<lb/>laris sit ad DI; quandoquidem laterum <lb/>numerus à binario tantum mensuratur, <lb/>non autem à quaternario. </s>

<s>Inscribantur iam duo coni; nempe ADG in semisolido, <lb/>habens altitudinem HD; conus verò AIG in hemisphaerio. <lb/>

<arrow.to.target n="marg106"></arrow.to.target><lb/>Erit igitur semisolidum ABCDEFG ad hemisphaerium ut <lb/>axis ad axem, nempe ut DH ad HI, hoc est ut conus ADG

<pb pagenum="69"/>ad conum AIG (cum sint in eadem basi); et permutando <lb/>semisolidum ad suum conum ADG, erit ut hemisphaerium <lb/>ad suum conum AIG; quare duplum erit. Propterea omne <lb/>solidum, quale dictum est duplum erit inscripti sibi rombi <lb/>solidi. Quod etc. </s>

<s><margin.target id="marg106"></margin.target>6. huius.</s>

<s><emph type="center"/><emph type="italics"/>Lemma.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si hemisphaerium ABC, et conus quicumque rectus DBE eandem <lb/>altitudinem habuerint FB; erit hemisphaerium ad praedictum conum <lb/>

<arrow.to.target n="fig74"></arrow.to.target><lb/>ut duplum basis hemisphaerij ad basim eius<lb/>dem coni. </s>

<s>Sit ut ponitur: Et inscribatur in hemi<lb/>sphaerio conus ABC. Erit ergo conus ABC ad <lb/>conum DBE ut basis AC ad basim DE; <expan abbr="sum-ptisq;">sum<lb/>ptisque</expan> antecedentium duplis, erit hemisphaerium ABC ad conum DBE <lb/>ut duplum basis AC ipsius hemisphaerij, ad DE basim coni. Quod <lb/>erat etc.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXIX.<emph.end type="center"/></s>

<s>Quodlibet sphaerale solidum circa diagonalem revolu<lb/>tum, cuius latera à quaternario mensurentur, ad inscri<lb/>ptum sibi rombum solidum, est ut superficies inscriptae <lb/>sibi sphaerae, ad semisuperficiem circumscriptae. </s>

<s>Sit solidum quale dictum est AB <lb/>

<arrow.to.target n="fig75"></arrow.to.target><lb/>CDE cui inscribatur semirombus, <lb/>hoc est conus ACE; ad altitudinem <lb/>verò hemisphaerij sit conus AFE, in <lb/>basi AE. <lb/>

<arrow.to.target n="marg107"></arrow.to.target></s>

<s><margin.target id="marg107"></margin.target>6. huius.</s>

<s>Erit ergo semisolidum ad hemis<lb/>phaerium ut axis ad axem, hoc est <lb/>ut CG ad GF, sive ut conus ACE, <lb/>ad conum AFE (sunt enim in eadem <lb/>basi) et permutando erit semisolidum ad suum conum <lb/>AGE, ut hemisphaerium ad alterum conum AFE, hoc est <lb/>per lemma praemissum, ut duo circuli ex HI, ad circulum <lb/>ex AE, vel sumptis duplis, ut quatuor circuli ex HI, ad <lb/>duos circulos ex AE; hoc est ut superficies inscriptae intra <lb/>solidum sphaerae, ad semisuperficiem circumscriptae. Pro-

<pb pagenum="70"/>pterea etiam dupla eandem rationem habebunt, hoc est <lb/>totum sphaerale solidum ad inscriptum sibi rombum soli<lb/>dum erit ut dictum est. Quod etc. <lb/><gap desc="SM"/></s>

<s>Poterat etiam concludi; solidum sphaerale praedictum esse ad in<lb/>scriptum sibi rombum, ut inscriptus in poligono circulus ad semicir<lb/>culum circumscriptum; vel ut quadratum cateti GH ad semiquadratum <lb/>diagonalis GA eiusdem poligoni.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Lemma.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si in triangulo aequilatero inscriptus fuerit circulus. Erit circulus <lb/>

<arrow.to.target n="fig76"></arrow.to.target><lb/>alter cuius diameter sit latus trianguli, triplus in<lb/>scripti circuli. </s>

<s>Inscribatur circulus ABC in triang. aequilatero <lb/>DEF. Sitque G punctum, centrum et circuli, et trian<lb/>guli; propterea DG dupla ipsius GC, hoc est ipsius <lb/>GA. Ergo quadr. DG quadruplum est quadrati ex GA <lb/>et quadratum DA triplum erit eiusdem GA; Quare <lb/>etiam circulus cuius semidiameter sit DA triplus erit circuli cuius semi<lb/>diameter sit CA. Quod erat etc.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXX.<emph.end type="center"/></s>

<s>Si circa circulum descriptum fuerit triangulum aequi<lb/>laterum et revolvatur figura, erit factus conus acquilaterus <lb/>ad inscriptam sibi sphaeram ut 9 ad 4. </s>

<s>Esto circa circulum ABC triangulum <lb/>

<arrow.to.target n="fig77"></arrow.to.target><lb/>aequilaterum DEF, et convertatur figura. <lb/>Dico factum conum aequilaterum esse <lb/>ad inscriptam sphaeram in proportione <lb/>dupla sesquiquarta, nempe ut 9 ad 4. </s>

<s>Concipiatur in hemisphaerio GAI co<lb/>nus GAI. Erit iam per lemma praecedens <lb/>circulus cuius diameter DF triplus circuli cuius diameter <lb/>GI; sed conus DEF ad conum GAI rationem habet com<lb/>positam ex ratione altitudinum EA ad AL; quae tripla <lb/>est: Et ex ratione basium, nempe circuli DF ad circulum <lb/>GI quae similiter tripla est: quare conus DEF ad conum <lb/>GAI erit ut 9 ad unum, <expan abbr="sumptisq;">sumptisque</expan> consequentium qua<lb/>druplis, erit conus DEF ad sphaeram sibi inscriptam, ut <lb/>9 ad 4. Quod erat etc. </s>

<s><emph type="center"/>PROPOSITIO XXXI.<emph.end type="center"/></s>

<s>Si circa eandem sphaeram descripti sint conus, et cy<lb/>lindrus, ambo aequilateri; erunt tria solida, nempe conus, <lb/>cylindrus, et sphaera in continua proportione sesquialtera. </s>

<s>Hoc autem patet. Posita enim sphaera ut 4 erit (per <lb/>Corollarium Prop. 30 p. partis) cylindrus ut 6; conus <lb/>autem ostensus est in praecedenti esse ut 9. Quare tria <lb/>solida erunt inter se in continua proportione sesquialtera. <lb/>Quod etc. </s>

<s><emph type="center"/>PROPOSITIO XXXII.<emph.end type="center"/></s>

<s>Sphaera ad inscriptum sibi conum aequilaterum est in <lb/>ratione numeri 32 at 9. </s>

<s>Sit in circulo cuius centrum A in<lb/>

<arrow.to.target n="fig78"></arrow.to.target><lb/>scriptum triangulum aequilaterum CBD <lb/>et convertatur figura circa CH. Dico <lb/>sphaeram esse ad factum conum aequi<lb/>laterum sibi inscriptum ut 32 ad 9. </s>

<s>Ducatur diameter EF ad angulos <lb/>rectos ipsi CH, et concipiatur in he<lb/>misphaerio conus ECF: Punctam A erit centrum tum cir<lb/>culi, tum etiam trianguli aequilateri BCD, propterea CH <lb/>sesquialtera erit ipsius CA. </s>

<s>Sed cum etiam ICL sit triangulum aequilaterum, erit <lb/>CA potentià tripla ipsius AI, ergò et circulus ex CA, sive <lb/>ex AE triplus erit circuli ex AI; <expan abbr="ideoq;">ideoque</expan> conus ECF, triplus <lb/>coni ICL videlicet ut 24 ad 8. Conus autem ICL ad conum <lb/>BCD ob similitudinem, est ut cubus AC ad cubum CH, <lb/>nimirum ut 8 ad 27. Quare ex aequo erit conus ECF ad <lb/>conum BCD ut 24 ad 27. Reductaque ratione ad minimos <lb/>terminos, erit conus ECF ad conum BCD ut 8 ad 9. Sumptis <lb/>igitur antecedentium quadruplis sphaera ad inscriptum sibi <lb/>conum aequilaterum erit ut 32 ad 9. Quod erat etc. </s>

<s><emph type="center"/>PROPOSITIO XXXIII.<emph.end type="center"/></s>

<s>Rombus solidus aequilaterus circa sphaeram descriptus <lb/>est ad ipsam sphaeram ut diameter quadrati ad latus <lb/>eiusdem. </s>

<s>Esto quadratum ABCD circa cir<lb/>

<arrow.to.target n="fig79"></arrow.to.target><lb/>culum cuius centrum E; et volvatur <lb/>figura circa diagonalem BD; Dico rom<lb/>bum solidum aequilaterum factum ex <lb/>revolutione, esse ad sphaeram ut dia<lb/>meter quadrati ad latus eiusdem. </s>

<s>Intelligatur in hemisphaerio conus <lb/>FGH, cuius basis FH, altitudo EG, et <lb/>ducatur IM. </s>

<s>Erit iam conus ABC cuius basis AC, similis cono FGH, <lb/>uterque enim rectus et rectangulus est. Ergo conus ABC <lb/>ad conum FGH erit ut cubus BE ad cubum EG, nempe <lb/>ut recta BE ad EL (sunt enim EB, EG, EI, EL in continua <lb/>ratione) sumptis autem consequentium duplis, erit conus <lb/>ABC ad hemisphaerium, ut BE ad EG, et propterea totus <lb/>rombus solidus ad totam sphaeram sibi inscriptam erit ut <lb/>BE ad EG, hoc est ut diameter alicuius quadrati ad latus <lb/>eiusdem. Quod etc. </s>

<s><emph type="center"/>PROPOSITIO XXXIV.<emph.end type="center"/></s>

<s>Sphaera ad inscriptum sibi cylindrum aequilaterum est <lb/>ut diameter quadrati ad 3 quart. lateris eiusdem. </s>

<s>Describatur intra circulum cuius <lb/>

<arrow.to.target n="fig80"></arrow.to.target><lb/>centrum A quadratum BCDE, et vol<lb/>vatur figura circa catetum AG. Dico <lb/>sphaeram ad cylindrum BCDE, esse ut <lb/>diameter alicuius quadrati ad 3 quart. <lb/>lateris eiusdem. </s>

<s>Intelligatur circa sphaeram alter <lb/>cylindrus aequilaterus FILM et pro<lb/>ducta AM iungantur AD, GO. Erunt ob similitudinem <lb/>triangulorum, in continua ratione FA, AD, AG, AP; Et

<pb pagenum="73"/>quia cylindri sunt similes, nempe aequilateri, erit cylindrus <lb/>IFML ad cylindrum BCDE ut cubus FM ad cubum CD, <lb/>hoc est ut cubus FD ad cubum DG, sive ut cubus FA ad <lb/>AD, hoc est ut recta FA ad quartam AP. Sumptisque <lb/>antecedentium subsequialteris, erit sphaera ad cylindrum <lb/>BCDE ut duae tert. ipsius FA ad AP; hoc est ut tota FA <lb/>ad sesqaialteram ipsius AP; sive (quod idem est) ut FA ad <lb/>3 quart. rectae AD. Constat ergo sphaeram ad inscriptum <lb/>sibi cylindrum aequilaterum esse ut FA ad 3 quar. ipsius <lb/>AD; hoc est ut diameter alicuius quadrati ad 3 quar. la<lb/>teris eiusdem. Quod etc. </s>

<s><emph type="center"/>PROPOSITIO XXXV.<emph.end type="center"/></s>

<s>Solidum exagonale, hoc est sphaerale solidum genitum <lb/>ab exagono circa catetum revoluto, septuplum est coni <lb/>eandem sibi basim, et altitudinem habentis. </s>

<s>Esto exagonum aequilaterum, et <lb/>

<arrow.to.target n="fig81"></arrow.to.target><lb/>aequiangulum ACDEFB et converta<lb/>tur circa catetum HI; <expan abbr="inscribaturq;">inscribaturque</expan> <lb/>conus AIB. Dico exagonale solidum <lb/>factum ex revolutione, septuplum esse <lb/>coni AIB. </s>

<s>Producantur CA, FB donec concur<lb/>rant in aliquo puncto L, eruntque ob <lb/>exagonum, quatuor triangula aequila<lb/>tera OCA, OAB, OBF, ABL, aequalia <lb/>inter se. Concipiatur ergo conus CLF perfectus; eritque <lb/>conus AIB duplus coni ALB, quandoquidem eandem habet <lb/>basim AB, sed altitudinem habet HI duplam ipsius HL. </s>

<s>Iam conus CLF ad conum ALB, erit ob similitudinem, <lb/>ut cubus CL ad cubum LA, nempe ut 8 ad 1; et dividendo <lb/>semisolidum CABF erit ad conum ALB, ut septem ad <lb/>unum. Propterea etiam dupla eandem rationem habebunt, <lb/>hoc est solidum exagonale integrum septuplum erit coni <lb/>AIB. Quod erat etc. </s>

<s><emph type="center"/>PROPOSITIO XXXVI.<emph.end type="center"/></s>

<s>Si circa circulum describatur exagonum, et revolvatur <lb/>figura circa catetum; erit sphaera sextupla coni, qui ean<lb/>dem basim, et eandem altitudinem cum solido habeat. </s>

<s>Esto circa circulum cuius centrum I <lb/>

<arrow.to.target n="fig82"></arrow.to.target><lb/>exagonum ABCDEF, et convertatur <lb/>circa catetum GH; inscribaturque in <lb/>facto solido exagonali conus AHF, qui <lb/>basim habeat circulum circa AF, alti<lb/>tudinem verò GH eandem cum solido. <lb/>Dico sphaeram sextuplam esse coni <lb/>AHF. </s>

<s>Concipiantur duo alij coni; nempe LHM in hemi<lb/>sphaerio, et AIF super basi AF constitutus ad cen<lb/>trum I. </s>

<s>Erit ergò propter exagonum, triangulum AIF aequila<lb/>terum, et ideo ipsa IG tripla erit potentià ipsius GA. <lb/>Constat igitur quod circulus cuius diameter LM (dupla <lb/>scilicet ipsius IG) triplus erit circuli cuius diameter AF, <lb/>et propterea conus LHM triplus erit coni AIF. Sphaera <lb/>autem duo decupla erit coni AIF, et ideo sextupla coni <lb/>AHF. Quod erat etc. </s>

<s><emph type="center"/>PROPOSITIO XXXVII.<emph.end type="center"/></s>

<s>Si circa circulum describatur exagonum, et volvatur <lb/>figura circa catetum; erit factum solidum ad factam sphae<lb/>ram sesquisextum. </s>

<s>Esto circa circulum cuius centrum I <lb/>

<arrow.to.target n="fig83"></arrow.to.target><lb/>exagonum ABCDEF et convertatur fi<lb/>gura circa catetum GH. Dico solidum <lb/>sphaerale factum, esse ad sphaeram <lb/>ut 7 ad 6. </s>

<s>Concipiatur enim in solido conus <lb/>AHF, ut in duabus praecedentibus pro<lb/>positionibus. </s>

<s>Erit ergo (per 35 huius) solidum exagonale ad conum

<pb pagenum="75"/>AHF ut 7 ad unum, conus autem AHF ad sphaeram est </s>

<s>

<arrow.to.target n="marg108"></arrow.to.target><lb/>ut 1 ad 6; quare ex aequo erit solidum ad sphaeram ut 7 <lb/>ad 6. Quod etc. </s>

<s><margin.target id="marg108"></margin.target>per praeced.</s>

<s><emph type="center"/><emph type="italics"/>Lemma.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Linea diagonalis exagoni potentià sesquitertia est cateti eiusdem. </s>

<s>Sit exagonum ABC cuius centrum D. Dico <lb/>

<arrow.to.target n="fig84"></arrow.to.target><lb/>diagonalem AC potentià esse sesquitertiam ca<lb/>teti EF. </s>

<s>Hoc autem patet. Nam ducta DB erit ABD <lb/>triangulam aequilaterum, ob exagonum; et AD la<lb/>tus erit potentià sesquitertium perpendicularis <lb/>DE; ergò sumptis lineis duplis, etiam AC sesqui<lb/>tertia erit potentià ipsius EF. Quod etc.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXXVIII.<emph.end type="center"/></s>

<s>Sphaera inscripti sibi solidi exagonalis circa diagonalem <lb/>revoluti, sesquitertia est. </s>

<s>Sit in circulo cuius centrum A de<lb/>

<arrow.to.target n="fig85"></arrow.to.target><lb/>scriptum exagonum BCDEFG; <expan abbr="iunctisq;">iunctisque</expan> <lb/>DH, DL, DM, DI, convertatur figura <lb/>circa diagonalem DG. Dico sphaeram <lb/>inscripti solidi exagonalis sesquitertiam <lb/>esse. Circulus enim, cuius diameter HI, <lb/>sesquitertius est circuli cuius diameter <lb/>LM (per lemma praecedens) ergo conus <lb/>HDI sesquitertius est coni LDM, sumptisque quadruplis, <lb/>erit sphaera sesquitertia solidi exagonalis. Quod erat etc. <lb/><gap desc="SM"/></s>

<s>Assumptum fuit solidum exagonale quadruplum esse coni LDM hoc <lb/>enim patet ex propositione 28 huius.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XXXIX.<emph.end type="center"/></s>

<s>Si idem exagonum dupliciter revolvatur, nempe circa <lb/>catetum, et circa diagonalem; Erit solidum circa catetum <lb/>revolutum, ad solidum circa diagonalem, in subduplicata <lb/>ratione numerorum 49 ad 48. Nempe ut radix <expan abbr="q.">que</expan> num. 49 <lb/>ad radicem <expan abbr="q.">que</expan> num. 48. </s>

<s>Esto exagonum aequiangulum, et aequilaterum ABC <lb/>DEF, quod <expan abbr="utroq;">utroque</expan> modo concipiatur revolutum, nempe <lb/>circa catetum HI et circa diagonalem <lb/>DA; ut inde fiant duo solida sphaeralia <lb/>

<arrow.to.target n="fig86"></arrow.to.target><lb/>inter se diversa specie; et intra <expan abbr="utrunq;">utrunque</expan> <lb/>intelligatur sphaera inscripta. Manife<lb/>stum iam est (per lemma Propositionis <lb/>praecedentis) diagonalem AD potentià <lb/>sesquitertiam esse cateti HI. Si ergo <lb/>ponatur HI rationalis 6 erit AD radix <lb/>quadrata nu meri 48. </s>

<s>Manentibus his. Solidum circa catetum revolutum, ad <lb/>

<arrow.to.target n="marg109"></arrow.to.target><lb/>inscriptam sphaer. est ut 7 ad 6; Sphaera autem ad soli<lb/>dum revolutum circa diagonalem est ut HI, ad AD, nempe <lb/>

<arrow.to.target n="marg110"></arrow.to.target><lb/>ut 6 ad rad. <expan abbr="q.">que</expan> num. 48. Quare ex aequo erit, solidum circa <lb/>catetum, ad solidum circa diagonalem ut 7 ad radicem <lb/>quadratam numeri 48. Nempe in subduplicata ratione nu<lb/>merorum 49, 48. Quod erat etc. </s>

<s><margin.target id="marg109"></margin.target>37. huius.</s>

<s><margin.target id="marg110"></margin.target>6. huius.</s>

<s><emph type="center"/><emph type="italics"/>Lemma.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si hemisphaerium altitudinem habuerit subduplam alicuius coni: <lb/>erit hemisphaerium ad conum praedictum, ut basis <lb/>ad basim. </s>

<s>Habeat haemisphaerium ABC altitudinem HB <lb/>subduplam altitudinis HE coni DEF. Dico hemis<lb/>phaerium ad conum DEF, esse ut circulus AC ad <lb/>circulum DF. </s>

<s>Concipiantur enim duo alij coni ABC in he<lb/>misphaerio, et DBF super basi DF. Erit ergò co<lb/>nus ABC ad conum DBF, ut basis AC ad basim <lb/>DF; <expan abbr="sumptisq;">sumptisque</expan> duplis, erit hemisphaerium ad conum DEF ut basis AC <lb/>ad basim DF. Quod erat etc.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XL.<emph.end type="center"/></s>

<s>Solidum parilaterum circa catetum revolutun ad in<lb/>scriptum sibi conum, rationem habet quam AB ad BC; <lb/>facto scilicet angulo DEB recto. </s>

<s>Esto poligonum FGHILE habens latera numero paria, <lb/>descriptum circa circulum cuius centrum D et conver-

<pb pagenum="77"/>tatur figura circa catetum CA, <expan abbr="fiatq;">fiatque</expan> angulus DEB rectus. <lb/>Dico solidum ad inscriptum sibi conum <lb/>

<arrow.to.target n="fig87"></arrow.to.target><lb/>FAE, esse ut AB ad BC. </s>

<s>Erit enim solidum ad sphaeram ut <lb/>

<arrow.to.target n="marg111"></arrow.to.target><lb/>BA ad AC, <expan abbr="sumptisq;">sumptisque</expan> consequentium <lb/>dimidijs, erit solidum ad hemisphae<lb/>rium ut BA ad DC, sed (per lemma <lb/>praece dens) hemisphaerium est ad <lb/>conum FAE, ut circulus ex DC ad <lb/>circulum ex CE; sive ut recta DC ad CB; ergò ex aequo <lb/>erit sphaerale solidum ad inscriptum sibi conum FAE, ut <lb/>AB ad BC. Quod erat etc. </s>

<s><margin.target id="marg111"></margin.target>12. huius.</s>

<s><emph type="center"/>PROPOSITIO XLI.<emph.end type="center"/></s>

<s>Conus inscriptus in solido circa catetum revoluto, ae<lb/>qualis est excessui quo solidum inscriptam sibi sphaeram <lb/>superat. </s>

<s>Manente figura et constructione praecedentis. Dico si <lb/>sphaera auferatur à solido FGHILE, quòd residuum, quod <lb/>superest, ablata sphaera, aequale erit cono FAE. </s>

<s>Est enim sphaerale solidum ad sphaeram ut BA ad AC; <lb/>

<arrow.to.target n="marg112"></arrow.to.target><lb/>et per conversionem rationis, solidum ad illud residuun <lb/>erit ut AB ad BC. Sed (per praecedentem) solidum ad in<lb/>scriptum sibi conum est ut AB ad BC. Aequalis est ergò <lb/>conus FAE, in solido sphaerali inscriptus, omnibus simul <lb/>solidulis annularibus quae circa sphaeram sunt; sive diffe<lb/>rentiae, quae est inter solidum inscriptamque in solido <lb/>sphaeram. Quod erat etc. </s>

<s><margin.target id="marg112"></margin.target>12. huius.</s>

<s><emph type="center"/>PROPOSITIO XLII.<emph.end type="center"/></s>

<s>Hemisphaerium ad excessum quo sua sphaera supe<lb/>ratur à solido sphaerali circa catetum revoluto, duplicatam <lb/>rationem habet diametri sphaerae ad latus poligoni, ex <lb/>cuius revolutione solidum genitum fuerat. </s>

<s>Manente praecedentium figura, et constructione. Dico <lb/>hemisphaerium, ad differentiam inter solidum, et inclusam <lb/>sphaeram, esse ut quadratum AC, ad quadratum FE.

<arrow.to.target n="marg113"></arrow.to.target></s>

<s><margin.target id="marg113"></margin.target>12. huius.</s>

<s>Est enim sphaera ad solidum circumscriptum ut CA <lb/>ad AB; et dividendo, sphaera ad differentiam inter sphae<lb/>ram et solidum, erit ut AC ad CB; sumptisque antece<lb/>dentium dimidijs, erit hemisphaerium ad praedictam dif<lb/>ferentiam, ut DC ad CB, hoc est ut quadratum DC ad <lb/>quadratum CE; vel ut quadratum AC ad quadratum FE. <lb/>Quod erat etc. </s>

<s><emph type="center"/><emph type="italics"/>Aliter.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Sphaera ad solidum est ut duo quadrata ex CD ad duo simul qua</s>

<s>

<arrow.to.target n="marg114"></arrow.to.target><lb/>drata CD, DE. Ergo dividendo erit sphaera ad differentiam inter ipsam <lb/>et solidum ut duo quadrata ex CD ad quadratum CE <expan abbr="sumptisq;">sumptisque</expan> ante<lb/>cedentium dimidijs, erit hemisphaerium ad differentiam inter sphaeram <lb/>et solidum, ut quadratum DC ad quadr. CE, sive ut quadratum AC ad <lb/>quadratum FE. Quod etc.<gap desc="/SM"/></s>

<s><margin.target id="marg114"></margin.target>13. huius.</s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Constat etiam hemisphaerium ad conum FAE inscriptum in sphaerali <lb/>solido, esse in duplicata ratione AC ad FE, nempe axis coni ad dia<lb/>metrum basis eiusdem. Quandoquidem conus FAE demonstratus est ae<lb/>qualis differentiae inter solidum sphaerale <expan abbr="inscriptamq;">inscriptamque</expan> sibi sphaeram.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XLIII.<emph.end type="center"/></s>

<s>Si exagono regulari simile exagonum inscribatur, ita <lb/>ut inscripti anguli puncta media circumscriptorum laterum <lb/>contingant, et convertatur figura circà catetum maioris <lb/>exagoni, erit solidum exagonale circumscriptum ad inscri<lb/>ptum ut 14 ad 9. </s>

<s>Sit ut ponitur: Convertaturque figura <lb/>

<arrow.to.target n="fig88"></arrow.to.target><lb/>circà AB; <expan abbr="circaq;">circaque</expan> AB diametrum conci<lb/>piatur sphaera, quae quidem maiori po<lb/>ligono inscripta erit, minori verò circum<lb/>scripta. <lb/>

<arrow.to.target n="marg115"></arrow.to.target></s>

<s><margin.target id="marg115"></margin.target>per 37. <lb/>huius.</s>

<s>Erit <expan abbr="itaq;">itaque</expan> solidum maius ad sphaeram <lb/>ut 7 ad 6 nempe ut 14 ad 12; sphaera </s>

<s>

<arrow.to.target n="marg116"></arrow.to.target><lb/>verò ad minus solidum erit ut 12 ad 9. Ergò ex aequo <lb/>solidum maius ad minus erit ut 14 ad 9. Quod erat etc. </s>

<s><margin.target id="marg116"></margin.target>38. huius.</s>

<s><emph type="center"/>PROPOSITIO XLIV.<emph.end type="center"/></s>

<s>Solidum sphaerale factum ex revolutione alicuius poli<lb/>goni circa diagonalem, ad solidum ex revolutione eiusdem <lb/>poligoni circà catetum; est ut rectangulum sub diagonali, <lb/>et cateto, bis sumptum, ad duo simul quadrata, quorum <lb/>alterum ex diagonali fit, alterum autem ex cateto. </s>

<s>Esto poligonum regulare quodcumque, <lb/>

<arrow.to.target n="fig89"></arrow.to.target><lb/>habens latera numero paria, cuius diago<lb/>nalis sit AB, catetus verò CD. Et conci<lb/>piatur poligonum converti duplici axe; <lb/>nempe primùm circà diagonalem AB; et <lb/>iterum circa catetum CD. Dico solidum <lb/>ex diagonali ad solidum ex cateto esse, <lb/>ut rectangulum BED bis sumptum, ad <lb/>quadrata ex BE, et ex ED: sive ut eorum quadrupla. </s>

<s>Fiat angulus EBH rectus, seceturq: bifariam DH in I; <lb/><expan abbr="eritq.">eritque</expan> EI media Aritmetica inter ED, EH: Iam solidum ex <lb/>

<arrow.to.target n="marg117"></arrow.to.target><lb/>diagonali ad inscriptam sibi sphaeram est, ut AB, ad CD; <lb/>sphaera verò ad solidum ex cateto, est ut CD, ad CH; ergò <lb/>

<arrow.to.target n="marg118"></arrow.to.target><lb/>ex aequo solidum ex diagon. ad solidum ex cateto, erit ut <lb/>AB ad CH, sive ut EB ad EI, (sunt enim semisses rectarum <lb/>AB, CH). Cum autem BE media Geometrica sit inter HE, <lb/>ED; ipsa verò EI media Aritmetica sit inter easd. erit so<lb/>lidum ex diagonali ad solidum ex cateto ut media Geomet. <lb/>ad mediam Aritmet. inter rectas HE, ED. Sed ratio rectae <lb/>HE ad ED, ead. est ac quadr. BE ad quadr. ED: propterea <lb/>erit solidum ex diagonali ad solidum ex cateto, ut spatium <lb/>medium proportionale Geometricum ad spatium medium <lb/>Aritmeticum inter quadrata BE, ED. Spatium autem me<lb/>dium Geometricum inter quadrata BE, ED est rectangu<lb/>lum BED; medium verò Aritmeticum est quadratum ED, <lb/>cum semisse quadrati DB. Ergo solidnm ex diagonali ad <lb/>solidum ex cateto erit ut rectangulum BED; ad quadra<lb/>tum ED cum semisse quadrati DB; Vel (sumptis duplis) ut <lb/>rectangulum BED, bis sumptum, ad quadratum ED bis, <lb/>cum integro quadrato DB. Sive ut rectangulum BED bis <lb/>sumptum, ad quadrata BE, ED. Quod erat etc.

<s><margin.target id="marg117"></margin.target>6. huius.</s>

<s><margin.target id="marg118"></margin.target>12. huius.</s>

<s>Assumpsimus rectangulum BED, medium proportionale esse inter <lb/>quadrata BE, ED. Hoc enim patet in propositis quibuscunque rectis <lb/>duabus lineis. </s>

<s>Assumpsimus etiam quadratum ED cum semisse quadrati DB, esse <lb/>medium Aritmeticum inter qnadrata BE, ED. Quod patet quadratum <lb/>enim BE superat quadratum ED quadrato BD.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Hic pro Corollario demonstrari potest, solidum ex diagonali factum <lb/>semper minus esse solido, quod fit ex cateto; quando idem poligonum <lb/>convertatur circa diagonalem, et circa catetum. Demonstratur hoc <lb/>modo. </s>

<s>Quoniam rectangulum BED bis sumptum, minus est duobus qua<lb/>dratis BE, ED (sunt enim in continua ratione quadratum EB, rectan<lb/>gulum DEB, et quadratum ED, <expan abbr="ideoq;">ideoque</expan> dupla mediae, minor est duabus <lb/>extremis magnitudinibus). Et est ut rectangulum BED bis sumptum <lb/>ad quadr. BE, ED simul, ità solidum ex diagonali ad solidum ex ca<lb/>teto; Erit solidum ex diagonali minus quam solidum ex cateto. Quod <lb/>erat etc. </s>

<s>Si quis autem quaerat, quo excessu solidum ex cateto superet soli<lb/>dum ex diagonali. Hoc modo illum proportione notum habebit. </s>

<s>Faciat ut duo quadrata BE, ED simul, ad quadratum quod fit ex <lb/>differentia rectarum BE, ED, ità maius solidum ad aliud: Et habebit <lb/>excessum quo maius solidum superat minus.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XLV.<emph.end type="center"/></s>

<s>Si intra poligonum regulare parilaterum inscribatur <lb/>simile poligonum, ità ut anguli inscripti bisectiones late<lb/>rum circumscripti contingant; <expan abbr="convertaturq;">convertaturque</expan> figura circa <lb/>catetum maioris poligoni: Erit maius <lb/>solidum sphaerale ad minus, ut sunt <lb/>

<arrow.to.target n="fig90"></arrow.to.target><lb/>duo simul quadrata duarum diagona<lb/>lium, ad duo quadrata minoris cateti. </s>

<s>Esto poligonum parilaterum ABC etc. <lb/>intra quod inscribatur simile poligonum <lb/>AIC etc. uti dictum est. <expan abbr="Convertaturq;">Convertaturque</expan> <lb/>figura circa AC catetum maioris poligoni. <lb/>Dico solidum sphaerale ABC, ad solidum AIC esse ut duo <lb/>quadrata simul duarum diagonalium, nempe BD, DC ad

<pb pagenum="81"/>duo quadrata minoris cateti DI. Circumscribatur solido <lb/>AIC sua sphaera, quae alteri solido inscripta erit. </s>

<s>Iam solidum ABC ad inscriptam sphaeram, est ut duo <lb/>quadrata simul BD, DC ad duplum quadrati DC (per 13 <lb/>huius). Sphaera verò ad inscriptum solidum est, ut duplum <lb/>quadrati DC ad duplum quadrati DI (per 7 huius). Ergo <lb/>ex aequo maius solidum sphaerale ad minus erit ut duo <lb/>simul quadrata BD, DC ad duplum quadrati DI. Quod <lb/>erat etc. </s>

<s><emph type="center"/>PROPOSITIO XLVI.<emph.end type="center"/></s>

<s>Iisdem positis: si convertatur figura circa diagonalem <lb/>maioris poligoni GC. Erit maius solidum ad minus, ut in<lb/>teger axis AC maioris solidi, ad utramque simul, nempe <lb/>semicatetum DG minoris, et quartam proportionalium GF; <lb/>si fiat ut semidiagonalis minoris ad semicatetum; ita se<lb/>micatetus ad tertiam, et tertia ad quartam. </s>

<s>Esto solidum quale positum est <lb/>

<arrow.to.target n="fig91"></arrow.to.target><lb/>ABCH cui inscriptum sit solidum <lb/>IBD uti dictum est. Ducatur, DE <lb/>perpendicularis ad GB, et EF ad GC; <lb/><expan abbr="eruntq;">eruntque</expan> in continua proportione CG, <lb/>GB, GD, GE, GF ob angulos rectos. </s>

<s>Iam solidum maius ad sphaeram <lb/>est ut AC ad HB (per 6 huius) sphaera <lb/>autem ad solidum minus est ut HB <lb/>ad utramque simul DG, GF (per 14 <lb/>huius). Quare ex aequo solidum maius <lb/>ad minus erit ut AC ad utramque simul DG, GF nempe <lb/>quod propositum fuerat. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Quando solida praedicta ab exagono genita fuerint: demonstratur <lb/>quod posita recta AC 32, DG et GF notae sunt. nempe DG 12 et GF 9. <lb/>Ergo in hoc casu solidum maius ad minus esset ut 32 ad 21. </s>

<s>Superest nunc ut solida sphaeralia absolutè considerata inter se <lb/>conferamus, et hoc quot modis fieri poterit: quemadmodum in proémio <lb/>operis nos esse facturos promiseramus.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XLVII.<emph.end type="center"/></s>

<s>Solida sphaeralia parilatera circa diagonalem revoluta, <lb/>inter se sunt ut parallelepipeda basi quadr. cateti, altitu<lb/>dine vero diagonali eorumdem. </s>

<s>Sint duo solida sphaeralia pari<lb/>

<arrow.to.target n="fig92"></arrow.to.target><lb/>latera circa diagonales AC, DF re<lb/>voluta. <expan abbr="Sintq;">Sintque</expan> HI, LV perpendicu<lb/>lares ad latera CB, FE. Dico solidum <lb/>sphaerale ABC ad solidum DEF esse <lb/>ut parallelepipedum basi quadrato <lb/>HI altitudine verò HC, ad parallelep. basi quadrato LV, <lb/>altitudine LF. </s>

<s>Intelligatur utrique circumscripta sphaera sua. Tunc <lb/>

<arrow.to.target n="marg119"></arrow.to.target><lb/>enim solidum ABC ad sphaeram suam erit ut quadratum <lb/>IH ad quadratum HC, sive (sumpta communi altitudine <lb/>CH) ut parallelepipedum basi quadrato IH, altitudine HC, <lb/>ad cubum HC. Sphaera autem ABC ad sphaeram DEF, <lb/>est ut cubus HC ad cubum LF. At sphaera DEF, (ut nuper <lb/>in altera ostendebamus) ad solidum suum DEF est ut <lb/>cubus LF, ad parallelepipedum basi quadato LV, altitu<lb/>dine LF: ergo ex aequo erit solidum ABC, ad solidum <lb/>sphaerale DEF, ut parallelepipedum basi quadrato HI, al<lb/>titudine HC; ad parallelepipedum basi quadrato LV, alti<lb/>tudine LF. Quod erat etc. </s>

<s><margin.target id="marg119"></margin.target>7. huius.</s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Idem concludetur etiam si concipiantur sphaera iuxta 6 huius intra <lb/>data solida inscriptae; sive altera tantum inscripta, altera verò cir<lb/>cumscripta iuxsta 6 et 7 huius sicut experienti patebit.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO XLVIII.<emph.end type="center"/></s>

<s>Solida sphaeralia parilatera circa catetum revoluta inter <lb/>se sunt, ut parallelepipeda basi quadrato diagonalis, alti<lb/>tudine verò quae sit aequalis cateto, et quartae proportio<lb/>nalium, si fiat ut diagonalis ad catetum, ita catetus ad <lb/>tertiam, et ita tertia ad quartam. </s>

<s>Sint duo solida sphaeralia circa catetos B, et D revo<lb/>luta. Continue turque ratio A ad B in quatuor terminis <lb/>A, B, E, F. Item ratio diagonalis C <lb/>

<arrow.to.target n="fig93"></arrow.to.target><lb/>ad catetum D continuetur in quatuor <lb/>terminis C, D, H, I. Dico, primum <lb/>solidum ad secundum esse ut paral<lb/>lelepipedum basi quadrato A, altitu<lb/>dine verò B et F; ad parallepipedum <lb/>basi quadrato C altitudine verò D <lb/>et I. </s>

<s>Nam primum solidum ad sphae<lb/>ram suam est ut B et F simul ad A bis sumptam: <expan abbr="acce-ptaq;">acce<lb/>ptaque</expan> communi basi quadrato A; erit solidum primum ad <lb/>sphaeram suam, ut parallelepipedum basi quadrato A, alti<lb/>tudine verò B et F simul, ad duos cubos A. Sphaera <lb/>autem prima ad secundam sphaeram est ut duo cubi A <lb/>ad duos cubos C. Sphaera tandem secunda ad solidum <lb/>suum, est ut duo cubi C, ad parallelepipedum basi qua<lb/>drato C altitudine verò D, et I simul (quod ostenditur ut <lb/>nuper factum est in prima sphaera) ergo ex aequo primum <lb/>solidum sphaerale ad secundum, erit ut parallelepipedum <lb/>basi quadrato A, altitudine B et F simul, ad parallele<lb/>pipedum basi quadrato C altitudine verò D et I simul. <lb/>Quod erat etc. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Idem concludi potest si sphaerae concipiantur intra ipsa solida <lb/>inscriptae iuxta Propositionem 13 huius; sive altera inscripta, altera <lb/>verò circumscripta iuxta 13 et 14 huius. Quando verò termini propor<lb/>tionis alij evadant à propositis, ut in hac, et in sequentibus, scias <lb/>proportionem semper eandem esse, in quibuscunque tandem terminis <lb/>eveniat.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO IL.<emph.end type="center"/></s>

<s>Solida sphaeralia imparilatera sunt inter se ut paralle<lb/>lepipeda, basi quadrato perpendicularis, quae ex centro <lb/>poligoni ducitur in latus eiusdem, altitudine verò aequali <lb/>praedictae perpendiculari, una cum dupla eius, quae ex

<pb pagenum="84"/>centro ad angulum poligoni ducitur, et cum tertia propor<lb/>tionalium ad duas praedictas. </s>

<s>Sint solida sphaeralia imparilatera, circa catetos B, et <lb/>D revoluta. Continuetur ratio per<lb/>

<arrow.to.target n="fig94"></arrow.to.target><lb/>pendicularis B ad radium poligoni <lb/>A in tribus terminis B, A, E. Item <lb/>ratio D ad C in tribus terminis <lb/>D, C, I, continuata sit. Dico soli<lb/>dum primum ad secundum esse ut <lb/>parallelepipedum basi quadrato B, <lb/>altitudine verò aequali B semel, <lb/>A bis, et E semel, <expan abbr="simulq;">simulque</expan> sum<lb/>ptis, ad parallelepipedum basi <lb/>quadr. D altitudine verò aequali D semel, C bis, et I <lb/>semel <expan abbr="simulq;">simulque</expan> sumptis. </s>

<s>Concipiatur in <expan abbr="utroq;">utroque</expan> solido sphaerali sua sphaera in<lb/>scripta, <expan abbr="eritq;">eritque</expan> solidum primum ad sphaeram suam ut B <lb/>

<arrow.to.target n="marg120"></arrow.to.target><lb/>et E simul cum dupla ipsius A ad quadruplam B sumpta<lb/>que communi basi quadrato B erit solidum primum ad <lb/>sphaeram suam ut parallelepipedum basi quadrato B alti<lb/>tudine verò B et E cum dupla A ad quatuor cubos B. <lb/>Sphaera autem prima ad secundam est, ut quatuor cubi <lb/>B ad quatuor cubos D; Sphaera tandem secunda ad so<lb/>lidum suum est, ut quatuor cubi D ad parallelepipedum <lb/>basi quadrato D altitudine D et I cum dupla ipsius C <lb/>(quod ostenditur ut nuper factum est) ergo ex aequo patet <lb/>quod propositum fuerat etc. </s>

<s><margin.target id="marg120"></margin.target>38 huius.</s>

<s><emph type="center"/>PROPOSITIO L.<emph.end type="center"/></s>

<s>Solidum sphaerale parilaterum diagonalem revolutum, <lb/>ac solidum sphaerale parilaterum circa catetum revolu<lb/>tum, est ut parallelepipedum basi quadrato cateti, alti<lb/>tudine diagonalis bis sumptum, ad parallelepipedum basi <lb/>quadrato cateti simul diagonalisque, altitudine verò ca<lb/>teti. </s>

<s>Sint duo solida sphaeralia, quorum alterum circa dia<lb/>gonalem A sit revolutum, alterum verò circa catetum C.

<pb pagenum="85"/>Dico solidum primum circa diagonalem, ad solidum se<lb/>cundum circa catetum, esse ut parallelepipedum basi <lb/>

<arrow.to.target n="fig95"></arrow.to.target><lb/>quadr. B altitudine A bis sumptum, ad parallelepipedum <lb/>basi aequali quadratis C, D, altitudine verò C. </s>

<s>Intelligatur in utroque solido inscripta sua sphaera. Et <lb/>

<arrow.to.target n="marg121"></arrow.to.target><lb/>erit solidum primum ad sphaeram suam, ut recta A, ad B; <lb/><expan abbr="sumptaq;">sumptaque</expan> eadem basi quadrato B; erit solidum primum ad <lb/>sphaeram suam, ut parallelepipedum basi quadrato B alti<lb/>tudine verò A, ad cubum B sive ut duplum dicti parallele<lb/>pipedi ad duos cubos B. Sphaera verò prima ad secundam <lb/>est, ut duo cubi B, ad duos cubos C. Sphaera tandem se<lb/>

<arrow.to.target n="marg122"></arrow.to.target><lb/>cunda ad solidum suum est, ut duo quadrata ex C, ad duo <lb/>quadrata C, et D; sumptaque communi altitudine C, est, <lb/>ut duo cubi C, ad parallelepipedum basi aequali quadratis <lb/>C et D altitudine verò C. Propterea ex aequo patet quod <lb/>propositum erat. </s>

<s><margin.target id="marg121"></margin.target>6. huius.</s>

<s><margin.target id="marg122"></margin.target>13. huius</s>

<s><emph type="center"/>PROPOSITIO LI.<emph.end type="center"/></s>

<s>Solidum sphaerale parilaterum circa diagonalem revo<lb/>lutum, ad solidum sphaerale imparilaterum est, ut paralle<lb/>lepipedum basi quadrato cateti, altitudine diagonali quater <lb/>sumptum; ad parallelepipedum basi quadrato rectae illius <lb/>quae ex centro poligoni imparila<lb/>

<arrow.to.target n="fig96"></arrow.to.target><lb/>teri perpendiculariter ducitur in <lb/>latus eiusdem; altitudine verò ae<lb/>quali praedictae perpendiculari, <lb/>una cum dupla illius quae ex cen<lb/>tro ad angulum ducitur, et cum <lb/>tertia proportionalium ad duas <lb/>praedictas. </s>

<s>Sint duo solida sphaeralia, nempe primum parilaterum <lb/>circa diagonalem A conversum, alterum verò imparilate-

<pb pagenum="86"/>rum circa catetum C revolutum. Continuetur ratio C ad <lb/>D in trib. terminis C, D, E. Dico primum solidum ad se<lb/>cundum esse, ut parallelepipedum basi quadrato B, altitu<lb/>dine A quater sumptum, ad parallelepipedum basi qua<lb/>drato C, altitudine verò aequali rectis C, et E cum dupla <lb/>D simul sumptis. </s>

<s>Nam solidum primum ad sphaeram suam est, ut recta <lb/>A ad B; sive sumpta communi basi quadrato B; ut pa<lb/>rallelepipedum basi quadrato B altitudine A, ad cubum B; <lb/>Vel ut parallelepipedum praedictum quater sumptum, ad <lb/>cubum B quater sumptum sphaera verò prima ad secun<lb/>dam est ut quatuor cubi B ad quatuor cubos C. Sphaera <lb/>denique secunda ad solidum suum (ut ostensum est in 49 <lb/>huius) est ut quatuor cubi C, ad parallelepipedum basi <lb/>quadrato C, altitudine verò aequali rectis C et E cum <lb/>dupla D simul sumptis. Propterea ex aequo patet quod <lb/>propositum erat. </s>

<s><emph type="center"/>PROPOSITIO LII.<emph.end type="center"/></s>

<s>Solidum sphaerale parilaterum circa catetum revolu<lb/>tum, ad solidum sph. imparilaterum, est ut parallelepipe<lb/>dum basi aequali quadratis diagonalis et cateti altitudine <lb/>cateti bis sumptum, ad parallelepipedum basi quadrato <lb/>lineae quae ex centro ducitur perpendiculariter in latus <lb/>poligoni imparilateri, altitudine verò aequali praedictae <lb/>lineae, una cum illa quae ex centro ad unum angulum per<lb/>ducitur, et cum tertia proportionalium ad duas praedictas. </s>

<s>Sint duo solida sphaeralia; al<lb/>

<arrow.to.target n="fig97"></arrow.to.target><lb/>terum parilaterum circa catetum <lb/>A revolutum; alterum imparilate<lb/>rum circa C conversum. Et ratio <lb/>C ad D, continuetur in tribus ter<lb/>minis C, D, E. Dico primum soli<lb/>dum ad secundum esse, ut paral<lb/>lelepipedum basi aequali quadratis <lb/>B et A, altitudine verò A, bis <lb/>sumptum; ad parallelepipedum basi quadrato C, altitudine <lb/>verò aequali C, et E, cum dupla ipsius D. </s>

<s>Nam solidum primum ad sphaeram suam est, ut duo <lb/>quadrata B et A, ad duplum quadrati A sive sumpta com<lb/>muni altitudine A ut parallelepipedum basi aequali qua<lb/>dratis B et A, altitudine A ad duos cubos A. Vel ut di<lb/>ctum parallelepipedum bis sumptum, ad quatuor cubos A. <lb/>Sphaera autem prima ad secundam, est ut quatuor cubi A <lb/>ad quatuor cubos C. Sphaera denique secunda ad solidum <lb/>suum est ut quatuor cubi C, ad parallelepipedum basi <lb/>quadrato C altitudine aequali C et E, cum dupla D (ut <lb/>ostensum fuit in Propos. 49 huius). Ergo ex aequo patet <lb/>quod propositum fuerat. </s>

<s><emph type="center"/>FINE DE'LIBRI <lb/>“ DE SPHAERA ET SOLIDIS SPHAERALIBUS ” <emph.end type="center"/></s>

<pb/>

<s><emph type="center"/>DE DIMENSIONE PARABOLAE<emph.end type="center"/></s>

<s><emph type="center"/>SOLIDIQUE HYPERBOLICI <lb/>PROBLEMATA DUO: <lb/>ANTIQUUM ALTERUM<emph.end type="center"/></s>

<s><emph type="center"/>IN QUO QUADRATURA PARABOLAE XX MODIS ABSOLVITUR, <lb/>PARTIM GEOMETRICIS, MECANICISQUE; PARTIM EX <lb/>INDIVISIBILIUM GEOMETRIA DEDUCTIS <lb/>RATIONIBUS: <lb/>NOVUM ALTERUM <lb/>IN QUO MIRABILIS CUIUSDAM SOLIDI AB HYPERBOLA GENITI, <lb/>ACCIDENTIA NONNULLA DEMONSTRANTUR.<emph.end type="center"/></s>

<s><emph type="center"/>CUM APPENDICE <lb/>DE DIMENSIONE SPATIJ CYCLOIDALIS, ET COCHLEAE.<emph.end type="center"/></s>

<pb/>

<s><emph type="center"/><emph type="italics"/>Ad Serenissimum Principem<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>LEOPOLDUM<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>ab Etruria<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="italics"/>Difficile reor, Serenissime Princeps Leopolde, ferrea hac <lb/>aetate libros conscribere; difficiliùs dedicare: quandoquidem <lb/>bonarum Artium studia ubique in bella degenerant, et Re<lb/>gnantes viri non exigunt ingeniorum vires, sed corporum.<emph.end type="italics"/></s>

<s><emph type="italics"/>Etrusca tamen Regia, non minus foecunda virtutum, <lb/>quàm Principum, mundum edocet, eandem esse Minervam <lb/>et Bellonam, unumque Apollinem qui arcum simul amat, et <lb/>citharam. Serenissima enim Celsitudo Tua (ut reliquos omit<lb/>tam) litterarum, et scientiarum omne genus perinde foret, <lb/>colitque, ac si mundus alta pace frueretur, pulsisque Furijs <lb/>solae Musae dominarentur. Verùm alia me maior difficultas <lb/>terret, dum ego tenuitatis meae conscius mecum ipse cogito, <lb/>libellum hunc ad eum Principem ire, qui illum non solum <lb/>protegere potest, sed etiam iudicare. Quicquid est, non acre <lb/>iudicium Sereniss. Celsitudinis Tuae, sed incomparabilem <lb/>humanitatem invoco, illam inquam humanitatem, quae nuper <lb/>amplissima in me beneficia contulit, e humi iacentem erexit <lb/>fortunam meam. Audiat preces meas Dominus Regnantium,<emph.end type="italics"/>

<pb/><emph type="italics"/>talemque Principem diu custodiat: siquidem divinitatis in<lb/>terest huiusmodi viros prosperari, ut aeterna Providentia <lb/>magis elucescat, et coniunctam aliquando cum potestate sa<lb/>pientiam in terris demonstrare valeamus.<emph.end type="italics"/></s>

<s><emph type="italics"/>Sereniss. Celsitud. Tuae<emph.end type="italics"/></s>

<s><emph type="italics"/>Humillimus, et obsequentiss. servus<emph.end type="italics"/></s>

<s><emph type="italics"/>Evangelista Torricellius.<emph.end type="italics"/></s>

<pb/>

<s><emph type="center"/>AD LECTOREM <lb/>PROEMIUM<emph.end type="center"/></s>

<s>Nullus in un universo Mathematicarum disciplinarum <lb/>Theatro fortasse tritior pulvis reperitur, quàm parabolae <lb/>quadratura. </s>

<s>Quarè ergò (inquis amice lector) circà tritum argu<lb/>mentum tàm diù desudasti? libenter equidem excipio obie<lb/>ctiones tuas; sed utinam ultimus desudaverim. Quam ta<lb/>men veniam mihi negas, scias eandem plurimis, et egregiè <lb/>laudatis Scriptoribus te denegare. Obiectum enim de pa<lb/>rabolae quadratura, quod nostra hac aetate confiteor mihi <lb/>nimis iam inveterasse, crediderim neque novum fuisse Ca<lb/>valerio, Galileo, Lucae Valerio, et alijs. Quin immò ipsum <lb/>Archimedem accusat, quicumque improbat lucubrationes <lb/>circà subiectum vetus institutas. Audiamus ipsum in proë<lb/>mio quadraturae parabolae, ubi scribens Dositheo inquit. <lb/><gap desc="SM"/>Eorum enim, qui antehac Geometriae operam dederunt, nonnulli id <lb/>investigare, et memoriae mandare studuerunt, circulo dato, vel circuli <lb/>portione quacunque, spactium rectilineum aequale illi posse inveniri. <lb/>Item spatium à coni totius rectanguli sectione compraehensum et <lb/>lineâ rectà, ad quadrati formam et mensuram reducere conati sunt; <lb/>sumentes non facilè concessibilia fundamenta.<gap desc="/SM"/> Quibus verbis diser<lb/>tissime fatetur Geometrarum Princeps argumentum libro<lb/>rum De dimensione circuli, e de quadratura parabolae, <lb/>neque suum fuisse, neque novum. Sed si quis attentè con<lb/>sideret Proëmialem epistolam, libro de lineis spiralibus <lb/>praefixam, intelliget praecipua Archimedis Theoremata,

<pb pagenum="94"/>aliorum inventa fuisse, et magna ex parte Cononis. Maxi<lb/>mae enim Propositiones librorum <gap desc="SM"/>De Sphaera et Cylindro; De <lb/>conoidibus et sphaeroidibus, et De lineis spiralibus<gap desc="/SM"/> (qui libri inter <lb/>opera Archimedis principem locum tenent (Cononis sunt: <lb/><gap desc="SM"/>Qui<gap desc="/SM"/> (ut inquit auctor) <gap desc="SM"/>non satis temporis ad haec excogitanda <lb/>sortitus, vitam permutavit, et ipsa reliquit inexplicata; cum illa inve<lb/>nisset, et alia quamplurima perquisisset, ac multum adeò Geometricas <lb/>facultates ampliasset.<gap desc="/SM"/> Si ergò licuit admirabili, ac propè di<lb/>vino Auctori, circa aliorum inventa laborare; quis negabit <lb/>ignoscendum ingeniolo meo mutuata theoremata contem<lb/>planti? Sed esto quod conclusio antiqua sit; argumenta <lb/>certè, quibus illa confirmabitur, ut plurimum nova erunt, <lb/>et inaudita: immo cum ad alteram partem libelli accede<lb/>mus, in quà de solido acuto hyperbolico dicendum est, non <lb/>solum ipsum Theorema inexcogitatum, et ut ita dicam pa<lb/>radoxicum erit, sed etiam demonstrandi ratio inusitata, et <lb/>penitus nova. Verùm (inquis) reliqui scriptores, qui huius<lb/>modi quadraturam aggressi sunt, vel singulas, vel ad sum<lb/>mum binas prodiderunt; neque tamen mediocrem laudem <lb/>consequuti sunt. Fateor; sed nec ego libellum hunc ex <lb/>professo institui, composuique: immo quod et alijs, mihi <lb/>quoque accidit; singulas hasce quadraturas diversis tem<lb/>poribus inveni, quas in unum collectas nunc demum vo<lb/>lentibus simul exhibeo. Tu tamen exclamas; heu nimis <lb/>est: quotus enim quisque reperietur tam famelicus Geo<lb/>metra, qui legat penè vicies repetitam propositionem, cum <lb/>numero lemmatum ferè duplo? Huic sanè obiectioni libet <lb/>contradicere. Cum enim libellus in Propositiones, ut plu<lb/>rimum non coherentes digestus sit; sed ita dispositas ut <lb/>ubicunque libuerit initium facere possis, et finem, dicam <lb/>cum Martiale </s>

<s><emph type="center"/><emph type="italics"/>. . . tibi carta plicetur <lb/>Altera; divisum sic breve fiet opus.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Si vero mavis probare consilium eorum qui unam, aut <lb/>alteram tantum quadraturam edidere; quis prohibet? Et <lb/>in hoc legere potes unam, aut alteram; si tamen hoc quo<lb/>que; nimis videbitur, nullam. Utilitatem exigis? concedo; <lb/>et in hanc partem libellum excusare non ausim. attamen

<pb pagenum="95"/>non deerit fortasse aliquis qui penitus inutilem non existi<lb/>met, cum Geometricus sit. Sola enim Geometria inter li<lb/>berales dixiplinas acritèr exacuit ingenium, idoneumque <lb/>reddit ad civitates exornandas in pace, et in bello defen<lb/>dendas; coeteris enim paribus, ingenium quod exercitatum <lb/>sit in Geometrica palestra, peculiare quoddam, et virile <lb/>robur habere solet: praestabitque semper, et antecellet, <lb/>circà studia Architecturae, rei bellicae, nauticaeque sive <lb/>mavis circà Aritmeticam, artemque metiendi, unde totum <lb/>civile commercium dependet, regiturque. Quinetiam circà <lb/>ministeria fluviorum, et aquarum stagnantium, unde non <lb/>nisi magna percipiuntur sive damna, sive beneficia; pro <lb/>ut bene, vel male intellecta fuerit huiusmodi rerum na<lb/>tura. Sed esto quòd inutilis penitus habeatur libellus; sive <lb/>quia Reipubl. nihil interest parabolae quadraturae; sive <lb/>quia multis ab hinc saeculis excogitata fuerat, et demon<lb/>strata. Huic verò obiectioni respondeat Reverendiss. D. Be<lb/>nedictus Castellius Magister meus. Ipse enim dicet, quòd <lb/>si Principes terrae, solam illam vulgarem, et apparentem <lb/>utilitatem in Artibus magni facerent, exigerentque, da<lb/>mnandi penitus essent sculptores, Celatoresque egregij <lb/>eijcendi civitatibus Pictores, Musici, Citharaedi, Poëtae, <lb/>atque id genus alij. Contrà verò ditandi essent, atque opi<lb/>bus, officijsque omnibus demerendi pistores, quorum utili<lb/>tati nulla alia par est; caupones, sutores et quicunque <lb/>artem colunt vitae hominum summoperè utilem. Quinetiam <lb/>si utilitas sola attendatur, damnandus erit vini usus, et <lb/>detestanda cultura vinearum. At in summo praetio ha<lb/>benda aqua, cuius utilitates tàm facilè est numerare, quàm <lb/>difficile sit ijs non indigere. </s>

<s>Utcumque ea res se se habeat, veniamus ad obiectiones <lb/>quae circà artis fundamenta versantur. Indignor equidem <lb/>Lucam Valerium, verè nostri saeculi Archimedem, cum <lb/>optimam causam suscepisset, pessimà defensione usum <lb/>fuisse. Solent ab eruditis culpari figurarum Geometricarum <lb/>dimensiones, quae Mecanicis fundamentis innixae stabiliun<lb/>tur, tam quàm duplex falsum supponant: alterum, <gap desc="SM"/>quòd <lb/>superficies gravitatem non habentes, habere tamen concipiuntur:<gap desc="/SM"/> alte<lb/>rùm verò, <gap desc="SM"/>quòd fila quae magnitudines ad libram suspendunt aequi-

<pb pagenum="96"/>distantis supponuntur, cum tamen in centro terrae concurrere debeant.<gap desc="/SM"/><lb/>Ego verò in ea sum sententia, vel nullam ex his supposi<lb/>tionibus esse falsam, vel reliqua omnia principia Geome<lb/>triae falsa existere eodem modo. Falsum enim est, quòd <lb/>circulus habeat centrum, sphaera superficiem, conus soli<lb/>ditatem. Loquor de figuris abstractis quales Geometria <lb/>considerare solet; non autem de fisicis et concretis. Ne<lb/>cesse igitur erit fateri quòd circuli centrum, superficies <lb/>spherae, soliditas coni, et reliqua huiusmodi non contro<lb/>versa, nullam aliam habeant existentiam, praeter illam <lb/>quam accipiunt per definitionem, et per intellectum. Eodem <lb/>prorsus modus gravitatis est in figuris Geometricis, quo<lb/>modo in ijsdem est centrum, perimeter, superficics, soli<lb/>ditas, etc. Laudarem igitur in Mecanicis contemplationibus <lb/>nova definitione figuras generare; hoc, aut alio non ab <lb/>simili modo. <lb/><gap desc="SM"/></s>

<s>Quadratum est quadrilaterum, quod, cum aequilaterum, et aequian<lb/>gulum sit, singula ipsius punta momentum habent procedendi versus <lb/>aliquam mundi plagam per lineas inter se parallelas.<gap desc="/SM"/></s>

<s>Huiusmodi enim definitio omnem demeret occasionem <lb/>dubitandi, illis, qui Mecanica Archimedis opera, secundum <lb/>ipsius mentem non accipiunt. Sed hucusque dictum sit <lb/>pro obliteranda primae falsitatis nota, quòd figurae Geo<lb/>metricae graves sint. </s>

<s>Venio nunc ad secundum (ut aliqui existimant) falsum. <lb/>Principiò, vulgatissima est etiam apud gravissimos viros <lb/>obiectio illa, videlicet. <gap desc="SM"/>Archimedem supposuisse aliquod falsum, <lb/>dum fila magnitudinum ex librà pendentium consideravit tanquam <lb/>inter se parallela, cum tamen re vera in ipso terrae centro concurrere <lb/>debeant.<gap desc="/SM"/> Ego verò, (quod pace clarissimorum virorum di<lb/>ctum sit) crediderim fundamentum Mecanicum longè alia <lb/>ratione esse considerandum. Concedo si Fisicae magnitu<lb/>dines ad libram liberè suspendantur, quòd fila materialia <lb/>suspensionum convergentia erunt; quandoquidem singula <lb/>ad centrum terrae respiciunt. Verum tamen si eadem libra, <lb/>licet corporea, consideretur non in superficie terrae, sed <lb/>in altissimis regionibus ultrâ orbem solis; tum fila (dum<lb/>modo adhuc ad terrae centrum respiciant) multò minùs <lb/>convergentia inter se erunt; sed quasi aequidistantia. Con-

<pb pagenum="97"/>cipiamus iam ipsam libram Mecanicam ultra stellatam li<lb/>bram firmamenti in infinitam distantiam esse provectam; <lb/>quis non intelligit fila suspensionum iam non ampliùs con<lb/>vergentia, sed exacte parallela fore? Quando ego considero <lb/>libram, figuras Geometricas ponderantem, non concipio <lb/>illam esse inter cartas librorum, in quibus depicta conspi<lb/>citur; neque suppons punctum, ad quod magnitudines <lb/>ipsius tendunt, esse centrum terrae; sed libram fingo in <lb/>infinitum remotam esse ab eo puncto, ad quod ipsius gravia <lb/>contendunt. Si posteà ibi conclusero triangulum aliquod <lb/>triplum esse cuiusdam spatij; retrahatur imaginatione ipsa <lb/>libra ad nostras regiones; concedo quòd retractâ librâ de<lb/>struetur aequidistantia filorum suspensionis, sed non ideò <lb/>destruetur proportio iam demonstrata figuraram. Peculiare <lb/>quoddam beneficium habet Geometra, cum ipse abstra<lb/>ctionis ope, omnes operationes suas mediante intellectu <lb/>exequatur. Quis igitur mihi hoc negaverit, si libeat con<lb/>siderare figuras appensas ad libram, quae quidem libra <lb/>ultra mundi confinium in infinitam distantiam remota <lb/>supponatur? Vel quis proibebit considerare libram in su<lb/>perficie terrae constitutam, cuius tamen abstracta ma<lb/>gnitudines tendant, non ad medium terrae punctum, sed <lb/>ad centrum caniculae, sive stellae polaris? Triangula et <lb/>parabolae, immò etiam spherae, cilindrique Geometrici, <lb/>cum nullam per se habeant motus differentiam, non magis <lb/>ad ipsius terrae, quam ad Saturni centrum contendunt. <lb/>Destruit ergò beneficium suum quisquis flguras illas, tam<lb/>quam ad unicum terrae centrum tendentes, contempla<lb/>tur. Cur denique non licebit mihi considerare puncta cu<lb/>iuscunque figurae eiusmodi virtute praedita, ut singula <lb/>versus eandem mundi plagam per lineas inter se parallelas <lb/>aequali momento contendant? His ita suppositis, quae vera <lb/>sunt, quemadmodum sunt verae passiones figurarum, quae <lb/>in definitionibus adhibentur, vera etiam erunt quaecunque <lb/>Theoremata per Mecanicas rationes ab ipsis astrahentibus <lb/>fuerint considerata, neque per falsas positiones demonstra<lb/>buntur. Tunc itaque falsum dici poterit fundamentum <lb/>Mecanicum, nempe fila librae parallela esse, quando ma<lb/>gnitudines ad libram appensae fisicae sint, realesque, et

<pb pagenum="98"/>ad terrae centrum conspirantes. Non autem falsum erit, <lb/>quando magnitudines (sive abstractae, sive concretae sint) <lb/>non ad centrum terrae, neque ad aliud punctum propin<lb/>quum librae respiciant; sed ad aliquod punctum infinitè <lb/>distans connitantur. </s>

<s>Coeterum, brevitatis, et facilitatis gratià à vocabulis <lb/>consuetis non discedemus; punctumque illud ad quod ma<lb/>gnitudines librae contendere supponuntur, Centrum terrae <lb/>nominabimus, Planum verò illud, quod erectum est ad <lb/>lineam connectentem praedictum punctum cum centro li<lb/>brae, Horizontem de more appellabimus. </s>

<s><emph type="center"/>SUPPOSITIONES, ET DEFINITIONES.<emph.end type="center"/></s>

<s>Ponatur eam esse centri gravitatis naturam, ut magnitudo liberè <lb/>suspensa ex quolibet sui puncto nunquam quiescat nisi cum centrum <lb/>gravitatis ad infimum suae sphaerae punctum pervenerit.<gap desc="/SM"/></s>

<s>Concipiamus figuram ABC, <lb/>

<arrow.to.target n="fig98"></arrow.to.target><lb/>suspensam ex sui puncto D, <lb/>mediante filo ED; liberè; hoc <lb/>est, ita ut in quamcumque <lb/>partem converti possit. Sit <lb/>centrum gravitatis F. pona<lb/>musque rectam EDG. perpen<lb/>dicularem esse ad horizontem. </s>

<s>Certum est, donec cen<lb/>trum F fuerit extrà perpen<lb/>diculum EG, figuram ipsam <lb/>nnnquam mansuram esse. Quando verò punctum F. fuerit <lb/>in perpendiculo suspensionis EG, tunc figura omninò quie<lb/>scet. Centrum enim gravitatis ipsius nusquàm poterit am<lb/>pliùs inferius descendere: Quin immò si figura moveretur, <lb/>centrum ipsum ascenderet, quod esse non potest. Si quis <lb/>enim centro E, intervallo EDF. tamquam unà recta linea, <lb/>sphaeram concipiat esse descriptam; ipsa erit sphaera, in

<pb pagenum="99"/>cuius superficie feretur punctum F, quando EDF. extensa <lb/>fuerit, et ad rectitudinem redacta. Certumque est infimum <lb/>punctum huiusmodi sphaerae esse in perpendiculo EG. </s>

<s>Aequiponderare sibi ipsi figura dicetur, quae ab aliquo sui puncto <lb/>liberè suspensa maneat, et ad nullam sui partem inclinetur.<gap desc="/SM"/></s>

<s>Aequiponderat sibi ipsi figura, quando (cum liberè suspensa sit) in <lb/>ipso suspensionis perpendiculo centrum gravitatis reperitur. Si enim <lb/>adhuc moveretur, centrum gravitatis ascenderet. Quod est impossibile.<gap desc="/SM"/></s>

<s>Centrum gravitatis tunc reperitur in ipso suspensionis perpendiculo, <lb/>quando figura liberè suspensa sibi ipsi aequiponderat. Alias enim figura <lb/>quiesceret, et centrum gravitatis ipsius posset adhuc inferiús descen<lb/>dere. Quod est absurdum.<gap desc="/SM"/></s>

<s>Centralitér ad illum librae punctum appendi figura dicetur, in quod <lb/>cadit perpendiculum, ex centro gravitatis figurae productum.<gap desc="/SM"/></s>

<s>Esto enim libra AB, cuius ful<lb/>

<arrow.to.target n="fig99"></arrow.to.target><lb/>crum sit C, et ad ipsam appensa <lb/>sit figura CEB, ita ut totum la<lb/>tus CB cohereat, et sit veluti ad <lb/>ipsam libram conglutinatum. Esto <lb/>centrum gravitatis figurae pun<lb/>ctum D, et ex D agatur perpendiculum DF ad horizontem <lb/>erectum. </s>

<s>Iam figura CEB dicetur, et considerabitur, tamquam <lb/>appensa centralitêr ad punctum F. Constat enim ex prae<lb/>dictis, quòd si figurae latus CB solvatur undique à brachio <lb/>librae, solumque remaneat filum connexionis DF, nihilo <lb/>tamen minus figura adhuc manebit ut prius manebat, ean<lb/>demque servabit versus libram positionem, quam antea <lb/>habebat. Vide Arch. Prop. 6. De Quadratura parabolae. </s>

<s>Aequalia gravia ex aequalibus distantijs aequiponderant, sive libra <lb/>ad horizontem parallela fuerit, sive inclinata. Et gravia eandem reci<lb/>procè rationem habentia, quam distantiae, aequiponderant, sive libra <lb/>sit ad horizontem parallela, sive inclinata.<gap desc="/SM"/></s>

<s>Haec sine alia explicatione praemitti poterant; quan<lb/>doquidem in doctrina aequiponderantium nunquam suppo<lb/>nitur libra horizonti aequidistans: Attamen quia ostendi <lb/>possunt, non omittendam censeo demonstrationem; prae<lb/>sertim cum nonnulli ex libra materiali male fabricata, <lb/>errorem susceperint, et intellectum admiserint. </s>

<s>Esto inclinata libra <lb/>

<arrow.to.target n="fig100"></arrow.to.target><lb/>AC, suspensa ex puncto <lb/>B ad filum BD. Sintque <lb/>magnitudines BFC, et G. <lb/>centraliter appensae ex <lb/>punctis E, et A. Et po<lb/>natur esse, ut magnitudo <lb/>BFG, ad magnitudinem <lb/>G, ita reciprocè distantia <lb/>AB ad BE. Dico libram <lb/>AC, quamvis inclinata, <lb/>magnitudinesque ab ipsa pendentes, penitùs conquiexere, <lb/>et aequiponderare. </s>

<s>Producantur enim perpendicula GAH, LEI, per centra <lb/>gravitatis figurarum G, et L, transeuntia, ducaturque ho<lb/>rizontalis libra CH, quae item appensa sit ad filum MD. <lb/>Quoniam igitur est per suppositionem, ut magnitudo BFC, <lb/>ad magnitudinem G, ita reciprocè AB ad BE; sive (ob <lb/>parallelas) HM ad MI; aequiponderabunt maguitudines <lb/>BFC, et G, ad libram horizontalem HC appensae. Ergo <lb/>commune centrum gravitatis erit omninò in perpendiculo <lb/>DF. Propterea magnitudines aequiponderabunt etiam dum <lb/>ad libram AC suspenduntur: aliàs, si moverentur, com<lb/>mune centrum gravitatis ipsarum, quod demonstratum est <lb/>esse in perpendiculo DF, ascenderet. Quod est impossibile. </s>

<s>Haec autem breviùs concludentur hoc modo. Conne<lb/>ctantur (in eadem figura) centra gravitatis ductâ rectâ GL. </s>

<s>Quoniam magnitudo <lb/>

<arrow.to.target n="fig101"></arrow.to.target><lb/>BFC ad magnitudinem <lb/>G, est ut AB ad BE, <lb/>sive (ob parallelas) ut <lb/>GN ad NL, erit N cen<lb/>trum commune gravitatis <lb/>magnitudinum appensa<lb/>rum. Si ergo libra AC <lb/>non quiesceret, centrum <lb/>gravitatis N, ascenderet. <lb/>Cum enim sit in perpendiculo DF, moveri non potest quin <lb/>ascendat. </s>

<s>Non me latet auctorum controversiam, circà libram <lb/>inclinatam, an redeat, maneatvè supponere centra magni<lb/>tudinum in ipsa libra esse collocata. Nos tamen, quia in <lb/>hoc libello, semper considerabimus magnitudines infrà <lb/>ipsam libram appensas, maluimus rei nostrae servire, quàm <lb/>aliorum controversiae demonstrationem accomodare. </s>

<s>Coeterum passiones parabolae quas in operis progressu <lb/>supponemus tamquam notas, vel ipsius Apollonij erunt, vel <lb/>Archimedis, vel saltem ex Apollonio ipso facili negotio <lb/>deducentur, cuiusmodi sunt hae, quae sequuntur. </s>

<s>Si Parabola rectam lineam tangentem habuerit, à qui<lb/>buslibet autem punctis ipsius tangentis rectae lineae usque <lb/>ad parabolam demittantur aequidistantes diametro, erunt <lb/>demissae inter se longitudine ut sunt portiones tangentis <lb/>potentià inter se. Deducitur enim hoc ex 20. prim. Conic. <lb/>Nam rectae illae demissae portionibus diametri respon<lb/>dent; at partes ipsius tangentis, ordinatim applicatis ae<lb/>quales sunt. </s>

<s>Item, si intrà parabolam à punctis quibuslibet rectae <lb/>illius ordinatim ductae, quae basis parabolae dicitur, rectae <lb/>lineae erigantur diametro parallelae. Erunt erectae inter <lb/>se ut sunt rectangula facta à portionibus basis, quae ab <lb/>ipsis erectis abscinduntur. Hoc enim et a Cavalerio, et a <lb/>nobis in secundo libro de motu ostenditur. </s>

<s><emph type="center"/>QUADRATURA PARABOLAE.<emph.end type="center"/></s>

<s><emph type="center"/>PLURIBUS MODIS PER DUPLICEM POSITIONEM, <lb/>MORE ANTIQUORUM ABSOLUTA.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Lemma Primum.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola duas tangentes habuerit, altera ex termino basis, alte<lb/>ram verò ex vertice: tangens, quae ad basim est, bifariam secabitur <lb/>ab illa, quae per verticem ducitur.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuius dia<lb/>

<arrow.to.target n="fig102"></arrow.to.target><lb/>meter BI, ordinatim verò appli<lb/>cata (sive basis) sit AC; tangens <lb/>ex termino basis sit CD; per ver<lb/>ticem verò tangens BE. Dico <lb/>ipsam CD bifariàm secari in pun<lb/>cto E. <lb/>

<arrow.to.target n="marg123"></arrow.to.target></s>

<s><margin.target id="marg123"></margin.target>35 primi <lb/>Conic.</s>

<s>Cum. N. CD sit tangens, DI diameter, erunt aequales <lb/>inter se DB. BI. Et quia AC ordinatim applicata est ad </s>

<s>

<arrow.to.target n="marg124"></arrow.to.target><lb/>diametrum BI, ipsa verò BE tangit in puncto B, erunt pa<lb/>rallelae AC, BE. Et ideo erit ut DB ad BI, ita DE ad EC. <lb/>

<arrow.to.target n="marg125"></arrow.to.target></s>

<s><margin.target id="marg124"></margin.target>per 32. eiusd.</s>

<s><margin.target id="marg125"></margin.target>per 2 sexti.</s>

<s>Quare aequales erunt etiam DE, EC, Quod erat osten<lb/>dendum etc. </s>

<s><emph type="center"/><emph type="italics"/>Lemma II.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola duas tangentes habuerit ex basis terminis; recta linea <lb/>quae ab occursu duarum tangentium ducitur diametro parallela, pro<lb/>positae parabolae liameter erit.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuius ex <lb/>punctis A et C, duae tangentes <lb/>sint AD, CD, concurrentes in D. <lb/>Ex puncto autem D recta duca<lb/>tur DE diametro parallela. Dico <lb/>ipsam DE propositae parabolae <lb/>diametrum esse. </s>

<s>

<arrow.to.target n="marg126"></arrow.to.target></s>

<s><margin.target id="marg126"></margin.target>per 35 2. Co<lb/>nicorum.</s>

<s>Sit enim, si possibile est. dia<lb/>meter FG. Erunt ergò ob tangentem AF aequales inter

<pb pagenum="103"/>se diametri portiones FB, BG. Iterum ob tangentem CI, <lb/>aequales erunt IB, BG. Et ideò aequales erunt inter se <lb/>ipsae FB, BI: totum et pars. quod fieri non potest. </s>

<s>Non est ergò alia diameter praeter ipsam DE. Quod <lb/>erat ostendendum etc. </s>

<s><emph type="center"/><emph type="italics"/>Lemma III.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola tres tangentes habuerit; duas ad basim, et tertiam per <lb/>verticem; erit triangulum sub tangentibus compraehensum octuplum <lb/>trianguli, quod oritur ex ductu quartae tangentis per verticem alte<lb/>rutrae semiparabolae.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuius basis AC, diameter BD; duae <lb/>tangentes ad basim AE, CE. Tangens per verticem sit FBG. <lb/>Demittatur ex F,concursu tangentium AF, GF, recta FI, </s>

<s>

<arrow.to.target n="marg127"></arrow.to.target><lb/>diametro parallela; eritque <lb/>

<arrow.to.target n="fig103"></arrow.to.target><lb/>FI, diameter parabolae AIB. <lb/>Ducatur denique LM, tan<lb/>gens semiparabolam AIB <lb/>per verticem I. Dico trian<lb/>gulum FEG, sub tangenti<lb/>bus compraehensum, octu<lb/>plum esse trianguli LFM, <lb/>quod nascitur ex ductu <lb/>quartae tangentis LM per <lb/>verticem I portionis AIB. </s>

<s><margin.target id="marg127"></margin.target>per lem. <lb/>praeced.</s>

<s>Iungatur AB basis parabolae AIB, eruntque parallelae <lb/>AB, LM; et cum sint aequales FL, LA, ob tangentem AF, <lb/>erit AF dupla ipsius FL; ideoque triangulum AFB quadru<lb/>

<arrow.to.target n="marg128"></arrow.to.target><lb/>plum trianguli sibi slmilis LFM. Ergò etiàm FBE quadru<lb/>

<arrow.to.target n="marg129"></arrow.to.target><lb/>plum trianguli LFM (sunt enim per lem. primum aequales <lb/>bases AF, FE) Propterea totum triangulum FEG octuplum <lb/>erit trianguli LFM. Quod erat ostendendum etc. </s>

<s><margin.target id="marg129"></margin.target>per lem. <lb/>primum.</s>

<s><emph type="center"/><emph type="italics"/>Corollarium Primum.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Ergò triangulum FEG, factum à primis tribus tangen<lb/>tibus, octuplum ostendetur eodem modo etiam trianguli <lb/>NGP. et propterea semper quadruplum erit duorum simul

<pb pagenum="104"/>triangulorum LFM, NGP; quae post ipsum (ductà utrinque <lb/>alia tangente) oriuntur. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium secundum.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Manifestum etiam est triangulum FEG sub tangentibus <lb/>contentum, auferre plusquàm dimidium ex trilineo mixto <lb/>ABCE: siquidem triangulum FEG, dimidium est duorum <lb/>simul triangulorum EBA, EBC. Ergo erit plusquàm dimi<lb/>dium trilinei mixti ABCE. </s>

<s>Hinc sequitur quòd possibile sit intrà figuram mixtam <lb/>

<arrow.to.target n="marg130"></arrow.to.target><lb/>ABCGF. figuram rectilineam inscribere per continuum du<lb/>ctum tangentium; quae quidem figura inscripta deficiat à <lb/>figura mixta, defectu minori quàm sit quaelibet data ma<lb/>gnitudo. </s>

<s><margin.target id="marg130"></margin.target>per primam <lb/>decimi.</s>

<s><emph type="center"/><emph type="italics"/>Lemma IV.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola duas tangentes habuerit ad basim: deinde per vertices <lb/>factarum portionum aliae tangentes ex ordine ducantur; et hoc fiat <lb/>quotiescunque libuerit; figura a tangentibus circumsepta, si ex vertice <lb/>parabolae suspendatur (posità diametro ad horizontem perpendiculari) <lb/>aequiponderabit.<gap desc="/SM"/></s>

<s>Esto parabola A <lb/>BC, cuius diameter <lb/>BD, et duae tangen<lb/>tes ad basim sint AE, <lb/>CE; per verticem <lb/>verò B tangens sit <lb/>FBG. Deinde, demis<lb/>sis (ut in praecedenti <lb/>lemmate) diametris <lb/>FH, GI, per vertices <lb/>portionum AHB, BI <lb/>C, tangentes ducan<lb/>tur LM, NO. </s>

<s>Iterumque per vertices reliquarum quatuor portionum <lb/>tangentes ducantur PQ, RS, TU, XZ; et sic semper donec <lb/>lituerit; Dico figuram; sive potiùs duas figuras rectilineas

<pb pagenum="105"/>a tangentibus PQRSFP, TUXZGT; circumseptas, ex pun<lb/>cto B aequiponderare: statuta priùs diametro BD ad ho<lb/>rizontem perpendiculari. </s>

<s>Ponatur itaque BD diameter parabolae ad horizontem <lb/>perpendicularis; et rectam FG, (quamcunque inclinationem <lb/>sortiatur) concipiamus esse libram, cuius fulcrum sit in B. </s>

<s>Quoniam igitur applicata AB bifariam secatur à dia<lb/>metro FH in puncto Y; suntque AB, LM, parallelae, erit <lb/>etiam LM secta bifariam in H; et ideò duorum triangu<lb/>lorum LFM, NGO, centra gravitatis sunt in FH, GI; sunt<lb/>que FH, GI ad horizontem perpendiculares, ideò appensa <lb/>centralitèr erunt dicta triangula ad libram FG. ex punctis <lb/>F et G. Aequiponderabuntque ex distantijs aequalibus <lb/>BF, BG. Cum ipsa quoque triangula sint aequalia; nempe <lb/>

<arrow.to.target n="marg131"></arrow.to.target><lb/>sub octupla eiusdem trianguli F e G. Eadem prorsus ra<lb/>tione posità librà LM, duo triangula PLQ, RMS appensa <lb/>erunt ex punctis et M; aequiponderabuntque ex puncto H, <lb/>et ideo appensa erunt ex puncto F. (quandoquidem filum <lb/>suspensionis FH perpendiculare est ad horizontem). Duo <lb/>verò triangula TNU, XOZ, praedictis aequalia (cum sint <lb/>singula suboctupla aequalium LFM, NGO.) ponderabunt <lb/>

<arrow.to.target n="marg132"></arrow.to.target><lb/>simul ambo ex puncto G. Ergo quatuor simul praedicta <lb/>triangula aequiponderabunt ex puncto B, nempe medio <lb/>totius librae FG: Eodem modo concludemus reliqua trian<lb/>gula, quotcunque sint, ex puncto B aequiponderare. Uni<lb/>versa ergo figura à tangentibus circumsepta ex puncto B <lb/>aequiponderabit. Quod etc. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium I.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hinc pro corollario animadvertemus centrum gravitatis <lb/>praedictae figurae, à tangentibus compraehensae, esse in <lb/>diametro parabolae. </s>

<s>Cum enim figura praedicta aequiponderet ex puncto B, <lb/>erit centrum gravitatis illius in linea quae ex puncto B du<lb/>citur perpendicularis ad horizontem; quapropter erit in <lb/>BD diametro parabolae. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium II.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Colligemus etiam centrum gravitatis omnium trilineo<lb/>rum mixtorum, quae sub linea parabolica ABC, et sub <lb/>omnibus tangentibus APQRSTUXZC, compraehenduntur, <lb/>semper in diametro parabolae existere. Patebit autem hoc <lb/>

<arrow.to.target n="marg133"></arrow.to.target><lb/>modo. Centrum trapetij AFGC, est in diametro; centrum <lb/>etiam parabolae est in diametro; ergò centrum reliquae <lb/>

<arrow.to.target n="marg134"></arrow.to.target><lb/>figurae mixtae erit in diametro. Si ergo centrum huiu<lb/>smodi figurae est in diametro, centrum etiam figurae à <lb/>

<arrow.to.target n="marg135"></arrow.to.target><lb/>tangentibus circumseptae demonstratum est esse in dia<lb/>metro; propterea centrum omnium simul trilineorum, quae <lb/>continentur sub tangentibus et linea parabolica, erit in <lb/>diametro per 8. prim. Aequipond. </s>

<s><margin.target id="marg133"></margin.target>15 primi ae<lb/>quiponder.</s>

<s><margin.target id="marg134"></margin.target>4 secundi <lb/>eiusd.</s>

<s><margin.target id="marg135"></margin.target>8 primi <lb/>eiusd.</s>

<s><emph type="center"/><emph type="italics"/>Lemma V.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola duas tangentes habuerit alteram per verticem, alteram <lb/>verò ad basim, et ex altera parte basis habeat parallelam diametro; <lb/>figura snb tribus praedictis rectis lineis, et curvà parabolicà comprae<lb/>hensa, aequiponderabit ex puncto tangentis verticalis, in quo ea sic <lb/>dividitur, ut per ad reliquam tangentem terminata, dupla sit illius <lb/>quae ad parallelam diametro terminatur.<gap desc="/SM"/></s>

<s>Esto parabola A <lb/>BC, cuius tangens ad <lb/>basim sit CD; per <lb/>verticem verò FBG; <lb/>et AG sit parallela <lb/>diametro. Secetur de<lb/>inde FG in E, ita ut <lb/>FE dupla sit reliquae <lb/>EG. Dico figuram <lb/>ABCFG (statutà dia<lb/>metro ad horiz. per<lb/>pendiculari) aequin<lb/>derare ex puncto C. <lb/>Concipiamus enim <lb/>diametr. parabolae esse horizonti perpendicularem (hoc

<pb pagenum="107"/>autem modo semper intelligendum est) quamcunque tan<lb/>dem inclinationem sortiatur libra GF. Et ducta tangente <lb/>AD (quae omnino transibit per E, ut infrà ostendemus) in<lb/>telligatur GF. libram esse, cuius fulcrum est E; ex qua <lb/>pendent ab una parte triangulum AGE; ab alterà verò, <lb/>figura mixta ABCFE. Quae quidem figurae si inter se non <lb/>aequiponderant, ponamus alteram ipsarum praeponderare. <lb/>Esto igitur; et praeponderet ABCFE, tanto excessu quan<lb/>tum est spatium K. </s>

<s>Inscribatur intrà ipsam alia figura à tangentibus <lb/>HILMNOPQFEH, terminata, ita ut reliquae portiunculae <lb/>sub dictis tangentibus et curva parabolica contentae, si<lb/>mul minores sint spatio K (quod fieri posse constat ex <lb/>Corollario secundo Lemmatis Tertij). Praeponderabit igitur <lb/>adhuc figura sub tangentibus compraehensa; quandoqui<lb/>dem pars ablata minor est excessu K, et in eodem pun<lb/>cto B ponderat simul cum tota magnitudine; tam enim <lb/>ablatae, quàm totius, centrum gravitatis est in diametro, <lb/>ut ostendimus ad Corollarium Secundum Lemmatis Quarti. </s>

<s>Accipiatur iàm GR quarta pars totius GA; ducaturque <lb/>ER. Sumatur etiam CL dupla reliquae LG; et ex pun<lb/>cto L centralitèr suspensum erit quodlibet triangulum ha<lb/>

<arrow.to.target n="marg136"></arrow.to.target><lb/>bens verticem in E puncto, et basim in recta GA, quae <lb/>ad horizontem recta ponitur. </s>

<s><margin.target id="marg136"></margin.target>Vide <lb/>Archim. De <lb/>Quadratura <lb/>Parab. Pro<lb/>pos. 6. 8. 10 <lb/>et 12.</s>

<s>Iam sic: duo triangula ZEX, UFT, ad triangulum EDF <lb/>sunt ut duo ad 8, et ad triangul. EBD ut 2. ad 4. et ad <lb/>aequale AGE. ut 2. ad 4. ergò ad triangulum ARE. erunt <lb/>

<arrow.to.target n="marg137"></arrow.to.target><lb/>ut 2 ad 3, nempe ut LE ad EG, hoc est ut LE ad EB <lb/>reciprocè. Quare in libra LB duo praedicta triangula ZEX, <lb/>

<arrow.to.target n="marg138"></arrow.to.target><lb/>UFT aequiponderant triangulo ARE ex puncto E. </s>

<s><margin.target id="marg137"></margin.target>per Corol. <lb/>prim.</s>

<s>Sumatur iterum GS quarta pars totius GR, et iunga<lb/>tur ES. Cum ergo GRE sit quarta pars ipsius GAE, vel <lb/>ipsius EDB, erit GRE octava pars totius EDF. Quapro<lb/>pter aequale erit GRE, alterutro ipsorum ZEX, UFT. Sed <lb/>quoniam HZI est octava pars ipsius ZEX, erunt quatuor <lb/>simul triangula HZI, LXM, NUO, <expan abbr="PTq.">PTque</expan> (quoniam aequalia <lb/>sunt inter se) ad triangulum ZEX ut 4. ad 8; sive ut 2. <lb/>

<arrow.to.target n="marg139"></arrow.to.target><lb/>ad 4: et propterea etiam ad triangulum GRE, erunt ut <lb/>2. ad 4; ad ipsum verò SRE erunt ut 2. ad 3, nempe LE

<pb pagenum="108"/>ad EB. Aequiponderant igitur ex puncto E hinc triangu<lb/>lum SRE, et inde quatuor praedicta triangula HZI, LXM, <lb/>NUO, <expan abbr="PTq.">PTque</expan> Eodem prorsus modo, si sub quatuor his <lb/>triang. alia fuerint in residuis portiunculis triang. ex or<lb/>dine descripta, ostendentur aequiponderare ex eodem pun<lb/>cto E, cum quodam triang. cuius vertex sit E, basis vero <lb/>contineat 3. quar. ipsius GS etc. Sed in nostro casu, cum <lb/>demonstratum sit prima duo triangula ZEX, UFT, aequi<lb/>ponderare triangulo ARE. Reliqua item quatuor triangula, <lb/>quorum unum est HZI. aequiponderare triangulo SRE; <lb/>Aequiponderabit tota simul figura ex praedictis triangulis <lb/>composita, triangulo AES, ex puncto E. Sed demonstra<lb/>tum fuit, eandem figuram praeponderare triangulo AEG; <lb/>necesse igitur est ut triangulum AEG minus sit trian<lb/>gulo AES; totum sua parte; Quod est impossibile. </s>

<s><margin.target id="marg139"></margin.target>Ex <lb/>lemmate 3.</s>

<s>Si vero ponamus praeponderare triangulum AEG figu<lb/>rae ABCFE. Esto; et sit excessus, quo praeponderat, spa<lb/>tium K. Accipiatnr GRE quarta pars trianguli AGE: et <lb/>iterum accipiatur GSE, quarta pars trianguli GRE; et hoc <lb/>

<arrow.to.target n="marg140"></arrow.to.target><lb/>semper fiat, donec veniatur ad aliquod triangulum GSE, <lb/>quod minus sit spatio K. Tunc enim triangulum ASE <lb/>adhuc praeponderabit figurae ABCFE. Sed eodem modo, <lb/>ut super demonstrabimus triangulum ipsum ASE aequi<lb/>ponderare alicui figurae rectilineae inscriptae intra figu<lb/>ram mixtam ABCFE. Necesse ergo iterum erit ut inscripta <lb/>figura rectilinea maior sit quàm figura mixta ABCFE, cui <lb/>ipsa inscribitur; pars suo toto. Quod est absurdum etc. </s>

<s><margin.target id="marg140"></margin.target>possibile est <lb/>per 1, De<lb/>cimi.</s>

<s>Aequiponderat ergo ex puncto E universa figura AB <lb/>CFG, quae sub curva parabolica, duabusqne tangentibus, <lb/>et linea ipsi diametro parallela continetur. Quod etc. </s>

<s>Quod assumptum est ita ostendemus: Nempe tangentem <lb/>AD transire per punctum E: hoc est, ita secare rectam <lb/>FG, ut pars FE, dupla sit reliquae EG. Secet enim AD <lb/>tangens rectam FG utcunque in E. Iam; cum parallelae <lb/>sint AG, BD, et aequales AE, ED; erunt aequales etiam <lb/>GE, EB, Sed aequales sunt FB, BE, ergo FE dupla est <lb/>ipsius EG. Ideo AD transit per illud E punctum, quod <lb/>ab initio dixeramus. </s>

<s><emph type="center"/>PROPOSITIO PRIMA<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Quaelibet parabola sesquitertia est trianguli eandem ipsi basim, et <lb/>eandem altitudinem habentis. </s>

<s>Esto parabola ABC, cuius dia<lb/>

<arrow.to.target n="fig104"></arrow.to.target><lb/>meter BD, iungaturque AB, BC, Dico <lb/>parabolam sesquitertiam esse trian<lb/>guli ABC, eandem cum ipsa basim, <lb/>et eandem altitudinem habentis, Du<lb/>cantur tangentes AE, CF, ad ba<lb/>sim: FH verò per verticem B; et AH sit ipsi diametro parallela. <lb/>concipiamusque parabolae diametrum erectam esse ad horizontem. <lb/>Iam secta HE in I, ita ut EI dupla sit ipsius IH, erit triangulum <lb/>HAE centralitèr appensum ad punctum I (habet enim centrum gra<lb/>vitatis in recta quae ex I ducitur parallela ad HA, et propterea ad <lb/>horizontem perpendiculari). Erit insuper figura mixta ABCFE centra<lb/>

<arrow.to.target n="marg141"></arrow.to.target><lb/>liter appensa ad punctum B. (quandoquidem habet centrum gravitatis <lb/>in diametro BD ad horizontem perpendiculari). Sed universa magni<lb/>

<arrow.to.target n="marg142"></arrow.to.target><lb/>tudo, composita ex dicto triangulo HAE, dictaque figura mixta, aequi<lb/>

<arrow.to.target n="marg143"></arrow.to.target><lb/>ponderat ex puncto E; erit ergo triangulum HAE ad figuram mixtam <lb/>ABCFE, ut reciprocè BE ad EI, nempe ut 3. ad 2. Propterca trapezium <lb/>AEFC sextuplum dicti trianguli, erit ad figuram mixtam ABCFE, ut <lb/>

<arrow.to.target n="marg144"></arrow.to.target><lb/>18. ad 2. et per conversionem rationis, ad parabolam erit ut 18, ad 16. <lb/>Qualium ergo partium parabula est 16, earum trapezium AEFC est 18. <lb/>et triangulum ABC 12. Quare parabola ad triangulum ABC erit ut 16, <lb/>

<arrow.to.target n="marg145"></arrow.to.target><lb/>ad 12 nempe sesquitertia. Quod erat ostendendum.<gap desc="/SM"/></s>

<s><margin.target id="marg141"></margin.target>Vide <lb/>Coroll, 1.</s>

<s><margin.target id="marg142"></margin.target>Lemmat. 4.</s>

<s><margin.target id="marg143"></margin.target>Lemm. <lb/>praeced.</s>

<s><margin.target id="marg144"></margin.target>ostendetur <lb/>inf.</s>

<s><margin.target id="marg145"></margin.target>ostenditur <lb/>inf.</s>

<s>Quod trapezium AEFC sextuplum sit trianguli HAE, <lb/>patet. Nam parallelogrammum HD, duplum est trianguli <lb/>HAB et propterea quadruplum trianguli HAE. ergò tra<lb/>pezium AEBD triplum erit trianguli HAE. totum verò <lb/>trapezium AEFC. sextuplum dicti trianguli HAE. Quod etc. </s>

<s>Cum autem trapezium AEFC sextuplum sit trianguli <lb/>HAE, erit sextuplum etiam trianguli EAB; et ideo tri<lb/>plum duorum EAB, BCF, Nempe ut 18. nd 6. Per con<lb/>versionem verò rationis, erit ad triangulum ABC ut 18. <lb/>ad 12. Quod etc. </s>

<s><emph type="center"/><emph type="italics"/>Lemma VI.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si duae parabolae utraqne duas tangentes ad basim habuerit; erunt <lb/>inter se trilinea mixta sub tangentibus, et curvis parabolicis contenta, <lb/>ut sunt ipsa triangula sub tangentibus compraehensa.<gap desc="/SM"/></s>

<s>Sint duae parabolae ABC, DEF. quarum utraqne duas <lb/>tangentes ad basim habeat AG, GC prioris, et DH, FH, <lb/>posterioris parabolae. Dico trilineum mixtum ABCG, ad <lb/>

<arrow.to.target n="fig105"></arrow.to.target><lb/>trilineum mixtum DEFH, esse ut triangulum AGC, ad <lb/>triangulum DHF. </s>

<s>Si enim non est ita: habebit alterum ex trilineis, puta <lb/>ABCG, ad reliquum, maiorem rationem quàm triangulum <lb/>AGC, ad DHF. Esto spatium K excessus, quo trilineum <lb/>ABCG, maius est quàm ut sit in ratione triangulorum. </s>

<s>Ducatur per verticem B tangens IL; demissisque ex <lb/>

<arrow.to.target n="marg146"></arrow.to.target><lb/>punctis I, et L lineis diametro parallelis (quae diametri <lb/>semiparabolarum erunt) ducantur tangentes OM, NP: et <lb/>ex punctis O; M; N, P, demittantur aliae diametri ut <lb/>supra; ducanturque aliae tangentes: et hoc semper fiat, <lb/>quousque reliquae simul omnes portiunculae, quae sub <lb/>tangentibus, et curva parabolica continentur; minores sint <lb/>

<arrow.to.target n="marg147"></arrow.to.target><lb/>spatio K. Tunc. N. universa figura tangentibus circum<lb/>

<arrow.to.target n="marg148"></arrow.to.target><lb/>septa, et in trilineo mixto ABCG inscripta, habebit adhuc <lb/>ad trilineum DEFH, rationem maiorem, quam triang. AGC, <lb/>ad triang. DHF. </s>

<s><margin.target id="marg147"></margin.target>Coroll I.</s>

<s>Inscribatur iam etiam in altero trilineo mixto DEFH.

<pb pagenum="111"/>figurae totidem laterum; ductis nimirùm tangentibus to<lb/>ties, quoties ductae fuerint in priori trilineo. </s>

<s>Quoniam verò est, ut triangulum IGL ad triangulum <lb/>QRH, ita duo simul triangula OIM, NLP, ad duo simul <lb/>UQS, TRZ. (sunt enim partes cum pariter multiplicibus in <lb/>

<arrow.to.target n="marg149"></arrow.to.target><lb/>eadem ratione). Et ut duo simul triangula OIM, NLP, ad <lb/>duo UQS, TRZ, ita quatuor triangula quae sunt infra <lb/>puncta O, M, N, P, ad quatuor illa quae sunt sob pun<lb/>ctis U, S, T, Z; ob eandem causam etc. Erunt etiam omnia <lb/>

<arrow.to.target n="marg150"></arrow.to.target><lb/>antecedentia simul (nempe figura inscripta in priori trilineo <lb/>mixto), ad omnia consequentia simul (nempe ad figuram <lb/>inscriptam in posteriori trilineo mixto) ut unum ad unum; <lb/>nempe ut IGL, ad QHR. Sive sumptis eorum quadruplis, <lb/>ut AGC ad DHF. Sedeadem inscripta figura habebat ad <lb/>trilineum DEFH maiorem rationem quam sit trianguli <lb/>AGC ad DHF. Minus ergo erit trilineum mixtum DEFH, <lb/>

<arrow.to.target n="marg151"></arrow.to.target><lb/>quam figura sibi in scripta; totum sua parte. Quod est <lb/>impossibile. Trilinea ergo sub tangentibus, et curvà para<lb/>bolica compraehensa, sunt inter se ut triangula sub ijsdem <lb/>tangentibus et basibus contenta. </s>

<s><margin.target id="marg149"></margin.target>25 quinti.</s>

<s><margin.target id="marg150"></margin.target>12 quinti.</s>

<s><margin.target id="marg151"></margin.target>X quinti</s>

<s><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Sit parabola ABC, cuius <lb/>

<arrow.to.target n="fig106"></arrow.to.target><lb/>diameter BD; et sit inscriptum <lb/>triangulum ABC. Dico parabo<lb/>lam sesquitertiam esse trian<lb/>guli ABC. </s>

<s>Ducantur duae tangentes <lb/>ad basim, quae sint AE, CE. <lb/>et FG tangat per verticem B. <lb/>Demissis deinde FI, GH dia<lb/>metro parallelis, ut sint dia<lb/>

<arrow.to.target n="marg152"></arrow.to.target><lb/>metri portionum AIB, BHC; <lb/>ducantur per I et H tangentes <lb/>LM, NO. </s>

<s>Erit ergo per Lemma praecedens, trilineum ABCE, ad trilineum <lb/>AIBF, ut est triangulum AEC, ad ABF. sive ad FBE. Idem verò trili<lb/>neum ABCE ad aliud trilineum BHCG, erit ut idem triangulum AEC,

<pb pagenum="112"/>ad triangulnm BGC; hoc est ad BGE, Coniunctim ergo, erit trilineum <lb/>

<arrow.to.target n="marg153"></arrow.to.target><lb/>ABCE ad duo trilinea AIBF, BHCG, ut triangulam AEC ad triangulum <lb/>FEG, nempe ut 4, ad unum, et dividendo, erit triangulum FEG ad duo <lb/>trilinea AIBF, BHCG, ut 3, ad unum. Trapezium autem AFCG, ad eadem <lb/>trilinea erit ut 9. ad unum; et per conversionem rationis, ad parabolam <lb/>erit ut 9. ad 8. et ad triangulum ABC ut 9. ad 6, Qualium ergò factium <lb/>parabola est 8, talium triangulum ABC est 6. Constat ergo parabolam <lb/>inscripti sibi trianguli sesquitertiam esse. Quod erat etc.<gap desc="/SM"/></s>

<s><margin.target id="marg153"></margin.target>24 quinti</s>

<s><emph type="center"/><emph type="italics"/>Lemma VII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si in parabola inscribatur triangulum: eandem habens cum parabola <lb/>basim, eandemque altitudinem. Inscribantur etiam pariter et in reliquis <lb/>portionibus duo alia triangula: Erit triangulum primò inscriptum, octu<lb/>plum alterutri posteriùs inscripti trianguli.<gap desc="/SM"/></s>

<s>Demonstratur hoc lemma ab Archimede prop. 21. De <lb/>Quadratura parabolae. </s>

<s><emph type="center"/><emph type="italics"/>Lemma VIII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si in parabola evidenter inscribatur figura ex triangulis constans. <lb/>Tam bina ipsius triangula (si prout sibi mutuò respondent ita sumantur) <lb/>quàm etiam tota inscripta figura aequiponderabit ex puncto medio <lb/>basis ipsius parabolae.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuios diameter sit BD; et inscripta <lb/>statuatur figura, ita ut diameter ad horizontem sit perpen<lb/>

<arrow.to.target n="fig107"></arrow.to.target><lb/>dicularis. Sectà deinde utraque AD, DC bifariam in EF; <lb/>iterunque sectis partibus bifariam in G, H, I, L. etc. </s>

<s>Ducantur GM, EN, HO, IP, FQ, LR, etc. Parallelae ad <lb/>diametrum. </s>

<s>Inscribaturque in parabola figura AMNOBPQRC. (quae <lb/>dicitur evidentèr inscribi). Dico triangula quae figuram

<pb pagenum="113"/>inscriptam componunt, si bina, et prout sibi mutuò respon<lb/>dent ita sumantur, aequiponderare ex puncto D. Praeterea <lb/>universam figuram inscriptam, ex ipsis triangulis compo<lb/>sitam, ab eodem puncto D, aequiponderare. </s>

<s>Sumantur enim duo triangula sibi mutuo respondentia, <lb/>puta NOB, BPQ, quae inter se aequalia erunt; cum trian<lb/>

<arrow.to.target n="marg154"></arrow.to.target><lb/>gula ANB, BQC suboctupla. sint eiusdem trianguli ABC; <lb/>ipsa vero NOB, BPQ, suboctupla sint aequalium triangu<lb/>lorum ANB, BQC. Habebunt insuper centra gravitatis in <lb/>rectis OS, PT, quae quidem ab angulis O, P, ducuntur ad <lb/>

<arrow.to.target n="marg155"></arrow.to.target><lb/>puncta media basium, NB, BQ, Cum verò OSH, PTI, rcctae <lb/>ad horizontem positae sint perpendiculares, erunt praedicta <lb/>triangula NOB, BPQ, centralitèr appensa ex punctis H, <lb/>

<arrow.to.target n="marg156"></arrow.to.target><lb/>et I. Quamòbrem ab aequalibus distantijs HD, DI, aequi<lb/>

<arrow.to.target n="marg157"></arrow.to.target><lb/>ponderabunt. Et sic de reliquis figurae triangulis. Quod <lb/>erat primò propositum. </s>

<s><margin.target id="marg155"></margin.target>13 primi ae<lb/>quiponder.</s>

<s><margin.target id="marg156"></margin.target>4. secundi <lb/>eiusd.</s>

<s><margin.target id="marg157"></margin.target>8. primi <lb/>eiusd.</s>

<s>Figura autem universa evidentèr inscripta componitur <lb/>ex partibus aequiponderantibus à puncto D; quarè etiam <lb/>ipsa ex D puncto aequiponderabit. Quod erat ostenden<lb/>dum, etc. </s>

<s><emph type="center"/><emph type="italics"/>Lemma IX.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Positis ijsdem. Si à parabola dematur universa figura evidentêr in<lb/>scripta, etiàm omnia segmenta parabolica, quae circumrelinquuntur, <lb/>ex puncto D. aequiponderabunt.<gap desc="/SM"/></s>

<s>Repetita enim eadem figura demonstratum est figuram <lb/>inscriptam aequiponderare ex puncto D. Ergo figura in<lb/>scripta centrum gravitatis habet in perpendiculo horizon<lb/>tali DB. (per 4. suppositionem) sed etiam parabola cen<lb/>trum gravitatis habet in diametro DB, (per 4. secundi <lb/>aequiponderantium) ergo centrum omnium reliquorum se<lb/>gmentorum erit in diametro DB. Quare ex puncto D. <lb/>aequiponderabunt. per 3. suppositionem. Quod etc. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium;<emph.end type="italics"/><emph.end type="center"/></s>

<s>Constat etiam eodem prorsus argumento, reliquum figu<lb/>rae evidentèr inscriptae, detracto priùs triangulo ABC,

<pb pagenum="114"/>aequiponderare ex puncto D. Item reliquum parabolae, <lb/>dempto triangulo ABC, aequiponderare ex D. </s>

<s><emph type="center"/><emph type="italics"/>Lemma X.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si ex parabola auferatur dimidium trianguli inscripti, tota reliqua <lb/>figura mixta aequiponderabit ex puncto basis reliqui trianguli, in quo <lb/>sic ea dividitur, ut pars ad curvam terminata quadrupla sit illius, quae <lb/>terminatur ad diametrum;<gap desc="/SM"/></s>

<s>Esto parabola <lb/>ABC inversa; eius<lb/>que diameter BD <lb/>ita statuatur ut <lb/>ad horizontem sit <lb/>perpendicularis; <lb/>Detractoque semi<lb/>triangulo inscripto <lb/>DBC; secetur AD <lb/>basis reliqui semitrianguli, in quinque partes aequales; <lb/>quarum una sit DE. Dico huiusmodi figuram ex puncto E <lb/>suspensam, aequiponderare. </s>

<s>Nisi enim aequiponderet; cum recta AD sit libra, cuius <lb/>fulcrum est in E, et magnitudo AFBGC, constans ex dua<lb/>

<arrow.to.target n="marg158"></arrow.to.target><lb/>bus portionibus parabolicis, appensa sit ad punctum D, <lb/>secundum centrum gravitatis ipsius: Reliquum autem <lb/>triangulum ABD altera magnitudo appensa sit ad pun<lb/>ctum H (sumpta DH tertia parte totius DA); Altera ex <lb/>his duabus magnitudinibus praeponderare necesse erit. </s>

<s><margin.target id="marg158"></margin.target>Coroll. <lb/>Lem. 9.</s>

<s>Ponamus primò praeponderare duas portiones AFB, <lb/>BGC; et sit excessus quo praeponderant aequalis spatio K. </s>

<s>Inscribatur evidentèr intrà duas portiones parabolicas <lb/>figura multilatera, ita ut omnia simul segmenta parabolica <lb/>circumrelicta minora sint spatio K. Tunc enim praeponde<lb/>rabit adhuc figura inscripta multilatera AIFLBMGNCBA. </s>

<s>Accipiatur DO quarta pars totius DB; et ductà AO, <lb/>non solum triangulum ABO, aequiponderabit sibi ipsi ex <lb/>puncto H; sed etiam quodcumque aliud triangulum ha<lb/>bens verticem in A et basim in recta DB, sibi ipsi aequi<lb/>ponderabit ex puncto eodem H. </s>

<s>Iam sic: Qualium partium AD est 15, DH est 5 et DE <lb/>est 3. Ergo DE ad EH erit ut 3. ad 2. Cum autem demon<lb/>

<arrow.to.target n="marg159"></arrow.to.target><lb/>stratum sit duo triangula AFB, BGC, aequiponderare ex <lb/>puncto D; triangulum verò BOA, ex puncto H; et cum <lb/>duo praedicta triangula sint ad totum triangulum ABD <lb/>ut duo ad 4.; erunt eadem ad triangulum ABO, ut 2. ad 3.; <lb/>

<arrow.to.target n="marg160"></arrow.to.target><lb/>nempe ut HE ad ED reciprocè. Quamobrem duo illa trian<lb/>gula AFB, HGC, cum triangulo ABO, aequiponderabunt <lb/>

<arrow.to.target n="marg161"></arrow.to.target><lb/>suspensa ex puncto E. </s>

<s><margin.target id="marg161"></margin.target>6. Aequi<lb/>pond.</s>

<s>Sumatur deinde DP quarta pars ipsius DO; ducaturque <lb/>AP. Iam; quia duo triangula FLB, BMG aequiponderant <lb/>ex D itemque duo AIF, GNC, aequiponderant ab eodem <lb/>

<arrow.to.target n="marg162"></arrow.to.target><lb/>puncto D; omnia simul praedicta quatuor triangula aequi<lb/>ponderabunt ex puncto D; Quatuor autem praedicta trian<lb/>gula ad triangulum AFB sunt ut duo ad 4. Sunt autem <lb/>AFB, AOD, subquadrupla eiusdem trianguli ABD, et pro<lb/>

<arrow.to.target n="marg163"></arrow.to.target><lb/>pterea ad triangulum AOP, erunt ut 2. ad 3. nempe ut <lb/>HE ad ED, reciprocè. Aequiponderant ergo quatuor illa <lb/>

<arrow.to.target n="marg164"></arrow.to.target><lb/>triangula cum triangulo AOP, ex puncto E. Ergo universa <lb/>simul figura evidenter inscripta AIFIB MGNCBA aequi<lb/>ponderat triangulo ABP. Sed eadem praeponderabat trian<lb/>gulo ABD, Minus ergò est triangulum ABD quam triangu<lb/>lum ABP. totum sua parte: quod est impossibile. </s>

<s><margin.target id="marg164"></margin.target>6. Aequi<lb/>pond.</s>

<s>Ponamus deinde praeponderare triangulum ABD; et <lb/>sit excessus quo praeponderat aequalis spatio K. </s>

<s>Accipiatur AOD quarta pars totius trianguli ABD; <lb/>iterumque sumatur APD quarta pars trianguli AOD; et <lb/>hoc semper fiat donec veniatur ad aliquod triangulum, <lb/>puta APD, quod minus sit spatio K. Tunc enim reliquum <lb/>ABP adhuc praeponderabit duabus portionibus parabolicis <lb/>AFB, BGC. Sed idem triangulum ostendetur (eodem pror<lb/>sus modo ut supra) aequiponderare alicui figurae intrà <lb/>parabolicas portiones inscriptae; necesse igitur erit quòd <lb/>portiones parabolicae minores sint quàm figura illa sibi <lb/>inscripta; totum sua parte. Quod est impossibile. Aequi<lb/>ponderant ergo parabola inversa (dempto semitriangulo <lb/>inscripto) ex puncto quod dictum est, Quod erat osten<lb/>dend. etc. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hinc inferre possumus, quòd si ex puncto E, recta <lb/>ducetur diametro aequidistans, centrum praedictae figurae <lb/>erit in producta. Siquidem figura ex puncto E aequipon<lb/>derat, et linea ex E ducta aequidistans diametro, est ad <lb/>horizontem perpendicularis. Posset etiam demonstrari, nisi <lb/>extrà rem esset, centrum praedictae figurae dictam paral<lb/>lelam ita secare, ut pars quae terminatur ad curvam sit <lb/>ad reliquam ut 11 ad 12. </s>

<s><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem sibi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, ex qua demptum <lb/>

<arrow.to.target n="fig108"></arrow.to.target><lb/>sit dimidium trianguli inscripti: Sum<lb/>ptaque DH, quae sit tertia pars totius <lb/>DA et DE quinta pars eiusdem; si pa <lb/>rabola huiusmodi statuatur inversa, ita <lb/>ut diameter sit horizonti perpendicu<lb/>

<arrow.to.target n="marg165"></arrow.to.target><lb/>laris, aequiponderabit figura ex puncto <lb/>E, Sed triangulum ABD appensum est <lb/>secundum centrum gravitatis ad punctum H librae HD. Duae autem <lb/>parabolicae portiones residuae appensae sunt secundum centrum gra<lb/>vitatis ad punctum D; Ergo triangulum ABD, ad duas reliquas por<lb/>tiones erit ut DE ad EH, reciprocè, nempe ut 3. ad 2: Sumptis <lb/>autem antecedentium duplis erit totum inscriptum triangulum ad reli<lb/>quas portiones ut 6. ad 2. Convertendo igitur, et componendo, erit <lb/>ipsa parabola ad inscriptum sibi triangulum ut 8, ad 6 Nempe sexqui<lb/>tertia. Quod etc.<gap desc="/SM"/></s>

<s><margin.target id="marg165"></margin.target>praeced. <lb/>Lem.</s>

<s>Libet hic demonstrare Lemma Lucae Valerij, nostro <lb/>tamen modo, diversisque penitùs Mechanicae principijs. <lb/>Ipse enim utitur propositione illa, quà ante demonstra<lb/>verat centrum gravitatis hemispherij. Nos autem simili <lb/>ratione, ac in praecedentibus, demonstrabimus et Lemma, <lb/>et ipsam Valerij conclusionem. </s>

<s><emph type="center"/><emph type="italics"/>Lemma XI.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Omnis semiparabola aequiponderat ex puncto basis, in quo sic ea divi<lb/>ditur ut pars ad curvam terminata sit ad reliquam ut quinque ad tria:<gap desc="/SM"/></s>

<s>Esto semiparabola ABC, cuius <lb/>

<arrow.to.target n="fig109"></arrow.to.target><lb/>diameter AB statuatur ad horizon<lb/>tem perpendicularis: Sectà deinde <lb/>AC in F, ita ut CF ad FA, sit <lb/>ut 5. ad 3. velut 15. ad 9. Dico <lb/>figuram ex puncto F suspensam <lb/>aequiponderare. </s>

<s>Secetur iterùm AC bifariàm <lb/>in D, et demissà DE parallela <lb/>diametro, erit ipsa DE diameter <lb/>parabolae BEC. Sumatur iam AI tertia pars totius AC. <lb/>Qualium igitur partium AC est 24. talium AD est 12. <lb/>AF 9. et AI 8, Ergo DF tres, et FI una. Iam si figura <lb/>non aequiponderat ex puncto F; Cum ID sit libra quae<lb/>dam cuius fulcrum est F, et ad punctum I appensum sit <lb/>

<arrow.to.target n="marg166"></arrow.to.target><lb/>triangulum ABC; ex puncto verò D appensa sit parabola <lb/>BEC; altera ex his figuris praeponderabit. </s>

<s><margin.target id="marg166"></margin.target>ex supp. <lb/>quinta.</s>

<s>Ponamus primò praeponderare parabolam BEC, sit ex<lb/>cessus quo praeponderat aequalis spatio K. </s>

<s>Inscribatur evidentèr intra parabolam BEC figura re<lb/>ctilinea, ita ut omnes simul residuae portiunculae quibus <lb/>parabola excedit inscriptam sibi figuram, minores sint <lb/>spatio K. Manifestum est, quod inscriptà evidentèr figura <lb/>adhuc praeponderabit triangulo ABC. </s>

<s>Accipiatur AHC quarta pars totius trianguli ABC. </s>

<s>Cum autem DE sit ad horizontem perpendicularis, et <lb/>

<arrow.to.target n="marg167"></arrow.to.target><lb/>triangulum BEC habeat centrum gravitatis in recta GE; <lb/>erit dlctum triangulum appensum ad D. Triangulum verò <lb/>BHC appensum ad punctum I; quandoquidem AI tertia <lb/>pars est totius AC, ipsa vero AB perpendicularis ad hori<lb/>zontem constituta est, Quoniàm autem BEC ad ABC est <lb/>ut unum ad 4., erit idem BEC ad HBC ut unum ad 3. <lb/>

<arrow.to.target n="marg168"></arrow.to.target><lb/>nempe reciprocè ut IF ad FD, Aequiponderant ergo ex <lb/>puncto F, triangula BEC et HBC. </s>

<s><margin.target id="marg167"></margin.target>13 primi ae<lb/>quipond.</s>

<s>Sumatur iterum ALC quarta pars trianguli AHC; Et <lb/>quoniam duo triangula BME, ENC aequiponderant ex G <lb/>

<arrow.to.target n="marg169"></arrow.to.target><lb/>(uti demonstratum est) aequiponderabunt etiam suspensa <lb/>ex D. Cum autèm duo dicta triangula BMC, ENC, sint ad <lb/>

<arrow.to.target n="marg170"></arrow.to.target><lb/>triangulum BEC, sive ad ipsi aequale AHC, ut unum ad 4.; <lb/>erunt ad LHC, ut unum ad 3; nempe reciprocè ut IF <lb/>ad FD. Aequiponderant ergo ex puncto F. duo triangula <lb/>BME, ENC, cum triangulo LHC. Figura ergò universa <lb/>evidentèr inscripta intrà parabolam BEC. aequiponderat <lb/>ex puncto F. cum triangulo LBC. Sed eadem praeponde<lb/>rabat triangulo ABC. Necesse igitur est quòd triangulum <lb/>ABC minus sit triangulo LBC. totum sua parte. Quod est <lb/>absurdum. </s>

<s>Ponamus deinde praeponderare triangulum ABC, et <lb/>sit excessus quo praeponderat aequalis spatio K. Sumatur <lb/>AHC quarta pars totius trianguli ABC. Iterùm sumatur <lb/>ALC quarta pars trianguli AHC, Et hoc semper fiat donec <lb/>ventum fuerit ad aliquod triangulum, puta ALC, minus <lb/>spatio K. Tunc enim triangulum LBC adhuc praeponde<lb/>rabit parabolae BEC. Sed eodem modo, quo supra, demon<lb/>strabimus dictum triangulum LBC aequiponderare cuidam <lb/>figurae evidentèr inscriptae intrà parabolam BEC. Unde <lb/>sequeretur ipsam parabolam BEC minorem esse aliqua <lb/>figurà sibi inscriptà; totum videlicet sua parte. Quod est <lb/>absurdum. Aequiponderat ergo semiparabola, uti dictum <lb/>est constituta, et ex puncto F suspensa. Quod etc. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hinc patet, quòd (cum semiparabola aequiponderet ex <lb/>puncto F.) si ab F demittatur recta ad horizontem per<lb/>

<arrow.to.target n="marg171"></arrow.to.target><lb/>pendicularis, in hac demissà erit centrum gravitatis semi<lb/>parabolae; aliàs enim non aequiponderaret ex F. Verùm <lb/>quoniam etiam diameter parabolae ad horizontem perpen<lb/>dicularis constituta est, concludemus; quòd recta quae ex <lb/>puncto F ducitur diametro aequidistans, transit per cen<lb/>trum semiparabolae. </s>

<s><margin.target id="marg171"></margin.target>Suppositio 4.</s>

<s><emph type="center"/>PROPOSITIO IV.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, cuius diameter <lb/>

<arrow.to.target n="fig110"></arrow.to.target><lb/>BD, triangulum verò inscriptum sit ABC, <lb/>Dico parabolam dicti trianguli esse ses<lb/>quitertiam. </s>

<s>Sumatur, qualium partium tota DC <lb/>est 24. talium DE 8.; DF 9; et DG, 12. <lb/>Eritque earundem EF una, et FG tres, Ductis verò EH, FI, GL, dia<lb/>

<arrow.to.target n="marg172"></arrow.to.target><lb/>metro parallelis, erit in EH centrum trianguli BDC; in FI centrum <lb/>semiparabolae DBMC, et in GL centrum portionis BMC, <lb/>

<arrow.to.target n="marg173"></arrow.to.target></s>

<s><margin.target id="marg172"></margin.target>13 primi ae<lb/>quip.</s>

<s><margin.target id="marg173"></margin.target>Lem. <lb/>praeced.</s>

<s>Ponatur centrum trianguli esse punctum quodcumque H. Item cen<lb/>trum semiparabolae esse punctum quodcumque I (quamquàm huius</s>

<s>

<arrow.to.target n="marg174"></arrow.to.target><lb/>modi puncta extrà ipsas figuras ubicunque libuerit sumantur, tamen <lb/>vero semper eodem modo inferemus.) iunctà deinde HI, et productà, in <lb/>

<arrow.to.target n="marg175"></arrow.to.target><lb/>ipsa HI erit centrum portionis parabolicae BMC; quod cum sit etiam <lb/>in recta GM producta, necessariò erit in communi concursu L. Para<lb/>bola ergò BMC ad triangulum DBC erit reciprocè ut HI ad IL, hoc <lb/>est, ut EF ad FG, nempe ut unum ad 3. Componendo ergo, sumptisque <lb/>duplis, erit tota parabola ad totum triangulum ut 4. ad 3. Nempe <lb/>sesquitertia. Quod erat propositum etc.<gap desc="/SM"/></s>

<s><margin.target id="marg174"></margin.target>4. sec. <lb/>aequip.</s>

<s><margin.target id="marg175"></margin.target>8. primi ae<lb/>quip.</s>

<s>Poterat haec demonstratio produci etiam hoc modo, <lb/>praemisso hoc Lemmate. </s>

<s>Si parabola ad extremum basis lineam habuerit dia<lb/>metro parallelam, ed diametri quadruplam, ductoque tertio <lb/>latere, compleatur triangulum. </s>

<s>Universa haec figura aequiponderabit ex puncto tertij <lb/>lateris, in quo sic dividitur ut pars ad <lb/>

<arrow.to.target n="fig111"></arrow.to.target><lb/>curvam terminata sit ad reliquam ut <lb/>5. ad 3. </s>

<s>Esto parabola ABC, cuius diameter <lb/>DB statuatur ad horizontem perpen<lb/>dicularis; considereturque ipsa para<lb/>bola inversa: Tum ad alterutrum basis <lb/>AC estremum, puta ad punctum A, <lb/>adiungatur recta AE, diametro aequi<lb/>distans, et ipsius diametri quadrupla. <lb/>Ducto deinde tertio latere EC trian<lb/>guli EAC, secetur in F, ita ut CF. ad FE sit ut 5. ad 3.

<arrow.to.target n="marg176"></arrow.to.target><lb/>Dico huiusmodi figuram ex puncto F aequiponderare. Quo<lb/>niam enim CE ordinatim applicatur ad diametrum AE; <lb/>erit tota figura EABC semiparabola. Ergo ijsdem ra<lb/>tionibus, eodemque progressu quo usi sumus in Lem<lb/>mate II ostendemus totam figuram aequiponderare ex F. <lb/>Sumatur iam EI octo earum partium, qualium tota EC <lb/>est 24. et EL 12. et EF 9, Eritque IF earundem una, et <lb/>FL 3. Ergo cum parabola ABC pendeat ex puncto L, ap<lb/>pensa ad ipsum secundum centrum gravitatis; triangulum <lb/>verò AEC ex puncto I; erit parabola ABC ad triangu<lb/>

<arrow.to.target n="marg177"></arrow.to.target><lb/>lum AEC ut reciprocè IF ad FL, nempe ut unum ad 3.; <lb/>sive ut 4. ad 12. et propterea ad triangulum ABC ut 4. <lb/>ad 3. etc. Est enim ABC quarta pars ipsius ACE etc. <lb/>Constat ergo parabolam sesquitertiam esse inscripti sibi <lb/>trianguli. </s>

<s><margin.target id="marg176"></margin.target>Ostendetur <lb/>infra.</s>

<s><margin.target id="marg177"></margin.target>ob commu<lb/>nem basim <lb/>AC altitu<lb/>dinem verò <lb/>in ratione <lb/>quadrupla.</s>

<s>Quod assumptum est, nempe rectam CE ordinatim ap<lb/>plicari ad diametrum AE, ostendemus hoc modo. </s>

<s>Si enim non est ordinatim applicata CE, applicetur or<lb/>dinatim CM; eritque MABC semiparabola; et quia sunt <lb/>

<arrow.to.target n="marg178"></arrow.to.target><lb/>aequales AD, DC, erit MC secta bifariam in N. Ergo MA <lb/>sesquitertia est ipsius NB; sed etiam EA ob constructio<lb/>

<arrow.to.target n="marg179"></arrow.to.target><lb/>nem sesquitertia est ipsius LB; ergo et reliqua EM sesqui<lb/>tertia est reliquae LN; et EC sesquitertia ipsius CL, quod <lb/>est impossibile. Est enim dupla, non autem sesquitertia. <lb/>Quare nulla alia praeter CE ex puncto C ordinatim ap<lb/>plicari potest ad diametrum AE. Quod etc. </s>

<s><margin.target id="marg178"></margin.target>ob semipara<lb/>bolam.</s>

<s><margin.target id="marg179"></margin.target>19 quinti.</s>

<s><emph type="center"/>PROPOSITIO V.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, eandemque <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, <lb/>

<arrow.to.target n="fig112"></arrow.to.target><lb/>cuius diameter BD, trian<lb/>gulum inscriptum ABC; <lb/>Dico parabolam esse ses<lb/>quitertiam trianguli AB <lb/>C. tibi inscripti. </s>

<s>Si enim ita non est, <lb/>neque triangulum ABC <lb/>erit triplum duarum si-

<pb pagenum="121"/>mul reliquarum portionum AEB, BFC. Sed erit vel magis quàm triplum, <lb/>sive minus quàm triplum. </s>

<s>Sit primò minus quam triplum, eruntque duae reliquae portiones <lb/>magis quàm tertià pars trianguli ABC. Estò excessus aequalis spatio K, <lb/>et inscribantur intrà portiones primùm trian gula AEB, BFC; iterumque <lb/>in reliquis portiuncolis quatuor triangula AGE, EHB, BIF, FLC; deinde <lb/>octo etc. et hoc semper donec excessus portionum supra inscriptas <lb/>evidentèr figuras sit minor spatio K. Tunc. <emph type="italics"/>n<emph.end type="italics"/> erunt inscriptae figurae <lb/>adhuc maiores quàm tertia pars trianguli ABC. </s>

<s>Sumatur iam triangulum ABM quarta pars totius trianguli ABC. <lb/>

<arrow.to.target n="marg180"></arrow.to.target><lb/>Et quoniam ABC quadruplum est tam trianguli ABM, quam triangu<lb/>lorum AEB, BFC simul sumptorum, aequale erit triangulum ABM, duo<lb/>bus simul triangulis AEB, BFC; et propterea triangulum MBC triplum <lb/>erit duorum simul triangulorum AEB, BFC. </s>

<s>Accipiatur iterùm triangulum ABN quarta pars totius trianguli ABM. <lb/>Cum ergò ABM quadruplum sit ipsius ABN; et duo AEB, BFC, qua<lb/>

<arrow.to.target n="marg181"></arrow.to.target><lb/>drupla sint quatuor simul subsequentium triangulorum AGE, EHB, BIF, <lb/>FLC; cumque antecedentia sint aequalia, aequalia erunt etiam conse<lb/>quentia; et propterea cum triangulum NBM triplum sit trianguli ABN, <lb/>triplum etiam erit idem triangulum NBM. quatuor simul triaugulorum, <lb/>AGE, EHB, BIF, FLC. Et ut unum ad unum ita omnia simul ad omnia. <lb/>

<arrow.to.target n="marg182"></arrow.to.target><lb/>Quare totum simul triangulum NBC, triplum erit figurarum evidentèr <lb/>intra portiones inscriptarum. Sed triangulum ABC minus erat quam <lb/>triplum earundem; Ergo ABC minus est quàm NBC totum sua parte. <lb/>Quod est absurdum etc. </s>

<s><margin.target id="marg182"></margin.target>12 Quinti.</s>

<s>Ponamus deinde triangulum ABC esse plus quam triplum duarum <lb/>simul reliquarum portionum. Esto; et excessui, quo est magis quàm <lb/>triplum, aequale sit spatium K. </s>

<s>Accipiatur ABM quarta pars totius trianguli ABC. Iterum sumatur <lb/>ABN quarta pars ipsius ABM. Et hoc semper fiat donec veniatur ad <lb/>aliquod triangulum, puta ABN, quod minus sit spatio K. Eritque adhuc <lb/>triangulum NBC magis quam triplum duarum portionum. Sed eàdem <lb/>prorsus ratione, et ordine quo supra, ostendemus triangulum NBC <lb/>triplum esse cuiusdam figurae intrà portiones evidentèr inscriptae; <lb/>necesse igitur erit quòd portiones ipsae minores sint quàm figurae <lb/>intrà ipsas descriptae: Totum sua parte. quod est impossibile. </s>

<s>Triangulum ergo ABC duarum reliquarum portionum triplum est; <lb/>et componendo; et per conversionem rationis parabola ad suum <lb/>triangulum erit ut 4. ad 3. Nempe sesquitertia. Quod erat proposi<lb/>tum etc.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Lemma XII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola tres tangentes habuerit, duas ad basim, tertiam verò <lb/>per verticem: Erit triangulum sub tangentibus compraehensum, reli<lb/>quae figurae (demptà parabolà) triplum.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuius diameter BD; tangentes ad <lb/>basim AE, CE; per verticem verò FBG. </s>

<s>Dico triangulum FEG, sub tangentibus compraehensum <lb/>reliquae figurae mixtae ABCGF (dempta scilicet parabola) <lb/>triplum esse. </s>

<s>Si enim non est triplum, erit certè vel magis, vel <lb/>minùs quàm triplum. </s>

<s>Sit primò minùs quàm triplum; eritque reliqua figura <lb/>mixta ABCGF, magis quàm tertia pars trianguli FEG. <lb/>Sit excessus K. Ducanturque per vertices abscissarum <lb/>portionum tangentes HI, LM; Iterumque per vertices sub<lb/>sequentium portionum, tangentes agantur NO, PQ, RS, TU. <lb/>et hoc semper; donec excessus figurae mixtae ABCGF, <lb/>supra figuram ex triangulis constantem NOPQRSTUGF, <lb/>minus aliquando relinquatur quàm spatium K. Tunc enim <lb/>erit adhuc figura ex triangulis inscripta maior quàm tertia <lb/>pars trianguli FEG. </s>

<s>Accipiatur triangulum FEI quarta pars trianguli FEG; <lb/>eritque triangulum FEI aequale duobus simul triangulis <lb/>HFI, LGM: (cum tam ista duo, quàm illud solum, sub<lb/>

<arrow.to.target n="marg183"></arrow.to.target><lb/>quadrupla sint eiusdem trianguli FEG) Ergò triangul. IEG <lb/>triplum erit duorum simul triangulorum HFI, LGM. </s>

<s>Sumatur iterùm triangulum FEX quarta pars ipsius <lb/>FEI. Cumque sit FEI quadruplum trianguli FEX, duo <lb/>verò triangula HFI, LGM quadrupla sint quatuor simul <lb/>triangulorum NHO, PIQ, RLS, TMU, et antecedentia ae<lb/>

<arrow.to.target n="marg184"></arrow.to.target><lb/>qualia; etiam consequentia aequalia erunt; eritque trian<lb/>gulum FEX aequale quatuor praedictis triangulis NHO, <lb/>PIQ, RLS, TMU; et propterea XEI triplum erit eorumdem <lb/>quatuor triangulorum. Cumque sit ut unum ad unum, ita <lb/>

<arrow.to.target n="marg185"></arrow.to.target><lb/>omnia ad omnia: erit totum simul triangulum XEG tri<lb/>plum universae figurae rectilineae intrà figuram mixtam <lb/>inscriptae. Sed eiusdem figurae inscriptae triangulum FEG <lb/>minus erat quam triplum; necesse igitur est ut triangu<lb/>lum FEG minus sit quàm ipsum XEG totum videlicet sua <lb/>parte. Quod est impossibile. </s>

<s><margin.target id="marg185"></margin.target>12 Quinti.</s>

<s>Ponamus deinde triangulum FEG esse plus quàm tri<lb/>plum reliquae figurae mixtae demptà parabolà. </s>

<s>Esto et sit excessus aequalis spatio K. accinietur trian<lb/>gulum FEI quarta pars totius FEG: et iterum sumatur <lb/>triangulum FEX quarta pars trianguli FEI: et hoc fiat <lb/>semper donec veniatur ad aliquod triangulum, puta FEX, <lb/>quod minus sit spatio K. Eritque triangulum XEG adhuc <lb/>maius quàm triplum reliquae figurae mixtae ABCGF. Sed <lb/>eadem penitùs ratione, atque ordine ut supra, ostendemus <lb/>triangulum XEG esse triplum cuiusdam figurae intrà figu<lb/>ram mixtam ABCGF. descriptae. Necesse ergò erit, ut <lb/>figura mixta ABCGF minor sit quam aliqua figura sibi <lb/>inscripta; totum sua parte. Quod est absurdum. </s>

<s>Si ergò parabola tres tangentes habuerit, ut positum <lb/>est, erit triangulum sub tangentibus contentum, reliquae <lb/>figurae, dempta parabola, triplum. Quod erat proposi<lb/>tum etc. </s>

<s><emph type="center"/>PROPOSITIO VI.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertiâ est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, cuius diameter <lb/>

<arrow.to.target n="fig113"></arrow.to.target><lb/>BD; duae tangentes AE, CE ad basim, <lb/>et tertia FBG per verticem. Dico para<lb/>bolam sesquitertiam esse inscripti sibi <lb/>trianguli ABC. </s>

<s>Triangulum enim FEG ad duo tri<lb/>linea mixta AFB, BGC per praecedens <lb/>Lemma, est ut 3. ad nnum. Ergò trape<lb/>zium. AFGC (cum triplum sit trianguli <lb/>FEG) ad duo eadem trilinea mixta erit ut 9. ad unum. Et ad para<lb/>bolam erit (per conversionem rationis) ut 9. ad 8. et ad triangulum <lb/>ABC, erit ut 9. ad 6. Qualium ergo partium parabola est octo, talium <lb/>triangulum ABC est 6; Quare parabola ad inscriptum sibi triangulum <lb/>est ut 8. ad 6. nempe sesquitertia. Quod erat etc.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Lemma XIII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola tres tangentes habuerit; duas ad basim, tertiam verò <lb/>per verticem, et ex universa figura dempta sit parabola, dimidiumque <lb/>trianguli sub tangentibus contenti. Reliqua figura aequiponderabit ex <lb/>quodam puncto, quod ita integram tangentem lateralem dividit ut pars <lb/>quae ad contactum curvae terminatur sit ad reliquam ut 9. ad unum.<gap desc="/SM"/></s>

<s>Esto parabola A <lb/>

<arrow.to.target n="fig114"></arrow.to.target><lb/>BC cuius diameter <lb/>BD concipiatur ad <lb/>horizontem perpendi<lb/>cularis; sintque duae <lb/>tangentes ad basim <lb/>AE, CD verticalis <lb/>verò tangens EBF. <lb/>Sectà deinde laterali <lb/>CD in H, ita ut CH <lb/>ad HD sit ut 9. ad <lb/>unum; Dico figuram <lb/>huiusmodi (demptà <lb/>parabolà, et semi<lb/>triangulo verticali EBD) aequiponderare ex puncto H. </s>

<s>Sumatur DI quinque partium earum, quarum DF est <lb/>15. sive quarum DH est 3. Eritque DH ad HI ut 3. ad 2. </s>

<s>Cum autem BD sit ad horizontem perpendicularis, por<lb/>tiones mixtae ABE, BCF, appensae erunt secundum cen<lb/>trum gravitatis ad punctum B, sive ad punctum D. Trian<lb/>gulum verò BDF ob eandem causam, et eodem modo <lb/>pendebit centraliter ex puncto I (quandoquidem FI dupla <lb/>est ipsius ID; et ipsa DB ad horizontem perpendicularis). <lb/>Iam si istae magnitudines non aequiponderant ex H pun<lb/>cto librae DI, altera ipsarum praeponderabit. Esto; et <lb/>praeponderent primò duae portiones mixtae ABE, BCF. <lb/>Sitque excessus quo praeponderant aequalis spatio K. </s>

<s>Inscribatur intrà mixtas portiones figura ex tangenti<lb/>bus, ut iam saepè factnm est. Donec excessus portionum <lb/>suprà figuram rectilineam inscriptam minur sit spatio K. <lb/>Tunc enim figura inscripta adhuc praeponderabit trian<lb/>gulo BDF. </s>

<s>Accipiatur triangulum DFG quarta pars totius trian<lb/>guli DFB; eritque triangulum DFG aequale triangulo <lb/>NFO (cum ambo sint subquadrupla eiusdem trianguli DFB) <lb/>

<arrow.to.target n="marg186"></arrow.to.target><lb/>et propterea triangulum DFG ad duo triangula TEM, <lb/>NFO, erit ut unum ad 2. ergò BFG ad duo triangula <lb/>LEM, NFO, erit ut 3. ad 2. nempe reciprocè ut DH ad HI. <lb/>Triangulum igitur BGF, et duo triangula LEM, NFO, ex <lb/>puncto H aequiponderant iuvicem. </s>

<s>Sumatur iterùm DFP quarta pars totius DFG, eritque <lb/>DFP aequale duobus simul triangulis quae sunt infrà pun<lb/>cta N et O. (Sunt enim quartae partes aequalium trian<lb/>

<arrow.to.target n="marg187"></arrow.to.target><lb/>gulorum DFG, NFO). Propterea triangulum DFP ad qua<lb/>tuor simul triangula L, M, N, O, erit ut unum ad 2. Sed <lb/>triangulum PFG, ad eadem quatuor triangula erit ut 3. <lb/>ad 2. nempe reciprocè ut DH ad HI. Aequiponderat igitur <lb/>triangulum PFG, cum quatuor dictis triangulis L, M, N, O, <lb/>ex puncto H. Quamobrem universa figura inter portiones <lb/>inscripta aequiponderabit cum triangulo BPF ex puncto H. <lb/>Sed eadem praeponderabat triangulo BDF. Necesse igitur <lb/>est ut triangulum BDF minus sit quam triangulum BPF, <lb/>totum sua parte. Quod esse non potest. </s>

<s>Ponamus deinde praeponderare triangulum BDF dua-

<pb pagenum="126"/>bus simul portionibus mixtis ABE, BCF; et ponatur ex<lb/>cessus quo praeponderat aequalis spatio K. </s>

<s>Accipiatur triangulum DFG quarta pars ipsius DFB. <lb/>et iterum sumatur DFP quarta pars ipsius DFG, et sic <lb/>semper donec veniatur ad aliquod triangulum, puta DFP <lb/>minus spatio K. Tunc enim reliquum triangulum adhuc <lb/>praeponderabit portionibus mixtis ABE, BCF. Sed osten<lb/>demus eodem penitus argumento, atque ordine ut supra <lb/>idem triangulum PFB praeponderare alicui figurae intrà <lb/>portiones ABE, BCF, descriptae. Necesse ergo erit quòd <lb/>ipsae duae portiones mixtae minores sint quàm aliqua sibi <lb/>inscripta figura; totum sua parte; quod est absurdum. </s>

<s>Constat ergo quod propositum fuerat. </s>

<s><emph type="center"/>PROPOSITIO VII.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia et trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto, ut in praecedenti Lemmate para<lb/>

<arrow.to.target n="fig115"></arrow.to.target><lb/>bola ABC. cum duabus tangentibus late<lb/>ralibus, sive ad basis AE, CD; atque EBF <lb/>per verticem. Concipiaturque diameter ad <lb/>horizontem perpendicularis; et ablatà pa<lb/>rabola dectractoque dimidio verticalis <lb/>trianguli; accipiatur DI tertia pars to<lb/>tius DF, et sit DH sesquialtera ipsius HI. <lb/>Aequiponderant ergo (per lemma praece<lb/>dens) ex puncto H librae DI, duae ma<lb/>gnitudines. Nempe hinc duae portiones ABCFE appensae ad pun<lb/>ctum D; inde verò triangulum BDF appensum ad punctum I. </s>

<s>Quamobrem DBF ad ABCFE, erit ut reciprocè DH ad HI, nempe <lb/>ut 3. ad 2. Sumptisque antecedentium duplis, erit totum verticale trian<lb/>gulum EDF ad reliquam figuram mixtam triplum, Propterea (ut in <lb/>Propositione sexta demonstratum est) parabola inscripti sibi trianguli <lb/>sesquitertia erit. Quod erat propositum demonstrare.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Lemma XIV.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si duorum conorum latera trianguli per axem secta fuerint in partes <lb/>aequales numero, et magnitudine, ductisque per puncta sectionum <lb/>planis basi parallelis, super sectionum circulis intelligantur cylindri

<pb pagenum="127"/>aeque alti intrà conos descripti: Erit ut primus conus ad secundum, <lb/>ita omnes cylindri primi coni, ad omnes cylindros secundi coni.<gap desc="/SM"/></s>

<s>Sint duorum conorum trian<lb/>

<arrow.to.target n="fig116"></arrow.to.target><lb/>gula per axem ABC, DEF, et <lb/>duo eorum latera, puta AB, <lb/>DE, secentur in partes numero <lb/>aequales; nempe in totidem <lb/>partes dividatur tàm AB, quàm <lb/>DE; sintque partes lateris AB <lb/>aequales inter se, et partes <lb/>DE item aequales inter se. Ductis deinde per singula <lb/>sectionum puncta planis GH, IL, etc. basi AC paral<lb/>lelis: item planis MN, OP, etc. basi DF parallelis; <lb/>Concipiantur cylindri AH, GL, etc. eiusdem altitudinis <lb/>intrà conum ABC descripti; itemque in altero cono <lb/>alij cylindri aequealti intelligantur; Dico esse ut conus <lb/>ABC ad conum DEF, ita omnes cylindros coni ABC <lb/>ad omnes cylindros coni ABC ad omnes cylindros coni <lb/>DEF. </s>

<s>Concipiantur duo coni GAH, MDN; quorum vertices <lb/>sint A et D, bases verò circuli GH, MN. </s>

<s>Iàm; Cylindrus AH ad conum GAH est ut cylindrus <lb/>DN ad conum MDN. (nempe in ratione tripla) conus verò <lb/>GAH ad conum GBH in eadem basi est ut AG ad GB; <lb/>sive (propter divisionem in constructione adhibitam) ut DM <lb/>ad ME, hoc est ut conus MDN ad conum MEN. Conus <lb/>denique GBH ad conum similem ABC est ut cubus GB <lb/>ad cubum BA; sivè (propter constructionem) ut cubus ME <lb/>ad cubum ED, nempe ut conus MEN ad conum similem <lb/>DEF. Quarè ex aequo cylindrus AH ad conum ABC, erit <lb/>ut cylindrus DN ad conum DEF. Et permutando cylin<lb/>drus AH ad cylindrum DN erit ut conus ABC ad co<lb/>num DEF. </s>

<s>Ulterius. Cylindrus etiam GL ad cylindrum MP. eodem <lb/>penitus modo demonstratur esse ut conus GBH ad conum <lb/>MEN, sive ut conus ABC ad conum DEF; et hoc modo <lb/>semper. Proptereà ut unus cylindrus AH ad unum DN, <lb/>

<arrow.to.target n="marg188"></arrow.to.target><lb/>ità quilibet antecedentium ad quemlibet consequentium, <lb/>

<arrow.to.target n="marg189"></arrow.to.target><lb/>ergò ut unus ad unum, nempe ut conus ABC ad conum

<pb pagenum="128"/>DEF, ità omnes simul cylindri coni ABC, ad omnes simul <lb/>cylindros coni ABC, ad omnes simul cylindros coni DEF. <lb/>Quod etc. </s>

<s><margin.target id="marg188"></margin.target>11 Quinti.</s>

<s><margin.target id="marg189"></margin.target>12 Quinti.</s>

<s><emph type="center"/><emph type="italics"/>Lemma XV.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Dato trilineo mixto, sub lineà parabolica, eiusque tangente, et alià <lb/>rectà diametro parallela compraehenso; possibile est in dato trilineo <lb/>figuram inscribere constantem ex parallelogrammis aequealtis, quae <lb/>figura deficiat à trilineo mixto minori differentià quàm sit quaecnmque <lb/>data magnitudo.<gap desc="/SM"/></s>

<s>Esto linea parabolica <lb/>

<arrow.to.target n="fig117"></arrow.to.target><lb/>ABC, cuius tangens CD, <lb/>et diametro aequidistans <lb/>sit AD. Dico intrà trili<lb/>neum mixtum ABCD. <lb/>describi posse figuram <lb/>constantem ex paralle<lb/>logrammis aequealtis, <lb/>quae figura deficiat à <lb/>trilineo mixto, minori <lb/>defectu quàm sit spa<lb/>tium quodcumque da<lb/>tum K. </s>

<s>Secetur enim DC bi<lb/>fariam in X; iterumque <lb/>partes bifariàm dividan<lb/>tur in H et in P; sem<lb/>perque hoc fiat donec <lb/>veniatur ad sectionem <lb/>aliquam, puta DE, eius<lb/>modi ut parallelogram. ADE, minus sit spatio K. (Quod <lb/>autem hoc fieri possit, patet. Si enim compleatur paralle<lb/>logrammum ADC, ex ipso per continuam bisectionem <lb/>semper detrahitur dimidium; ergò tandem remanebit AE <lb/>minus quolibet dato spatio). Ducantur deinde ex punctis <lb/>sectionum rectae EF, HG, etc. aequidistantes ipsi DA; <lb/>per puncta autem I, B, etc. ubi parallelae secant para<lb/>bolam, ducantur LG, MN, etc. aequidistantes tangenti CD. <lb/>Et factum erit quod oportebat. </s>

<s>Parallelogrammum enim CO, aequale est ipsi OP, et <lb/>

<arrow.to.target n="marg190"></arrow.to.target><lb/>addito communi OI, erunt duo CO, OR, aequalia ipsi RQ, <lb/>sive ipsi RS: additoque communi RT, erunt tria CO, OR, <lb/>RT, aequalia ipsi TP, hoc est ipsi TX, additoque com<lb/>muni TZ et sic semper procedendo erunt denique omnia <lb/>simul parallelogramma CORTZYBIA aequalia ipsi paral<lb/>lelogrammo AE. nempe minora spatio K. Multò igitur <lb/>minor erit defectus figurae inscriptae ex parallelogrammis <lb/>aequealtis compositae, à trilineo mixto ABCD, quàm sit <lb/>propositum spatium K. Quod erat etc. </s>

<s><margin.target id="marg190"></margin.target>36 Primi.</s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hinc notabimus quod eodem prorsus modo, eàdemque <lb/>operatione, figura etiam circumscribitur dato trilineo mixto, <lb/>constans ex parallelogrammis aequealtis, ita ut excessus <lb/>figurae circumscriptae suprà ipsum trilineum, minor sit <lb/>quocumque spatio dato K. </s>

<s><emph type="center"/><emph type="italics"/>Lemma XVI.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola tangentem habuerit: et insuper duas rectas diametro <lb/>parallelas, quae duo trilinea abscindant sub tangente, et lineà para<lb/>bolica compraehensa; Erit figura ex parallelogrammis aequealtis con<lb/>stans in maiori trilineo descripta, ad figuram eiusdem speciei in minori <lb/>trilineo descriptam, ut cubus maioris tangentis ad cubum minoris.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuius tangens CD; et diametro <lb/>parallela sit utraque DA, EF; ut fiant duo trilinea mixta

<pb pagenum="130"/>ABCD maius, et FBCE minus. Dico, si in utroque tri<lb/>lineo inscribatur figura constans ex parallelogrammis ae<lb/>qualibus utrimque numero, (ut in praecedenti lemmate <lb/>expositum est) figuram trilinei ABCD, ad figuram trilinei <lb/>FBCE, esse ut cubus DC ad cubum CE. </s>

<s>Concipiamus, (ad evitandam linearum multitudinem, et <lb/>confusionem) triangulum GEC cum sua portione parabolae <lb/>intercepta FBC, transferri, et esse idem quod positum est <lb/>sub signis HIL. trilineumque FBCE esse idem cum tri<lb/>lineo MNLI. </s>

<s>Inscribatur iam in utroque trilineo ABCD, et MNLI, <lb/>(quod quidem repraesentat ipsum FBCE translatum) figura <lb/>constans ex parallelogrammis aequealtis; et sit idem nu<lb/>merus parallelogrammorum in utroque trilineorum. Intel<lb/>ligatur etiam conus, cuius vertex C, sive L; et diameter <lb/>basis sit, hinc quidem AD, inde verò HI. Sintque in sin<lb/>gulis coni segmentis cylindri aequealti OP, QR etc. <lb/>

<arrow.to.target n="marg191"></arrow.to.target></s>

<s><margin.target id="marg191"></margin.target>1. Sexti.</s>

<s>Iam parallelogrammum BP ad SD, est ut recta BR <lb/>ad SP, hoc est ut quadratum RC ad CP; hoc est ut qua</s>

<s>

<arrow.to.target n="marg192"></arrow.to.target><lb/>dratum RT ad quadratum PU: hoc est ut cylindrus QR ad <lb/>cylindrum OP. Eodem modo erit parallelogrammum XR <lb/>ad SD, ut cylindrus YR ad UD. Ergo erunt duo simul <lb/>

<arrow.to.target n="marg193"></arrow.to.target><lb/>parallelogramma BP, XR, ad SD; ut duo simul cylindri <lb/>TP, YR, ad cylindrum UD. Procedendo itaque semper hoc <lb/>modo, et denique componendo, erit tota inscripta figura <lb/>ex parallelogrammis constans in trilineo ABCD, ad paral<lb/>lelogrammum SD, ut omnes simul cylindri qui in cono <lb/>ACD, ad cylindrum UD. </s>

<s><margin.target id="marg192"></margin.target>ob parabo<lb/>lam.</s>

<s><margin.target id="marg193"></margin.target>24. Quinti.</s>

<s>Amplius; parallelogrammum SD ad NI compositam <lb/>habet rationem, ex ratione rectae SP ad NZ, sive qua<lb/>drati PC ad ZI (sunt enim duae figurae, sed circa ean<lb/>dem parabolam translatam) sive quadrati PU ad ZK; et <lb/>ex ratione rectae DP ad IZ. Est ergò parallelogrammum <lb/>SD ad NI ut cylindrus UD ad KI, Denique parallelo<lb/>

<arrow.to.target n="marg194"></arrow.to.target><lb/>grammum NI ad totam figuram inscriptam intrà trilineum <lb/>MNLI, est ut cylindrus KI ad omnes cylindros inscriptos <lb/>intrà conum HLI, Propterea ex aequo erit figura ex pa<lb/>rallelogrammis constans inscripta in maiori trilineo ABCD, <lb/>ad figuram ex parallelogrammis inscriptam in minori tri-

<pb pagenum="131"/>lineo MNLI, ut omnes cylindri in cono ACD ad omnes <lb/>cylindros in cono HLI. Nempe ut conus ACD ad conum <lb/>

<arrow.to.target n="marg195"></arrow.to.target><lb/>HLI, hoc est ad conum GCE (qui idem est). Nempe ut <lb/>cubus DC ad cubum CE. Quod erat etc. </s>

<s><margin.target id="marg194"></margin.target>ostendetur <lb/>eodem mo<lb/>do ut in al<lb/>tera figura.</s>

<s><margin.target id="marg195"></margin.target>Lemma. 14.</s>

<s><emph type="center"/><emph type="italics"/>Lemma XVII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola tangentem habuerit, et insuper duas diametro paral<lb/>lelas rectas lineas, quae duo trilinea mixta abscindant; erunt inter se <lb/>abscissa reilinea ut cubi suarum tangentium.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuius <lb/>

<arrow.to.target n="fig118"></arrow.to.target><lb/>tangens CD: et diametro paral<lb/>lela sit utraque DA, EB. Dico <lb/>trilineum mixtum ABCD ad tri<lb/>lineum mixtum BCE, esse ut <lb/>cubus tangentis DC, ad cubum <lb/>tangentis CE. </s>

<s>Si enim ita non est, sit alte<lb/>rum illorum, si possibile est, <lb/>maius quam ut habeat dictam <lb/>proportionem ad reliquum; et <lb/>ponamus illud esse ABCD, maius <lb/>quàm quod esse deberet ex<lb/>cessu K. </s>

<s>Inscribatur intrà trilineum ABCD figura ex parallelo<lb/>grammis aequealtis constans; ita ut à trilineo deficiat mi<lb/>nori defectu quàm sit spatium K (haec autem fieri posse <lb/>

<arrow.to.target n="marg196"></arrow.to.target><lb/>ostendimus). Habebitque adhuc figura inscripta ad reli<lb/>quum trilineum BCE maiorem rationem quàm cubus DC <lb/>ad cubum CE. </s>

<s>Inscribatur intrà alterum trilineum BCE figura eiusdem <lb/>speciei, et eiusdem numeri parallelogrammorum cùm de<lb/>scripta intra trilineum ABCD. Erit ergò figura inscripta <lb/>trilineo ABCD ad figuram inscriptam trilineo BCE ut cu<lb/>

<arrow.to.target n="marg197"></arrow.to.target><lb/>bus DC ad cubum CE. Sed eadem figura inscripta trilineo <lb/>ABCD ad trilineum BCE habet maiorem rationem quàm <lb/>cubus DC ad CE. Minus ergo est trilineum BCE quàm <lb/>inscripta sibi figura. totum sua parte. Quod est impossi<lb/>bile. Constat ergo propositum. </s>

<s><margin.target id="marg197"></margin.target>Lemma <lb/>praecedens.</s>

<s><emph type="center"/>PROPOSITIO VIII.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, cuius diameter <lb/>

<arrow.to.target n="fig119"></arrow.to.target><lb/>BE, tangentes verò AF, CF, productae <lb/>eousque donec occurrant ipsis AD, CH, <lb/>diametro parallelis. Iungaturque rectae <lb/>lineae AB, BC. (licet in figura omissae <lb/>

<arrow.to.target n="marg198"></arrow.to.target><lb/>sint). Dico parabolam trianguli ABC esse <lb/>sesquitertiam. <lb/>

<arrow.to.target n="marg199"></arrow.to.target></s>

<s><margin.target id="marg198"></margin.target>Lem. praece<lb/>dens</s>

<s><margin.target id="marg199"></margin.target>2. Sexti</s>

<s>Erit enim ABCD ad trilineum BCF, <lb/>ut cubus DC ad cubum CF, nempe ut <lb/>octo ad unum. (cum enim sit ut AE ad <lb/>EC, ita DF ad FC, erit DF aequalis ipsi <lb/>FC, cubusque DC octuplus cubi CF). Item trilineum CBAH ad trilineum <lb/>BAF est ut octo ad unum. Coniuctim ergò erunt duo trilinea ABCD, <lb/>CBAH, ad spatium ABCF. ut octo ad unum. Et dividendo bis, erunt <lb/>duo triangula AFD, CFH, ad spatium ABCF, ut 6. ad unum. Quamobrem <lb/>triangulum AFD, sive AFC ad spatium ABCF, erit ut 3, ad unum; et <lb/>ad parabolam erit ut 3. ad 2. vel ut 6. ad 4. Propterea parabola erit ad <lb/>triangulum ABC ut 4. ad 3. Nempe sesquitertia. Quod erat propositum <lb/>demonstrare etc.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Lemma XVIII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si fuerit ut prima magnitudo ad secundam, ita tertia ad quartam; <lb/>Et hoc quotiescumque libuerit. Fuerintque omnes primae inter se, item <lb/>omnes tertiae magnitudines inter se aequales. Erunt omnes primae <lb/>simul ad omnes secundas, ut sunt omnes tertiae simul ad omnes quar<lb/>tas magnitudines.<gap desc="/SM"/></s>

<s>Esto ut A prima ad B secundam, <lb/>

<arrow.to.target n="fig120"></arrow.to.target><lb/>ita C tertia ad D quartam. Et iterum <lb/>ut E prima ad F secundam, ita G <lb/>tertia ad H quartam; et sic quotie<lb/>scunque libuerit. Sintque omnes pri<lb/>mae A, E, I, etc. item omnes tertiae <lb/>C, G, M, etc. inter se aequales. </s>

<s>Dico omnes primas simul ad omnes <lb/>secundas simul, ita esse ut sunt omnes <lb/>simul tertiae, ad omnes quartas ma<lb/>gnitudines. </s>

<s>Quoniam enim convertendo est ut B

<pb pagenum="133"/>ad A ita D ad C. Item ut F ad E; sive ad aequalem A, <lb/>ita H ad G, sive ad C; erunt simul BF ad A, ut sunt DH <lb/>slmul ad C. Hoc modo procedendo, ostendemus omnes </s>

<s>

<arrow.to.target n="marg200"></arrow.to.target><lb/>secundas simul esse ad A, ut sunt omnes quartae simul <lb/>ad ipsam C. Ipsa verò A ad omnes est ut C ad omnes <lb/>tertias (sunt enim aeque submultiplices). Ergo ex aequo <lb/>

<arrow.to.target n="marg201"></arrow.to.target><lb/>omnes secundae ad omnes primas, sunt ut omnes quartae <lb/>simul ad omnes tertias. Convertendo igitur constat quod <lb/>erat propositum demonstrare. </s>

<s><margin.target id="marg200"></margin.target>24. Quinti.</s>

<s><margin.target id="marg201"></margin.target>15. Quinti.</s>

<s><emph type="center"/><emph type="italics"/>Lemma XIX.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola tangentem habuerit ad basim; ex alia verò parte rectam <lb/>diametro parallelam. Erit triangulum sub tangente, et parallela dia<lb/>metro, ipsaque basi compraehensum, ipsius parabolae triplum.<gap desc="/SM"/></s>

<s>Esto parabola ABC, <lb/>

<arrow.to.target n="fig121"></arrow.to.target><lb/>cuius tangens CD, paral<lb/>lela diametro sit AD; <lb/>Dico triangulum ADC <lb/>esse parabolae ipsius AB <lb/>C, triplum. </s>

<s>Si enim non est tri<lb/>plum parabolae, per con<lb/>versionem rationis, non <lb/>erit sesquialterum trilinei <lb/>ABCD; et propterea (du<lb/>plicato antecedente) to<lb/>tum parallelogrammum AE non erit triplum trilinei ABCD. </s>

<s>Trilineum ergo ABCD erit vel plus, vel minus quam <lb/>tertia pars parallelogrammi AE. Ponatur primùm <emph type="italics"/>s<emph.end type="italics"/>sse plus <lb/>quàm tertia pars, et sit excessui aequale spatium K. </s>

<s>Inscribatur intrà trilineum ABCD, figura constans ex <lb/>parallelogrammis aequealtis, deficiensque ab ipso trilineo <lb/>minori defectu quàm sit ipsum spatium K. Et inscripta <lb/>iam sit eiusmodi figura. Erit ergò adhuc figura inscripta <lb/>plus quàm tertia pars parallelogrammi AE. </s>

<s>Concipiatur circa rectam AD, circulus, qui sit basis <lb/>cuiusdam coni verticem habentis in puncto C. et super <lb/>eàdem basi intelligatur cylindrus AE eiusdem altitudinis

<pb pagenum="134"/>cum ipso cono; sectusque sit tam conus quàm cylindrus <lb/>planis basi parallelis per singulas rectas FG, HI, LM, etc. <lb/>ductis. Concipiantur etiam intrà conum ACD cylindri ae<lb/>quealti PO, OI, etc. <lb/>

<arrow.to.target n="marg202"></arrow.to.target></s>

<s><margin.target id="marg202"></margin.target>1. Sexti. <lb/>ob parabo<lb/>lam.</s>

<s>Iam sic; parallelogrammum AF ad ND, est ut recta <lb/>DA ad ON: nempe ut quadratum DC ad quadratum CO; <lb/>sive, ut quadratum DA ad quadratum OG, nempe, ut cy</s>

<s>

<arrow.to.target n="marg203"></arrow.to.target><lb/>lindrus AF ad cylindrum PO, Et sic, semper. Suntque <lb/>omnes primae magnitudines aequales parallelogrammo AF, <lb/>et ideo aequales inter se; omnes autem tertiae magnitu<lb/>dines aequales cylindro AF, atque ideo inter se. Erunt ergo <lb/>omnes primae simul, hoc est parallelogrammum AQ, ad <lb/>omnes secundas simul, nempe ad figuram inscriptam in <lb/>trilineo ABCD, ut sunt omnes tertiae simul, nempe cylin<lb/>drus AQ, ad omnes quartas simul, hoc est ad omnes cy<lb/>lindros intrà conum ACD descriptos. Convertendo igitur; <lb/>erit figura trilineo inscripta ad parallelogrammum AQ <lb/>ut omnes cylindri intra conum ACD ad cylindrum <expan abbr="Aq.">Aque</expan> <lb/>Parallelogrammum verò AQ ad parallelogrammum AE <lb/>est ut DQ ad DE, hoc est ut cylindrus AQ ad cylindrum <lb/>AE. Propterea ex aequo, figura inscripta in trilineo ad <lb/>totum parallelogrammum AE, erit ut omnes cylindri in <lb/>cono inscripti ad cylindrum AE. Sed figura inscripta <lb/>in trilineo (ex iam dictis) plus quàm tertia pars paral<lb/>lelogrammi AE, ergò omnes cylindri in cono descripti <lb/>erunt plusquam tertia pars cylindri AE, nempe maiores <lb/>quàm conus ACD. pars videlicet suo toto. Quod est im<lb/>possibile. </s>

<s><margin.target id="marg203"></margin.target>ob similitud. <lb/>triang.</s>

<s>Sed ponamus nunc trilineum ABCD esse minus quam <lb/>tertia pars parallelogrammi AE; sitque defectus aequalis <lb/>

<arrow.to.target n="marg204"></arrow.to.target><lb/>spatio K. Circumscribatur trilineo ABCD figura constans <lb/>ex parallelogrammis aequealtis excedensque minori ex<lb/>cessu quàm sit spatium K; et erit figura circumscripta <lb/>adhuc minor quàm tertia pars parallelogrammi AE. </s>

<s><margin.target id="marg204"></margin.target>Coroll. <lb/>Lem. 15.</s>

<s>Concipiatur iterum circa rectam AD circulus pro basi <lb/>coni, qui verticem habeat C; itemque pro basi cylindri <lb/>ACED eiusdem altitudinis cum ipso cono ACD. </s>

<s>Intelligatur insuper circa conum descripta figura solida <lb/>constans ex cylindris aequealtis AQ, GI, etc. </s>

<s>Iam parallelogram<lb/>

<arrow.to.target n="fig122"></arrow.to.target><lb/>mum AF ad paralle<lb/>logrammum AQ (ob <lb/>aequalitatem) est ut <lb/>cylindrus ADFG ad <lb/>cylindrum ADQR. Am<lb/>plius. Parallelogram<lb/>mum GH ad paralle<lb/>logrammum LI est ut <lb/>

<arrow.to.target n="marg205"></arrow.to.target><lb/>GF, sive AD ad <expan abbr="Lq;">Lque</expan> <lb/>nempe ut quadratum <lb/>DC ad quadratum CQ, <lb/>sive ut quadratum DA, <lb/>vel FG, ad quadratum <expan abbr="Gq;">Gque</expan> nempe ut cylindrus GH ad <lb/>cylindrum GI. etc. et hoc modo semper. Suntque omnes <lb/>singillatim primae magnitudines aequales parallelogrammo <lb/>AF, et ideò inter se: item omnes tertiae aequales cylindro <lb/>AF, et ob id inter se; ergo erunt omnes primae simul, hoc <lb/>est parallelogrammum AE, ad omnes secundas simul, <lb/>hoc est ad figuram trilineo circumscriptam, ut omnes ter<lb/>tiae simul, nempe cylindrus AE, ad omnes quartas simul, <lb/>nempe ad cylindros conum ACD circumscribentes. Con<lb/>vertendo igitur, erit figura circumscripta trilineo, ad pa<lb/>rallelogrammum AE, ut omnes cylindri circumscribentes <lb/>conum ad cylindrum AE. Sed figura trilineo circumscripta <lb/>minor est quàm tertia pars parallelogrammi AE; ergo <lb/>etiam omnes cylindri circumscribentes conum, minores <lb/>erunt quàm tertia pars cylindri AE; nempe minores cono <lb/>ACD. Totum sua parte: quod esse non potest. Triangulum <lb/>ergo ADC ipsius parabolae omninò triplum erit. Quod pro<lb/>positum fuerat. </s>

<s><margin.target id="marg205"></margin.target>1. sexti. <lb/>ob parabo<lb/>lam. <lb/>ob similitud. <lb/>triangul.</s>

<s><emph type="center"/>PROPOSITIO IX.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, cuius diameter EB, <lb/>

<arrow.to.target n="fig123"></arrow.to.target><lb/>triangulum inscriptum sit ABC. Dico para<lb/>bolam trianguli ABC esse sesquitertiam; </s>

<s>Ducatur enim tangens CD, et sit recta <lb/>AD diametro aequidistans: </s>

<s>Erit ergò per praecedens lemma, triangu<lb/>lum ACD parabolae triplum; et propterea <lb/>erit parabola partes quatuor earum, quarum <lb/>triangulum ADC est duodecim, nempe qualium <lb/>triangulum ABC est tres. (triangulum enim <lb/>ABC aequale est triangulo EFC, cum utrum<lb/>que duplum sit trianguli EBC, ergo triangu<lb/>lum ABC quarta pars erit totius ADC). Constat ergo parabolam ad <lb/>inscriptum sibi triangulum esse ut 4. ad 3. Nempe sesquitertiam. <lb/>Quod etc.<gap desc="/SM"/></s>

<s><emph type="center"/>PROPOSITIO X.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem sibi basim, et eandem <lb/>alritudinem habentis. </s>

<s>Esto parabola ABC, cuius dia<lb/>

<arrow.to.target n="fig124"></arrow.to.target><lb/>meter BD. Dico parabolam ABC <lb/>inscripti sibi trianguli esse sesqui<lb/>tertiam. </s>

<s>Compleatur parallelogrammum <lb/>ADBE, et nisi parabola sesquitertia <lb/>sit trianguli sibi inscripti, neque <lb/>(sumptis dimidijs) semiparabola A <lb/>BD, sesquitertia erit trianguli ABD; <lb/>neque eadem semiparabola ABD <lb/>erit 2. tertii parallelogrammi ED, <lb/>sed vel plus vel minus quam 2. tertii <lb/>eiusdem. </s>

<s>Esto primùm si fieri potest semiparabola ABD magìs quàm 2. tert. <lb/>parallelogrammi ED; et ponatur excessus aequalis spatio K. Ipsique <lb/>semiparabolae figura inscribatur constans ex parallelogrammis ae<lb/>qualtis (more apud Geometras usitato, prout factum est Lemmate XV.) <lb/>ita ut differentia inter figuram inscriptam; et ipsam semiparabolam <lb/>minor sit spatio K. Tunc enim inscripta figura adhuc maior erit quàm <lb/>2. tert. parallelogrammi ADBE. </s>

<s>Ducatur circa diametrum AC semicirculus AXC, completoque re<lb/>ctangulo, sive quadrato AFXD. ducantur GL, HM, IO, perpendiculares <lb/>ad AC, et compleantur rectangula DL, GM, HO; Tum intelligatur figura <lb/>AFXD circumverti circa axem AD; ita ut quadrans ADX, hemisphae<lb/>rium describat, quadratum verò AFXD, cylindrum; et rectangula in <lb/>quadrante inscripta totidem cylindros faciant in ipso hemisphaerio <lb/>compraehensos. </s>

<s>Iam parallelogrammum BG ad PD, est ut BD ad GP, sive ut re<lb/>ctangulum CDA ad rectangulum CGA; sive ut quadratum XD ad qua<lb/>dratum LG; sive ut cylindrus XG ad LD. Et hoc modo semper. Suntque <lb/>omnes primae magnitudines aequales parallelogrammo BG, et omnes <lb/>tertiae aequales cylindro XG. Ergo erunt omnes primae simul, hoc est <lb/>

<arrow.to.target n="marg206"></arrow.to.target><lb/>parallelogrammum TD, ad omnes secundas simul, nempe ad figuram <lb/>inscriptam in semiparabola, ut sunt omnes tertiae simul, nempe cylin<lb/>drus VD, ad omnes quartas simul, hoc est ad omnes cylindros in he<lb/>misphaerio inscriptos. Parallelogrammum verò TD ad ED est ut cy<lb/>lindrus VD ad FD, ergo ex aequo, erit parallelogrammum ED ad <lb/>figuram in semiparabola inscriptam ut cylindrus FD ad omnes cylindros <lb/>in ipso hemisphaerio compraehensos. Sed parallelogrammum ED minus <lb/>est quam sesqnialterum figurae intra semiparabolam inscriptae; Ergò <lb/>cylindrus FD minor erit quàm sesquialter omnium cylindrorum in he<lb/>misphaerio descriptorum. Quod est absurdum. Scimus enim dictum <lb/>cylindrum hemisphaerij esse sesquialterum. </s>

<s>Esto deinde (si fieri potest) semiparabola minus quàm 2. tert. ipsius <lb/>parallelogrammi ED. Ponaturque defectus aequalis spatio K. </s>

<s>Tum ipsi semiparabolae figura quaedam circumscribatur, constans <lb/>ex parallelogrammis aequealtis (more solit, ut factum est in Lem<lb/>mate XV. eiusque Corollario) ita ut differentia inter circumscriptam <lb/>figuram ipsamque semiparabolam minor sit spatio K. Tunc enim ma<lb/>nifestum est, quòd figura circumscripta adhuc minor erit quàm 2. tert. <lb/>parallelogrammi ED. </s>

<s>Fiat circa diametrum AC se<lb/>

<arrow.to.target n="fig125"></arrow.to.target><lb/>micirculus, ut in escriptione prae<lb/>cedentis constructionis, completo<lb/>que quadrato AOFD, perficiantur <lb/>reliqua rectangula FL, GM, HN, <lb/>IA. circa quadrantem descripta. <lb/>Tum revolvatur figura AF circa <lb/>axem AD, ita ut solida generen<lb/>tur iam dicta; nempe hemisphae<lb/>rium ex quadrante, cylindrus ex <lb/>quadrato AF; totidemque cylin<lb/>dri quot rectangula erunt ipsi <lb/>quadranti circumscripta. Iàm pa<lb/>rallelogrammum BL ad se ipsum <lb/>est ut cylindrus factus ex FL ad se ipsum. Amplius. Parallelogram-

<pb pagenum="138"/>mum QM ad PM; est ut QL ad LP; sive BD ad LP, sive ut rectang. <lb/>CDA ad CLA, sive ut quadratum FD ad LG, sive ut quadratum RL <lb/>ad LG; nempe ut cylindrus factus ex RM ad cylindrum ex GM: et hoc <lb/>modo semper. </s>

<s>Suntque omnes primae magnitudines aequales parallelogrammo BL, <lb/>

<arrow.to.target n="marg207"></arrow.to.target><lb/>omnesque tertiae aequales cylindro facto ex FL. Ergo erunt omnes <lb/>primae simul nempe parallelogrammum AB ad omnes simul secundas, <lb/>nempe ad figuram semiparabolae circumscriptam, ut sunt omnes ter<lb/>tiae simul; nempe cylindrus ex OD factus, ad omnes quartas, nempe <lb/>ad cylindros hemisphaerio ciscumscriptos. Sed parallelogrammum ED <lb/>magis est quàm sesquialterum figurae circumscriptae ad semipara<lb/>bolam, ergo cylindrus ex OD magis quàm sesquialter erit ad omnes <lb/>cylindros hemisphaerio circumscriptos. </s>

<s>Quod est absurdum: Scimus enim cylindrum hemisphaerio circum<lb/>scriptum ipsius hemispherii esse sesquialterum. Patet itaque paralle<lb/>logr. ED sesquialternm esse ad semiparabolam ABD; et ideo semiparab. <lb/>sesquitertia trianguli ABD.<gap desc="/SM"/></s>

<pb/>

<s><emph type="center"/>QUADRATURA PARABOLAE<emph.end type="center"/></s>

<s><emph type="center"/>PER NOVAM INDIVISIBILIUM GEOMETRIAM<emph.end type="center"/></s>

<s><emph type="center"/>PLURIBUS MODIS ABSOLUTA<emph.end type="center"/></s>

<s>Hactenus de dimensione parabolae more antiquorum <lb/>dictum sit; Reliquum est ut eandem parabolae mensuram <lb/>nova quedam, sed mirabili ratione aggrediamur; ope sci<lb/>licet Geometriae Indivisibilium, et hoc diversis modis: <lb/>Suppositis enim praecipui Theorematib. antiquorum tàm <lb/>Euclidis, quàm Archimedis, licet de rebus inter se diver<lb/>sissimis sint, mirum est ex unoquoque eorum quadraturam <lb/>parabolae facili negotio elici posse; et vice versa. quasi <lb/>ea sit commune quoddam vinculum veritatis. Posito enim <lb/>quòd cylindrus inscripti sibi coni triplus sit, hinc sequitur <lb/>parabolam inscripti sibi triangoli esse sesquitertiam: Si <lb/>verò mavis praemittere cylindrum inscriptae sibi sphaerae <lb/>esse sesquialterum, continuò parabolae quadratura infertur. <lb/>Eadem concluditur supposita demonstratione, quae probat <lb/>centrum gravitatis coni positum esse in axe, ita ut pars <lb/>quae ad verticem est, reliquae sit tripla. Parabola non <lb/>minus quadratur etiam supponendo spatium à linea spirali <lb/>in prima revolutione descripta, et à recta quae initium <lb/>est revolutionis, compraehensum, subtriplum esse primi <lb/>circuli. Contrà verò: supposità parabolae quadratura, prae<lb/>dicta omnia Theoremata facilè demonstrari possunt. Quod <lb/>autem haec indivisibilium Geometria novum penitus in-

<pb pagenum="140"/>ventum sit, equidem non ausim affirmare. Crediderim po<lb/>tius veteres Geometras hoc metodo usos in inventione <lb/>Theorematum difficillimorum, quamquam in demonstratio<lb/>nibus aliam viam magis probaverint, sive ad occultandum <lb/>artis arcanum, sive ne ulla invidis detractoribus profer<lb/>retur occasio contradicendi. Quicquid est, certum est hanc <lb/>Geometriam mirum esse pro inventione compendium, et <lb/>innumera quasi imperscrutabilia Theoremata, brevibus, di<lb/>rectis, affirmativisque demonstrationibus confirmare; quod <lb/>per doctrinam antiquorum fieri minimè potest. Haec enim <lb/>est in Mathematicis spinetis via verè Regia, quàm primus <lb/>omnium aperuit, et ad pubblicum bonum complanavit mi<lb/>rabilium inventorum machinator Cavalerius. </s>

<s><emph type="center"/>PROPOSITIO XI.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, cuius tangens CD, <lb/>

<arrow.to.target n="fig126"></arrow.to.target><lb/>et diametro aequidistans sit AD. Perficiatur <lb/>parallelogrammum AE; et circa diametrum <lb/>AD intelligatur circulus, qui sit basis coni <lb/>cuiusdam verticem habentis in puncto C, et <lb/>item sit basis cylindri alicuius ACED eiusdem <lb/>altitudinis cum dicto cono. </s>

<s>Ducantur iam quaelibet recta FG paral<lb/>lela ad AD, et per ipsum intelligatur tran<lb/>

<arrow.to.target n="marg208"></arrow.to.target><lb/>sire planum parallelum circulo AD. Erit ergò <lb/>FG ad IB ut recta DA ad IB hoc est ut <lb/>quadratum DC ad quadratum CI, sive ut <lb/>quadratum DA ad IG, sive ut circulus DA ad circulum IG, nempe ut <lb/>circulus FG ad eundem IG. Et hoc semper; suntque omnes primae ma<lb/>gnitudines aequales rectae DA. Et ideò inter se; omnes etiam tertiae <lb/>aequales circulo DA et ob id inter se; ergo per Lemma 18 erunt <lb/>omnes primae simul, nempe parallelogrammum AE, ad omnes secundas <lb/>simul, nempe ad trilineum ABCD, ut sunt omnes tertiae simul, nempe <lb/>cylindrus AE, ad omnes quartas simul hoc est ad conum ACD. Est <lb/>igitur parallelogrammum AE triplum trilinei ABCD. Sumptoque di<lb/>midio, erit triangulum ACD sesquialterum trilinei ABCD; et per con-

<pb pagenum="141"/>versionem rationis, erit triangulum ACD triplum ipsius parabolae. <lb/>Propterea, ex demonstratione propositionis 9. erit parabola inscripti <lb/>sibi trianguli sesquitertia. Quod erat etc.<gap desc="/SM"/></s>

<s><margin.target id="marg208"></margin.target>ob parabo<lb/>lam. <lb/>ob similitud. <lb/>triang.</s>

<s>Alia quoque ratione parabolam quadrabimus, demon<lb/>stratis priùs, quà fieri poterit brevitate, indivisibilium prin<lb/>cipijs. Declinabimus autem ab immenso Cavalerianae Geo<lb/>metriae oceano, minori audacia radentes terram. Qui volet, <lb/>haec omnia videre poterit (in fonte dicam, an in pelago?) <lb/>circa medium secundi libri Geometriae indivisibilium Ca<lb/>valerij. </s>

<s><emph type="center"/><emph type="italics"/>Lemma XX.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Quadrata omnium partium cuiuscunque rectae lineae subtripla sunt <lb/>totidem quadratorum totius.<gap desc="/SM"/></s>

<s>Esto quaelibet recta linea AB. Dico omnia simul qua<lb/>drata omnium partium rectae AB esse subtripla totidem <lb/>quadratorum eiusdem rectae lineae AB. </s>

<s>Fiat enim quadratum ACDB, ductàque <lb/>

<arrow.to.target n="fig127"></arrow.to.target><lb/>diametro AD, convertatur figura circa axem <lb/>AB donec in eum locum redeat unde cepit <lb/>moveri. Manifestum est, quod à quadrato <lb/>cylindrus CH describetur, à triangulo verò <lb/>ABD conus DAH, qui verticem habebit in A. <lb/>Ducatur iam quaelibet EF parallela ipsi CA, <lb/>eritque AF, sive FG, (sunt enim aequales) <lb/>una ex infinitis partibus totius AB. </s>

<s>Iam; quadratum totius AB, ad quadratum partis AF, <lb/>est, ob aequalitatem, ut quadratum EF ad FG, nempe ut <lb/>circulus diametro EL factus, ad circulum diametro GI. <lb/>

<arrow.to.target n="marg209"></arrow.to.target><lb/>Et sic erit semper. Suntque primae magnitudines sin<lb/>gulae aequales quadrato AB, et tertiae semper aequales <lb/>circulo DH. Ergo omnes primae simul, hoc est tot qua<lb/>

<arrow.to.target n="marg210"></arrow.to.target><lb/>drata lineae AB, quot ipsa habet partes, ad omnia quadrata <lb/>partium, erunt ut omnes tertiae simul, hoc est ut cyliudrus <lb/>CH, ad omnes quartas simul, nempe ad conum DAH. Sunt <lb/>ergò tot quadrata alicuius lineae quot ipsa habet partes, <lb/>ad omnia quadrata partium ipsius ut cylindrus CH ad <lb/>conum DAH, nempe tripla. Et convertendo constat pro<lb/>positum quod demonstrandum fuerat etc. </s>

<s><margin.target id="marg209"></margin.target>2. duodecimi.</s>

<s><emph type="center"/><emph type="italics"/>Lemma XXI.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Omnia rectangula, quae continentur sub aliqua recta linea cum <lb/>singulis suis partibus, et reliquis partibus sub sesquialtera sunt totidem <lb/>quadratorum eiusdem rectae lineae.<gap desc="/SM"/></s>

<s>Assumpta praecedentis Lemmatis figura, acceptum sit <lb/>in recta AB quodlibet punctum F. Rectangulum sub BAF <lb/>tanquàm una recta linea, et sub FB. contentum, erit unum <lb/>ex omnibus praedictis rectangulis, (unum enim latus com<lb/>ponitur ex tota AB, cum parte AF; alterum verò est FB, <lb/>nimirum reliqua pars). Rectangulum autem praedictum, <lb/>sub BAF tamquam una recta et sub FB contentum, idem <lb/>est, ob aequalitatem laterum, ac rectangulum EIL. Et hoc <lb/>semper verum erit hoc modo, ubicunque sit punctum F. <lb/>Sed omnia rectangula sub rectis interceptis in trapezio <lb/>CAHD (qualium una est EI) et sub reliquis, qualium una <lb/>est IL; una cum omnibus quadratis intermediarum sectio<lb/>num (qualium una est FI) aequantur (propter V secundi <lb/>elementorum) omnibus quadratis dimidiarum, qualium una <lb/>est FL. Omnia verò quadrata intermediarum sectionum, <lb/>

<arrow.to.target n="marg211"></arrow.to.target><lb/>(qualium una est FI) ad omnia quadrata dimidiarum (qua<lb/>lium una est FL) sunt ut unum ad 3. Si ergo demantur <lb/>omnia quadrata intermediarum, remanebunt omnia rectan<lb/>gula, quorum unum est EIL, sive omnia rectangula con<lb/>tenta sub AB cum singulis suis partibus, et reliquis par<lb/>tibus, subsesquialtera omnium quadratorum, quae fiunt à <lb/>dimidiis, sive totidem quadratorum totius AB. Quod fuerut <lb/>ostendendum etc. </s>

<s><margin.target id="marg211"></margin.target>Lem. <lb/>praeced.</s>

<s><emph type="center"/>PROPOSITIO XII.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertiam est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, cuius diameter <lb/>

<arrow.to.target n="fig128"></arrow.to.target><lb/>BE, et circa parabolam sit parallelo<lb/>grammum DC. Ducatur quaelibet FG <lb/>diametro parallela; eritque FG. ad GI, <lb/>ut BE ad GI, sive ut rectangulum CEA, <lb/>ad CGA, hoc est ut quadratum CE ad rectangulum CGA. Et hoc modo

<pb pagenum="143"/>semper; suntque primae magnitudines aequales semper rectae BE; ter<lb/>tiae autem semper aequales quadrato CE. Ergo omnes primae simul, <lb/>hoc est parallelogrammum AB, ad omnes secundas simul, nempe ad <lb/>

<arrow.to.target n="marg212"></arrow.to.target><lb/>semiparabolam AIBE; erunt ut omnes simul tertiae, videlicet tot <lb/>quadrata lineae CE quot ipsa habet partes, ad omnes quartas simul, <lb/>nempe ad omnia rectangula sub CE cum singulis suis partibus, et <lb/>sub reliquis partibus. Ergo (ex praecedenti Lemmate) parallelogram<lb/>mum AB er<emph type="italics"/>i<emph.end type="italics"/>t ipsius semiparabolae sesquialterum; Totumque paralle<lb/>logrammum DC erit totius parabolae sesquialterum, nempe ut 6. ad <lb/>4. Propterea parabola ad inscriptum sibi triangulum (quod quidem <lb/>parallelogrammi DC sub duplum est) erit ut 4. ad 3. Nempe sesqui<lb/>tertia. Quod erat etc.<gap desc="/SM"/></s>

<s>Possumus sine molestia illorum lemmatum, parabo<lb/>lam quadrare eadem argumento, diversis tamen principijs, <lb/>nempe per suppositionem proportionis, quam cylindrus ha<lb/>bet ad sphaeram sibi inscriptam; quae quidem proportio <lb/>sesquialtera est, ut ostenditur ex Archimede; libro primo <lb/>de Sphaera et Cylindro. </s>

<s><emph type="center"/>PROPOSITIO XIII.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, circa quam sit pa<lb/>

<arrow.to.target n="fig129"></arrow.to.target><lb/>rallelogrammum AD; et circa diametrum <lb/>AC fiat semicirculus, circa quem sit rectan<lb/>gulum AE. Tum manente axe AC, intelli<lb/>gatur circumverti ipsum semicirculum, ita <lb/>ut ex ipsius revolutione Sphaera circum<lb/>scribatur: ex conversione verò rèctang. AE <lb/>cylindrus nascatur. </s>

<s>Sumpto iam quolibet puncto G. ducatur <lb/>recta GF parallea diametro HB; et per idem punctum G agatur pla<lb/>num GL erectum ad axem AC. </s>

<s>Erit recta FG ad GI, ut BH ad GI (ob aequalitatem) hoc est rectan<lb/>gulum GHA, ad rectangulum CGA, sive ut quadratum HN ad quadra<lb/>tum GM (ob circulum) sive ut quadratum GL ad quadratum GM; nempe <lb/>ut circulus ex semidiametro GL in cylindro, ad circulum ex semidia<lb/>metro GM in sphaera. Et hoc semper, ubicunque sumatur punctum G. <lb/>Sunt autem aequales inter se tàm omnes primae, quàm omnes tertiae <lb/>magnitudines. Ergò omnes primae, nempe parallelogrammum AD ad <lb/>

<arrow.to.target n="marg213"></arrow.to.target><lb/>omnes secundas, nempe ad parabolam ABC, erunt ut omnes tertiae, <lb/>hoc est cylindrus, ad omnes simul quartas, videlicet ad sphaeram. Sed

<pb pagenum="144"/>cylindrus ad sphaeram est sesquialter; ergò parallelogrammum etiam <lb/>AD parabolae sesquialterum erit: et ipsa parabola inscripti sibi trian<lb/>guli sesquitertia; ut in praecedenti conclusum est. Quod etc.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Lemma XXII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si magnitudines quotcunque ad libram appensae fuerint ex quibus<lb/>cunque punctis: totidemque magnitudines alterius ordinìs ex iisdem <lb/>punctis pendeant, pariter cum praedictis magnitudinibus proportionales. <lb/>Erit unum idemque librae punctum centrum aequilibrij utriusque or<lb/>dinum magnitudinum.<gap desc="/SM"/></s>

<s>Sint ad libram AB magni<lb/>

<arrow.to.target n="fig130"></arrow.to.target><lb/>tudines primi ordinis quotcun<lb/>que C,D,E,F, ex quibuscunque <lb/>punctis appensae. Totidemque <lb/>magnitudines G, H, I, L, se<lb/>cundi ordinis pendeant ex ijsdem punctis; et sint pro<lb/>portionales: nempe: Ut C ad D, ita sit G ad H. Iterum <lb/>ut C ad E, ita sit G ad I. etc. Dico idem punctum librae <lb/>esse centrum commune aequilibrij utriusque ordinis ma<lb/>gnitudinum suspensarum. </s>

<s>Cum enim sit ut C ad D, ita G ad H, ex eodem puncto <lb/>aequiponderabunt, tam duae magnitudines C et D, quam <lb/>duae G et H. </s>

<s>Ampliùs. Cum sit ut C ad D, ita G ad H, erit conver<lb/>tendo et componendo DC ad C, ut HG ad G. C autem <lb/>ad E est ut G ad I; ergò ex aequo CD simul ad E erit <lb/>ut GH simul ad I. Quare magnitudines CD, et E, ex eodem <lb/>puncto aequiponderabunt, ex quo aequiponderant duae <lb/>GH et I. </s>

<s>Ulterius. Cum autem per iam dicta, sit ut CD ad E, <lb/>ita GH ad I, erit componendo CDE ad E, ut GHI ad I. <lb/>Sed E ad C est ut I ad G; et C ad F, ut G ad L. </s>

<s>Quare ex aequo CDE simul ad F, erit ut GHI simul <lb/>ad L. Ergo duae magnitudines CDE et F. habebunt idem <lb/>punctum aequilibrij, quod habent duae magnitudines GHI <lb/>et L. Et sic etiam si sint plures magnitudines, usque in <lb/>infinitum, quod erat propositum etc. </s>

<s><emph type="center"/><emph type="italics"/>Lemma XXIII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si parabola tangentem habuerit ad basim, ex altera verò parte <lb/>lineam diametro parallelam. Trilineum compraehensum sub curvà pa<lb/>rabolicà, sub tangente, et sub parallela praedictà, aequiponderabit ex <lb/>puncto tangentis ubi ea sic dividitur, ut pars ad contactum terminata <lb/>reliquae sit tripla.<gap desc="/SM"/></s>

<s>Esto parabola ABC, cuius tangens <lb/>

<arrow.to.target n="fig131"></arrow.to.target><lb/>ad basim sit CD; aequidistans diametro <lb/>sit AD. Dico trilineum mixtum ABCD <lb/>aequiponderare ex puncto tangentis CD, <lb/>ubi ea dividitur ut pars versus conta<lb/>ctum C, reliquae sit tripla. </s>

<s>Concipiatur figura ita ut DA ad ho<lb/>rizontem sit perpendicularis; et circa <lb/>diametrum DA intelligatur circulus, <lb/>qui sit basis coni verticem habentis in puncto C. Sumpto <lb/>iam quolibet puncto E, ducatur EF aequidistans ipsi DA; <lb/>et per ipsam transeat planum parallelum basi coni. </s>

<s>Erit ergò recta DA ad EB, ut quadratum DC ad CE; <lb/>

<arrow.to.target n="marg214"></arrow.to.target><lb/>sive ut quadratum DA ad EF, hoc est ut circulus DA <lb/>ad EF. Et hoc semper, tubicunque sit punctum E. Ergò <lb/>cum ad libram DC pendeant ab ijsdem punctis magnitu<lb/>dines duorum ordinum proportionales ut in praecedenti <lb/>lemmate imperatum est, habebunt omnes magnitudines <lb/>simul primi ordinis (hoc est omnes lineae trilinei ABCD, <lb/>sive ipsum trilineum) idem punctum aequilibrij, quod ha<lb/>bent omnes magnitudines simul secundi ordinis (hoc est <lb/>omnes circuli coni ACD, sive idem conus). Conus autem <lb/>aequiponderat ex puncto quod secat CD ita ut pars ad C <lb/>reliquae sit tripla, quandoquidem recta DA est ad ho<lb/>rizontem perpendicularis; ergo etiam trilineum ABCD <lb/>aequiponderabit ex eodem puncto. Quod erat proposi<lb/>tum etc. </s>

<s><margin.target id="marg214"></margin.target>ob parabo<lb/>lam.</s>

<s><emph type="center"/>PROPOSITIO XIV.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC, cuius diameter DE <lb/>

<arrow.to.target n="fig132"></arrow.to.target><lb/>intelligatur ad horizontem perpendicularis; <lb/>sintque CF et AD tangentes; ipsa vero AF <lb/>diametro aequidistans. Sumatur deinde FH <lb/>quarta pars totius FC; et ex puncto H (per <lb/>lemma praecedens) aequiponderabit trilineum <lb/>mixtum ABCF. Accipiatur etiam FI tertia pars <lb/>totius FC, et ex I aequiponderabit totum trian<lb/>gulum AFC. Parabola vero, cum babeat cen<lb/>trum in diametro, aequiponderat ex D. Ergo <lb/>trilineum ABCF ad ipsam parabolam erit re<lb/>ciprocè ut DI ad IH, nempe duplum (qualium <lb/>enim partium FC est 12. talium ipsa FD est 6. <lb/>FI verò 4. et FH 3. et ideó DI 2. et IH una). Propterea componendo <lb/>erit totum triangulum AFC, parabolae triplum. Reliquum quadraturae <lb/>absolvitur ut in Propositione IX. factum est. Quod erat etc.<gap desc="/SM"/></s>

<s><emph type="center"/><emph type="italics"/>Aliter.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Positis ijsdem, ut suprà, sumatur FH, quarta pars to<lb/>tius FC, aequiponderabitque ex puncto H trilineum mi<lb/>xtum ABCF. Sumatur etiam FI, ter<lb/>

<arrow.to.target n="fig133"></arrow.to.target><lb/>tia pars ipsius FD; tunc enim aequi<lb/>ponderabit ex puncto I triangulum <lb/>FDA. </s>

<s>Trilineum verò mixtum ABCD ae<lb/>quiponderat ex puncto D. (nam trian<lb/>gulum totum ADC aequiponderat ex <lb/>puncto D; parabola etiam ablata ex <lb/>eodem puncto D aequiponderat, ergò <lb/>etiam reliquum trilineum ABCD ex <lb/>puncto D aequiponderare necesse est). <lb/>Erit itaque triangulum FDA ad trilineum ABCD ut reci<lb/>procè DH ad HI, nempe ut 3. ad unum; et per conver<lb/>sionem rationis triangulum ADC ad parabolam erit ut 3. <lb/>ad 2. sive ut 6. ad 4. Quarè parabola ad triangulum ABC

<pb pagenum="147"/>est ut 4. ad 3. Nempe sesquitertia. Quod erat propositum <lb/>demonstrare etc. </s>

<s>Alijs etiam principijs parabolae quadraturam aggredia<lb/>mur, praemissa sequenti progressionum Geometricarum <lb/>speculatione. </s>

<s><emph type="center"/><emph type="italics"/>Lemma XXIV.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Si duae rectae lineae invicem concurrant, et inter ipsas descriptum <lb/>sit quoddam flexilineum constans ex lineis alternatim parallelis; erunt <lb/>omnes lineae quae inter se parallelae sunt, in continua proportione.<gap desc="/SM"/></s>

<s>Concurrant invicem due <lb/>

<arrow.to.target n="fig134"></arrow.to.target><lb/>rectae lineae AB, CB in pun<lb/>cto B; et inter ipsas descri<lb/>ptum sit flexilineum CADE <lb/>FG. etc. ita ut CA, DE, FG, <lb/>etc. sint inter se parallelae; item AD, EF, et reliquae <lb/>vicisim sumptae inter se parallelae sint. Dico AC, ED, GF, <lb/>esse in continua proportione. </s>

<s>Est enim, ob parallelas, ut AC ad ED, ita AB ad BE, <lb/>sive DB ad BF, hoc est ED ad GF. Constat ergò quod <lb/>

<arrow.to.target n="marg215"></arrow.to.target><lb/>propositum fuerat. </s>

<s><margin.target id="marg215"></margin.target>2. et 4. sexti.</s>

<s><emph type="center"/><emph type="italics"/>Lemma XXV.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Positis duabus rectis lineis invicem concurrentibus, ut suprà; si <lb/>inter ipsas fuerint duae parallelae AC, DE, et iunctà CD, continuatum <lb/>intelligatur flexilineum ACDE in infinitum usque ad pnnctum concur<lb/>sus B. Dico in huiusmodi flexilineo esse omnes, et singulos ad unguem <lb/>terminos qui sunt in progressione proportionis AC ad DE, in infinitum <lb/>continuatae.<gap desc="/SM"/></s>

<s>Ponatur F aequalis ipsi <lb/>

<arrow.to.target n="fig135"></arrow.to.target><lb/>AC, et G aequalis ipsi DE: <lb/>Et concipiatur propositio F <lb/>ad G continuata in infinitis <lb/>suis terminis FH. </s>

<s>Iam si possibile est, ali<lb/>quem, sive aliquos terminos <lb/>esse in progressione FH, qui non reperiantur in flexilineo.

<pb pagenum="148"/>Esto: et sit maximus terminus I, illorum, qui cum sint <lb/>in progressione FH, non sunt in flexilineo. Erit ergò ter<lb/>minus I ipsi praecedens, in flexilineo. Sit ille MN. Et quo<lb/>niam L ad I est ut F ad G, sive ut AC ad DE, sive ut <lb/>NM ad PO proximè sequentem, suntque aequales L, et NM; <lb/>erunt aequales etiam I et PO. Terminus ergo I qui pone<lb/>batur non esse in flexilineo, in eodem repertus est. </s>

<s>Eodem penitus modo demonstrabimus nullum terminum <lb/>esse in flexilineo, qui non sit etiam in progressione FH. <lb/>etc. Concludemus igitur esse in flexilineo omnes precisè <lb/>terminos proportionis AC ad DE in infinitum continuatae, <lb/>cum demonstratum sit nullum in flexilineo terminum de<lb/>siderari qui sit in progressione FH; neque ullum supera<lb/>bundare, qui non reperiatur etiam ia progressione FH. etc. </s>

<s><emph type="center"/><emph type="italics"/>Lemma XXVI.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Suppositis infinitis rectis lineis continua proportione maioris inae<lb/>qualitatis, rectam lineam, quae praedictis omnibus sit aequalis reperire.<gap desc="/SM"/></s>

<s>Ponantur primae duae lineae datae progressionis esse <lb/>A, B: quib. ponantur aequales; CD maiori A, et EF mi<lb/>nori B. Sintque CD, EF parallelae; et iungantur DF, CE, <lb/>quae necessariò concurrent. Concurrant. itaque in puncto <lb/>G, et ductà CF, ipsi aequidistans sit GL. </s>

<s>Dico rectam DL aequalem <lb/>

<arrow.to.target n="fig136"></arrow.to.target><lb/>esse omnibus infinitis termi<lb/>nis progressionis ABM simul <lb/>sumptis. </s>

<s>Concipiatur enim conti<lb/>nuatum flexilineum DCFE <lb/>etc. in infinitum, usque ad <lb/>punctum G, eruntque in ipso <lb/>omnes lineae, sive termini <lb/>datae progressionis ABM. </s>

<s>Producantur iam HE, NI, <lb/>et reliquae ipsis parallelae <lb/>

<arrow.to.target n="marg216"></arrow.to.target><lb/>usque ad DL. Eritque EF. <lb/>aequalis ipsi CP, et HI ae<lb/>qualis ipsi <expan abbr="Pq;">Pque</expan> et NO ipsi QR;

<pb pagenum="149"/>et sic de singulis. Qualibet enim linea quae sit in flexilineo, <lb/>habebit suam portiunculam respondentem in rectà DL, sibi <lb/>aequalem; donec flexilineum pervenerit ad ultimum pun<lb/>ctum G: Tunc autem neque de flexilineo, neque de linea <lb/>DL quidquam supererit; sed tam ipsum flexilineum, quàm <lb/>etiam recta DL penitus absumpta erit: Est enim ipsa GL, <lb/>quae ab ultimo flexilinei puncto G ducitur, ultima omnium <lb/>parallelarum, quae producuntur usque ad DL. Ergò omnes <lb/>simul lineae flexilinei, quarum prima est CD, alternatim <lb/>sumptae (hoc est omnes lineae poogressionis ABM) ae<lb/>quales sunt omnib. portiunculis rectae DL simul sumptis; <lb/>hoc est ipsi DL. Quod erat ostendendum etc. </s>

<s><margin.target id="marg216"></margin.target>34. primi.</s>

<s><emph type="center"/><emph type="italics"/>Lemma XXVII.<emph.end type="italics"/><emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Suppositis infinitis magnitudinibus in continua proportione Geome<lb/>trica maioris inaequalitatis, erit prima magnitudo media proportionalis <lb/>inter primam differentiam et inter aggregatum omnium.<gap desc="/SM"/></s>

<s>Assumptà enim praecedenti <lb/>

<arrow.to.target n="fig137"></arrow.to.target><lb/>

<arrow.to.target n="marg217"></arrow.to.target><lb/>constructione, ducatur FU aequi<lb/>distans ipsi GC: et erit DU prima <lb/>differentia. Sed DU ad primam <lb/>magnitudinem DC est ut FD ad <lb/>DG, hoc est ut DC ad DL aggre<lb/>gatum omnium. Quod erat demo<lb/>strandum etc. </s>

<s><margin.target id="marg217"></margin.target>4. Sexti.</s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hoc esse verum etiam in numeris, et cuiuscunque ge<lb/>neris magnitudinibus non dubitabimus affirmare. Afferre<lb/>mus etiam universaliorem demonstrationem, praecipuè cum <lb/>admodum brevis sit. Huius veritatis conclusis cum à nobis <lb/>obiter celeberrimo Cavalerio collata fuisset, ipse etiam <lb/>idem Theorema sequenti demonstratione, quae à nobis <lb/>iam in prima inventione adhibita fuerat, confirmavit. </s>

<s>Praemittitur hoc. Quod si fuerint quotcunque magni<lb/>tudines sive finitae numero, sive infinitae, quarum ante-

<pb pagenum="150"/>cedens semper sequente maior sit, erit prima omnium <lb/>magnitudo aequalis omnibus differentijs simul cum ipsa <lb/>minima magnitudine sumptis. </s>

<s>Notum est hoc apud Geometras, demonstraturque ut à <lb/>nobis factum est in Lemmate 15. Ubi ostendimus parallelo<lb/>grammum AE aequale esse omnibus differentis inter se<lb/>quentia parallelogramma, et minimo parallelogrammo OC. </s>

<s>Supponantur iam infinitae numero magnitudines in con<lb/>tinua proportione Geometrica maioris inaequalitatis; ma<lb/>nifestnm est quod minima omnium magnitudo vel non <lb/>erit, vel punctum erit. Ergo in hoccasu erit prima magni<lb/>tudo aequalis omnibus tantum differentijs. </s>

<s>Cum autem ponantur magnitudines in continua pro<lb/>portione Geometrica, erunt etiam differentiae in eadem <lb/>ratione proportionales; et ideo (factà conversione) erit ut <lb/>prima differentia ad primam magnitudinem, ita secunda <lb/>differentia ad secundam magnitudinem, et sic semper. <lb/>Propterea ut una ad unam, ita collectim erunt omnes ad <lb/>omnes. Nempe ut prima differentia ad primam magnitu<lb/>dinem, ita erunt omnes simul differentiae (hoc est ipsa <lb/>prima magnitudo) ad omnes magnitudines simul. </s>

<s>Constat ergò primam magnitudinem mediam propor<lb/>tionalem esse inter primam differentiam, et aggregatum <lb/>omnium </s>

<s><emph type="center"/>PROPOSITIO XV.<emph.end type="center"/><lb/><gap desc="SM"/></s>

<s>Parabola sesquitertia est trianguli eandem ipsi basim, et eandem <lb/>altitudinem habentis. </s>

<s>Esto parabola ABC in quà inscriptum <lb/>

<arrow.to.target n="fig138"></arrow.to.target><lb/>sit triangulum ABC. Dico parabolam <lb/>trianguli ABC esse sesquitertiam. </s>

<s>Inscribantur enim etiam in reliquis <lb/>portioniaus ADB, BEC, duo triangula <lb/>

<arrow.to.target n="marg218"></arrow.to.target><lb/>ADB, BEC. Eritque triangulum ABC, quadruplum duorum simul trian<lb/>

<arrow.to.target n="marg219"></arrow.to.target><lb/>gulorum ADB, BEC. Concipiantur etiam in reliquis quatuor portiun<lb/>culis AD, DB, BE, EC, inscripta quatuor triangula; eruntque duo simul <lb/>triangula ADB, BEC, quadrupla praedictorum simul quatuor subsequen<lb/>tium triangulorum; et hoc modo semper. Parabola igitur nihil aliud <lb/>est quàm aggregatum quoddam infinitarum numero magnitudinum in <lb/>proportione quadrupla, quarum prima est triangulum ABC, secunda

<pb pagenum="151"/>verò constat ex duobus triangulis ADB, BEC. Propterea prima magni<lb/>tudo ABC media proportionalis erit inter primam differentiam, et ag<lb/>gregatum omnium, nempe parabolam. </s>