Blancanus, Josephus, Sphaera mvndi, sev cosmographia demonstratiua , ac facile methodo tradita : in qua totius Mundi fabrica, vna cum nouis, Tychonis, Kepleri, Galilaei, aliorumq' ; Astronomorum adinuentis continentur ; Accessere I. Breuis introductio ad geographiam. II. Apparatus ad mathematicarum studium. III. Echometria, idest Geometrica tractatio de Echo. IV. Nouum instrumentum ad Horologia

Table of Notes

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        <div xml:id="echoid-div11" type="section" level="1" n="10">
          <p>
            <s xml:id="echoid-s364" xml:space="preserve">
              <pb o="2" file="0018" n="18" rhead="Apparatus ad Sphœram."/>
            terum extende ad punctum B. </s>
            <s xml:id="echoid-s365" xml:space="preserve">atque hac apertura, nota punctum q, in ſemidiametro A C. </s>
            <s xml:id="echoid-s366" xml:space="preserve">in quo, ſito eodẽ
              <lb/>
            crure, alterum dilata vſq; </s>
            <s xml:id="echoid-s367" xml:space="preserve">ad B. </s>
            <s xml:id="echoid-s368" xml:space="preserve">atque hoc interuallo, manente eodem crure in B. </s>
            <s xml:id="echoid-s369" xml:space="preserve">altero imprime in circuli
              <lb/>
            peripheria punctum r, erit enim arcus B r, quinta pars totius circuli, ſeu graduum 72. </s>
            <s xml:id="echoid-s370" xml:space="preserve">cum autem B p, ſit gr.
              <lb/>
            </s>
            <s xml:id="echoid-s371" xml:space="preserve">75. </s>
            <s xml:id="echoid-s372" xml:space="preserve">eritarcus r p, gr. </s>
            <s xml:id="echoid-s373" xml:space="preserve">3. </s>
            <s xml:id="echoid-s374" xml:space="preserve">eo igitur accuratè diuiſo in 5. </s>
            <s xml:id="echoid-s375" xml:space="preserve">partes æquales, totidem gradus ac proinde gradũ etiam
              <lb/>
            vnum obtinebis, &</s>
            <s xml:id="echoid-s376" xml:space="preserve">c. </s>
            <s xml:id="echoid-s377" xml:space="preserve">huius praxis demõſtratio eſt apud Ptolem. </s>
            <s xml:id="echoid-s378" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s379" xml:space="preserve">1. </s>
            <s xml:id="echoid-s380" xml:space="preserve">Magnæ conſtr. </s>
            <s xml:id="echoid-s381" xml:space="preserve">quã refert etiam P. </s>
            <s xml:id="echoid-s382" xml:space="preserve">Cla-
              <lb/>
            uius ad 16. </s>
            <s xml:id="echoid-s383" xml:space="preserve">quarti Elem. </s>
            <s xml:id="echoid-s384" xml:space="preserve">Schol. </s>
            <s xml:id="echoid-s385" xml:space="preserve">2.</s>
            <s xml:id="echoid-s386" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s387" xml:space="preserve">Porrò quoniam Aſtronomi vnum gradum diuidunt in 60. </s>
            <s xml:id="echoid-s388" xml:space="preserve">particulas, quas alij Minuta, alij Prima, appel-
              <lb/>
            lant: </s>
            <s xml:id="echoid-s389" xml:space="preserve">propterea ſi circulus datus ſit adeo magnus, vt vlteriori partitioni ſufficiat, valde è re noſtra erit, ſin-
              <lb/>
            gulos gradus in 60. </s>
            <s xml:id="echoid-s390" xml:space="preserve">huiuſmudi prima ſeu minuta diſſecuiſſe; </s>
            <s xml:id="echoid-s391" xml:space="preserve">aut ſaltem in 5. </s>
            <s xml:id="echoid-s392" xml:space="preserve">partes æquas, quarum ſingu-
              <lb/>
            læ 12. </s>
            <s xml:id="echoid-s393" xml:space="preserve">minuta contineant. </s>
            <s xml:id="echoid-s394" xml:space="preserve">ſtatuunt præterea Aſtronomi, vnum primum continere 60. </s>
            <s xml:id="echoid-s395" xml:space="preserve">particulas, quas ſe-
              <lb/>
            cunda appellant, pariter vnum ſecundum continere 60. </s>
            <s xml:id="echoid-s396" xml:space="preserve">Tertia, &</s>
            <s xml:id="echoid-s397" xml:space="preserve">c. </s>
            <s xml:id="echoid-s398" xml:space="preserve">ſcribunt autem breuitatis cauſa, hu-
              <lb/>
            iuſmodi particulas hoc compendio, v. </s>
            <s xml:id="echoid-s399" xml:space="preserve">g.
              <lb/>
            </s>
            <s xml:id="echoid-s400" xml:space="preserve">
              <note position="right" xlink:label="note-0018-01" xlink:href="note-0018-01a" xml:space="preserve">
                <lb/>
              G. # |. # ||. # |||.
                <lb/>
              3. # 4. # 7. # 15.
                <lb/>
              </note>
            ideſt gr. </s>
            <s xml:id="echoid-s401" xml:space="preserve">3. </s>
            <s xml:id="echoid-s402" xml:space="preserve">Prima 4. </s>
            <s xml:id="echoid-s403" xml:space="preserve">ſecunda 7. </s>
            <s xml:id="echoid-s404" xml:space="preserve">& </s>
            <s xml:id="echoid-s405" xml:space="preserve">tertia 15. </s>
            <s xml:id="echoid-s406" xml:space="preserve">&</s>
            <s xml:id="echoid-s407" xml:space="preserve">c. </s>
            <s xml:id="echoid-s408" xml:space="preserve">pro numero enim apicum |. </s>
            <s xml:id="echoid-s409" xml:space="preserve">||. </s>
            <s xml:id="echoid-s410" xml:space="preserve">|||. </s>
            <s xml:id="echoid-s411" xml:space="preserve">ſuperpoſitionem de-
              <lb/>
            nominantur Prima, Secunda, Tertia, &</s>
            <s xml:id="echoid-s412" xml:space="preserve">c.</s>
            <s xml:id="echoid-s413" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s414" xml:space="preserve">Hocloco illud quoq; </s>
            <s xml:id="echoid-s415" xml:space="preserve">non ignorandum: </s>
            <s xml:id="echoid-s416" xml:space="preserve">ſi plures circuli ex eodem centro ſint deſcripti, ſeu ſint concen-
              <lb/>
            trici, atq; </s>
            <s xml:id="echoid-s417" xml:space="preserve">ex centro duę rectæ lineæ vſque ad vltimum circulum producantur, erunt arcus omnium circu-
              <lb/>
            lorum concentricorum inter cas intercepti ſimiles inuicem. </s>
            <s xml:id="echoid-s418" xml:space="preserve">ideſt tot gradus erunt in arcu minoris circuli,
              <lb/>
            quot in arcu maioris, vt in præcedenti figuræ duæ rectæ A C. </s>
            <s xml:id="echoid-s419" xml:space="preserve">A F. </s>
            <s xml:id="echoid-s420" xml:space="preserve">comprehendunt duos arcus F C. </s>
            <s xml:id="echoid-s421" xml:space="preserve">R S.
              <lb/>
            </s>
            <s xml:id="echoid-s422" xml:space="preserve">totq; </s>
            <s xml:id="echoid-s423" xml:space="preserve">gradus ſunt in vno atq; </s>
            <s xml:id="echoid-s424" xml:space="preserve">in altero: </s>
            <s xml:id="echoid-s425" xml:space="preserve">in minori quidem minores, in maiori vero maiores pro ratione cir-
              <lb/>
            culorum; </s>
            <s xml:id="echoid-s426" xml:space="preserve">quod ex ſe manifeſtum videtur, & </s>
            <s xml:id="echoid-s427" xml:space="preserve">experientia comprobari poteſt, & </s>
            <s xml:id="echoid-s428" xml:space="preserve">P. </s>
            <s xml:id="echoid-s429" xml:space="preserve">Clauius in ſcholio pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s430" xml:space="preserve">33. </s>
            <s xml:id="echoid-s431" xml:space="preserve">ſexti Elem. </s>
            <s xml:id="echoid-s432" xml:space="preserve">illud Geometricè demonſtrauit. </s>
            <s xml:id="echoid-s433" xml:space="preserve">Gradus dicti ſunt a gradiendo, quod præcipuè inſer-
              <lb/>
            uiant in cognoſcendis ſolis ac reliquorum Planetarum gradibus.</s>
            <s xml:id="echoid-s434" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div14" type="section" level="1" n="11">
          <head xml:id="echoid-head13" style="it" xml:space="preserve">Qua ratione Angulorum quantitates menſurentur. # Propoſ. 2.</head>
          <p>
            <s xml:id="echoid-s435" xml:space="preserve">GEometræ tantum dicunt eſſe quemlibet angulum, quantus arcus, qui ex ſummitate anguli tãquam cen-
              <lb/>
            tro deſcribitur, quique inter duas lineas angulum illum facientes intercipitur, eique ſubtenditur, vt in
              <lb/>
            ſuperiori figura angulus B A F. </s>
            <s xml:id="echoid-s436" xml:space="preserve">erit gr. </s>
            <s xml:id="echoid-s437" xml:space="preserve">60. </s>
            <s xml:id="echoid-s438" xml:space="preserve">quia arcus B F. </s>
            <s xml:id="echoid-s439" xml:space="preserve">illi ſubtenſus, deſcriptuſque ex A. </s>
            <s xml:id="echoid-s440" xml:space="preserve">extremitate
              <lb/>
            eiuſdem anguli B A F. </s>
            <s xml:id="echoid-s441" xml:space="preserve">eſt pariter gr. </s>
            <s xml:id="echoid-s442" xml:space="preserve">60. </s>
            <s xml:id="echoid-s443" xml:space="preserve">ſimiliter angulus F A C. </s>
            <s xml:id="echoid-s444" xml:space="preserve">erit gr. </s>
            <s xml:id="echoid-s445" xml:space="preserve">30. </s>
            <s xml:id="echoid-s446" xml:space="preserve">quia arcus F C. </s>
            <s xml:id="echoid-s447" xml:space="preserve">illi ſubtenſus eſt
              <lb/>
            gr. </s>
            <s xml:id="echoid-s448" xml:space="preserve">30. </s>
            <s xml:id="echoid-s449" xml:space="preserve">angulus vero B A C. </s>
            <s xml:id="echoid-s450" xml:space="preserve">qui rectus eſt co@tinet quantitatem gr. </s>
            <s xml:id="echoid-s451" xml:space="preserve">90. </s>
            <s xml:id="echoid-s452" xml:space="preserve">quia arcus B C. </s>
            <s xml:id="echoid-s453" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s454" xml:space="preserve">pariter 90. </s>
            <s xml:id="echoid-s455" xml:space="preserve">conti-
              <lb/>
            net; </s>
            <s xml:id="echoid-s456" xml:space="preserve">vnde omnes recti anguli ſunt gr. </s>
            <s xml:id="echoid-s457" xml:space="preserve">90. </s>
            <s xml:id="echoid-s458" xml:space="preserve">ratio, huius eſt quia v@ex vltima ſexti Elem. </s>
            <s xml:id="echoid-s459" xml:space="preserve">Patet, arcus habent
              <lb/>
            eandem inter ſe proportionem, quam habent anguli quos ſubrendunt, v. </s>
            <s xml:id="echoid-s460" xml:space="preserve">g. </s>
            <s xml:id="echoid-s461" xml:space="preserve">ita eſt arcus B F. </s>
            <s xml:id="echoid-s462" xml:space="preserve">ad F C. </s>
            <s xml:id="echoid-s463" xml:space="preserve">vti eſt
              <lb/>
            angulus B A F. </s>
            <s xml:id="echoid-s464" xml:space="preserve">ad angulum F A C. </s>
            <s xml:id="echoid-s465" xml:space="preserve">quod etiam facilè patere poteſt; </s>
            <s xml:id="echoid-s466" xml:space="preserve">ſi conſideremus arcum illum, qui angu-
              <lb/>
            lo opponitur, augeri, & </s>
            <s xml:id="echoid-s467" xml:space="preserve">minui ad diuaricationem, ac conſtrictionem linearum; </s>
            <s xml:id="echoid-s468" xml:space="preserve">ac proinde tantum eſſe,
              <lb/>
            quantus eſt angulus illi inſiſtens.</s>
            <s xml:id="echoid-s469" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s470" xml:space="preserve">Debetautem angulus, quando per tres litera@ eſt nominandus, ita nominari vt litera illa, quæ eſt ad an-
              <lb/>
            guli apicem medio loco proferatur, v. </s>
            <s xml:id="echoid-s471" xml:space="preserve">g. </s>
            <s xml:id="echoid-s472" xml:space="preserve">angulus, qui ad A. </s>
            <s xml:id="echoid-s473" xml:space="preserve">efficitur à duabus lineis B A. </s>
            <s xml:id="echoid-s474" xml:space="preserve">F A. </s>
            <s xml:id="echoid-s475" xml:space="preserve">nominandus
              <lb/>
            eſt angulus B A F. </s>
            <s xml:id="echoid-s476" xml:space="preserve">aut F A B. </s>
            <s xml:id="echoid-s477" xml:space="preserve">non autem B F A. </s>
            <s xml:id="echoid-s478" xml:space="preserve">aut F B A.</s>
            <s xml:id="echoid-s479" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div15" type="section" level="1" n="12">
          <head xml:id="echoid-head14" style="it" xml:space="preserve">Omne triangulum habere tres angulos continentes gr. 180. hoc est, aquales eſſe
            <lb/>
          duobus rectis angulis. # Propoſ 3.</head>
          <p>
            <s xml:id="echoid-s480" xml:space="preserve">EVclides hoc Geometricè docet ad 32. </s>
            <s xml:id="echoid-s481" xml:space="preserve">primi, quæ vel Geometrię Tyronibus notiſſima eſt. </s>
            <s xml:id="echoid-s482" xml:space="preserve">quod ſi lecto@
              <lb/>
            adeo Geometria leuiter imbutus ſi@, vt eam nondum perceperit liceat nobis in gratiam eius, huius pro-
              <lb/>
            poſitionis tale experimentum afferre. </s>
            <s xml:id="echoid-s483" xml:space="preserve">Sit trianguum quodcũque A B C. </s>
            <s xml:id="echoid-s484" xml:space="preserve">Dico tres ipſius angulos A B C.
              <lb/>
            </s>
            <s xml:id="echoid-s485" xml:space="preserve">
              <figure xlink:label="fig-0018-01" xlink:href="fig-0018-01a" number="5">
                <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0018-01"/>
              </figure>
            ſimul ſumptos eſſe æquales duobus rectis angulis, ſeu continere gradus 180
              <lb/>
            ex prima enim propoſitione ſuperiori manifeſtum eſt duos angulos rectos
              <lb/>
            continere gr. </s>
            <s xml:id="echoid-s486" xml:space="preserve">180. </s>
            <s xml:id="echoid-s487" xml:space="preserve">Facto ig@tur cẽtro in A. </s>
            <s xml:id="echoid-s488" xml:space="preserve">deſcribitur arcus D E. </s>
            <s xml:id="echoid-s489" xml:space="preserve">qui per pri-
              <lb/>
            mam pro poſitionem huius apparatus, expendatur. </s>
            <s xml:id="echoid-s490" xml:space="preserve">quot gradus continat per
              <lb/>
            diuiſionem circuli vel ſemicirculi, vel quadrant@s, cu@us pars ſit ipſe arcus.
              <lb/>
            </s>
            <s xml:id="echoid-s491" xml:space="preserve">idem faciendum erit, circa reliquos angulos B C. </s>
            <s xml:id="echoid-s492" xml:space="preserve">vt eorũ gradus inueſtigen-
              <lb/>
            tur: </s>
            <s xml:id="echoid-s493" xml:space="preserve">Inuentis igitur gradibus ſingulorũ angulorum, ij ſimul addantur, ſum-
              <lb/>
            mainque ſemper efficient gr. </s>
            <s xml:id="echoid-s494" xml:space="preserve">180. </s>
            <s xml:id="echoid-s495" xml:space="preserve">quæ eſt quanti as d@orũ rectoium: </s>
            <s xml:id="echoid-s496" xml:space="preserve">Et hoc
              <lb/>
            erat probandum. </s>
            <s xml:id="echoid-s497" xml:space="preserve">idem experiri poteris hoc modo, nam ſi componantur ſi-
              <lb/>
            mul tres arcus tribus angulis ſubtenſi, ſemicirculum conflabunt. </s>
            <s xml:id="echoid-s498" xml:space="preserve">Aliter ſic
              <lb/>
            idem experieris: </s>
            <s xml:id="echoid-s499" xml:space="preserve">ſacto diligenter triangulo ex charta; </s>
            <s xml:id="echoid-s500" xml:space="preserve">ipſius angulos reſcin-
              <lb/>
            de, eoſque ad centrum A. </s>
            <s xml:id="echoid-s501" xml:space="preserve">circuli figuræ Propoſ. </s>
            <s xml:id="echoid-s502" xml:space="preserve">1. </s>
            <s xml:id="echoid-s503" xml:space="preserve">alterum apud alterum
              <lb/>
            applica, ita vt mucrones angulorum ſimul conuen@ant ad A. </s>
            <s xml:id="echoid-s504" xml:space="preserve">lateraque eo-
              <lb/>
            rum ſe mutuo contingant: </s>
            <s xml:id="echoid-s505" xml:space="preserve">ſtatim enim videbis eos occupare ſpatium, cui in
              <lb/>
            periferia ſubtenduntur gr. </s>
            <s xml:id="echoid-s506" xml:space="preserve">180. </s>
            <s xml:id="echoid-s507" xml:space="preserve">ſiue ſemicirculus; </s>
            <s xml:id="echoid-s508" xml:space="preserve">& </s>
            <s xml:id="echoid-s509" xml:space="preserve">latera extrema efficient
              <lb/>
            lineam rectam, quæ cum diametro congruet.</s>
            <s xml:id="echoid-s510" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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