Clavius, Christoph
,
Geometria practica
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LIBER OCTAVVS.
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<
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">DE DIFFERENTHS QVADRATO-
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rum & cuborum, & de continuationeta-
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bulæ eorundem.</
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<
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<
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">Qvoniam</
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>
quadrati numeri creantur per continuam additionem nume-
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note-415-01
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note-415-01a
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xml:space
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">Differentiæ
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quadratorum</
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>
rorum imparium, vt Arithmetici demonſtrant: </
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<
s
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xml:space
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">fit vt differentia inter quemlibet
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quadratum, & </
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<
s
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echoid-s17819
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xml:space
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">proximè inſequen-
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<
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note-415-02
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note-415-02a
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Numeri \\ impares. # Quadra- \\ ti. # Radices.
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1 # 1 # 1
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3 # 4 # 2
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5 # 9 # 3
<
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7 # 16 # 4
<
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9 # 25 # 5
<
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11 # 36 # 6
<
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13 # 49 # 7
<
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15 # 64 # 8
<
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17 # 81 # 9
<
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19 # 100 # 10
<
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21 # 121 # 11
<
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23 # 144 # 12
<
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</
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tem ſit duplaradicis minoris, addi-
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ta inſuper vnitate. </
s
>
<
s
xml:id
="
echoid-s17820
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xml:space
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">Itaque duobus
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modis tabula quadratorum com-
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poni poteſt, & </
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>
<
s
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="
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">continuari. </
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<
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">Vno
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<
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xlink:label
="
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="
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xml:space
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">Compoſitio ta
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unsure
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-
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bulæ quadra-
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torum.</
note
>
modo, ſi omnes numeri impares
<
lb
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ordine ponantur, initio ſumpto ab
<
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/>
1. </
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<
s
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="
echoid-s17823
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xml:space
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">Nam 1. </
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>
<
s
xml:id
="
echoid-s17824
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xml:space
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">dat primum quadratum
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1. </
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>
<
s
xml:id
="
echoid-s17825
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xml:space
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">Et ex 1. </
s
>
<
s
xml:id
="
echoid-s17826
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xml:space
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">& </
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>
<
s
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="
echoid-s17827
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">3. </
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>
<
s
xml:id
="
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">fit ſecundus 4. </
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>
<
s
xml:id
="
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xml:space
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">cui ſi
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addatur ſequens impar 5. </
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>
<
s
xml:id
="
echoid-s17830
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xml:space
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">fit tertius
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9. </
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<
s
xml:id
="
echoid-s17831
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xml:space
="
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">& </
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>
<
s
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="
echoid-s17832
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xml:space
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">ſi addatur impar ſequens 7. </
s
>
<
s
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echoid-s17833
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xml:space
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">fit
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quartus quadratus 16. </
s
>
<
s
xml:id
="
echoid-s17834
"
xml:space
="
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">at que ita de-
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inceps. </
s
>
<
s
xml:id
="
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xml:space
="
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">Habet autem quilibet qua-
<
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/>
dratusra dicem tot vnitatum, quot
<
lb
/>
numeri impares ipſum conficiunt.
<
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/>
</
s
>
<
s
xml:id
="
echoid-s17836
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xml:space
="
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">Vt quia ſolus impar 1. </
s
>
<
s
xml:id
="
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xml:space
="
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">dat primum
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quadratum 1. </
s
>
<
s
xml:id
="
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xml:space
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">propterea eius radix
<
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/>
eſt 1. </
s
>
<
s
xml:id
="
echoid-s17839
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xml:space
="
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">Deinde quia duo impares 1. </
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>
<
s
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="
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<
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& </
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<
s
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="
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">3. </
s
>
<
s
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="
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xml:space
="
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">conficiunt ſecundum quadra-
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tum 4. </
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>
<
s
xml:id
="
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xml:space
="
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">erit eius radix 2. </
s
>
<
s
xml:id
="
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xml:space
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">Sic quia
<
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duo decim numeri impares 1. </
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>
<
s
xml:id
="
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xml:space
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">3. </
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>
<
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="
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">5. </
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<
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="
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">7. </
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>
<
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="
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">
<
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9. </
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<
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="
echoid-s17849
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">11. </
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<
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">13. </
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<
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">15. </
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<
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">17. </
s
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<
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="
echoid-s17853
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">19. </
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<
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="
echoid-s17854
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">21. </
s
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<
s
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="
echoid-s17855
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">23. </
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<
s
xml:id
="
echoid-s17856
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xml:space
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">compo-
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nunt quadratum 144. </
s
>
<
s
xml:id
="
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xml:space
="
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">erit eius ra-
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dix 12. </
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<
s
xml:id
="
echoid-s17858
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xml:space
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">& </
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<
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">ſic de cæteris. </
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">Atq; </
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<
s
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">in hũc
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/>
modum ſi ſemper ſequens nume-
<
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us impar adiiciatur ad quadratum præcedentem, conflatur ſequens numerus
<
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/>
quadratus, continuabiturque tabula in infinitum: </
s
>
<
s
xml:id
="
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xml:space
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">ſitamen prius ſeries nume-
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rorum imparium continuetur. </
s
>
<
s
xml:id
="
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xml:space
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">Radices ſerie numerorum naturali progre-
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diuntur.</
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<
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xml:space
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</
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<
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<
s
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<
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="
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modo condi poterit tabula quadratorum, & </
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>
<
s
xml:id
="
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xml:space
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">in infinitum continua-
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ri, ſine numerorum imparium ſerie, ſi omnes radices ponantur ordine, vt in tabu-
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la vides. </
s
>
<
s
xml:id
="
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xml:space
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">Cum enim primus quadratus ſit 1. </
s
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<
s
xml:id
="
echoid-s17868
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xml:space
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">cuius radix 1. </
s
>
<
s
xml:id
="
echoid-s17869
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xml:space
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">ſi hæc radix dupli-
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cata, addita inſuper 1. </
s
>
<
s
xml:id
="
echoid-s17870
"
xml:space
="
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">addatur primo quadrato 1. </
s
>
<
s
xml:id
="
echoid-s17871
"
xml:space
="
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">fit ſecundus 4. </
s
>
<
s
xml:id
="
echoid-s17872
"
xml:space
="
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">cuius ra-
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dix 2. </
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<
s
xml:id
="
echoid-s17873
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xml:space
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">Hæc duplicata, & </
s
>
<
s
xml:id
="
echoid-s17874
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xml:space
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">inſuper addita 1. </
s
>
<
s
xml:id
="
echoid-s17875
"
xml:space
="
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">ſi adiiciatur ſecundo quadra-
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to 4. </
s
>
<
s
xml:id
="
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xml:space
="
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">fit tertius 9. </
s
>
<
s
xml:id
="
echoid-s17877
"
xml:space
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">cuius radix 3. </
s
>
<
s
xml:id
="
echoid-s17878
"
xml:space
="
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">quæ duplicata, & </
s
>
<
s
xml:id
="
echoid-s17879
"
xml:space
="
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">inſuper addita 1. </
s
>
<
s
xml:id
="
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"
xml:space
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">facit 7.
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</
s
>
<
s
xml:id
="
echoid-s17881
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xml:space
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">Si igitur addantur 7. </
s
>
<
s
xml:id
="
echoid-s17882
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="
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">ad quadratum 9. </
s
>
<
s
xml:id
="
echoid-s17883
"
xml:space
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">fit quartus quadratus 16. </
s
>
<
s
xml:id
="
echoid-s17884
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xml:space
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">& </
s
>
<
s
xml:id
="
echoid-s17885
"
xml:space
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">ſic in infi-
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nitum.</
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<
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</
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<
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<
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<
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style
="
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">Nvmeri</
emph
>
autem cubi gignuntur quoque ex additione numerorum impa-
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rium, hoc modo. </
s
>
<
s
xml:id
="
echoid-s17888
"
xml:space
="
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">Deſcripta ſerie imparium numerorum ab 1. </
s
>
<
s
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">incipientium, </
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>
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