Transcription of Appropinquatio circuli per radices dyadice expressas [1676-1716(?)]
$4\sqrt{2}$ est ambitus quadrati circulo inscripti, $8\sqrt{2-\sqrt{2}}$ octanguli, $16\sqrt{2-\sqrt{2+\sqrt{2}}}$ sedecanguli, et $32\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$ est ambitus $32$anguli, et ita porro, vel brevius exprimendo Ambitus quadrati, $8$guli, $16$guli erit $4\sqrt{2-\textcircled{\small{0}}} 8\sqrt{2-\textcircled{\small{1}}} 16\sqrt{2-\textcircled{\small{2}}}$ et generaliter $2^v \sqrt{2- \textcircled{\small{v.2}}}$ aeq.$\mathcal{D}$. Ergo $2^{2v} . \overline{2-\textcircled{\small{v.2}}}$ aeq. $\mathcal{D}^2$.
Quoniam autem in progressione notarum numericarum dyadica, multiplicatio per potentias ipsius $2$ fit sola adjectione nullarum seu zero, hinc tantum opus erit ad progressionem inveniendam, qua continue ad ambitum circuli accedatur ut extrahatur radix ex $2$, ex $2+\sqrt{2+\sqrt{2}}$ &c. Radixque illa quaevis detrahatur ex $2$, residuum continue accedet ipsi quadrato circumferentiae. Ut autem subtractio sit facilior tantum pro $2$ seu $10.00000$ ponitur 1.111111111. Aggrediamur jam opere ipso radicis extractionem ex $1.1111111$
$11111$ in $11111$ est minus quam $100000$ in $100000$ sed $111111$ est maximus in suo gradu. Ergo characteres nunquam magis quin duplicantur multiplicando in seipsum uno ad summum maximi
$L$ aequ. $1$ et $a$ aeq. $1$ ergo $b$ aeq. $0$ ergo $\begin{array}{ccccccccccccccc} &+ & b &+ & c &+ & d &+ & e &+ & f &+ & g & &\\ a &+ & b &+ & c &+ & d &+ & e &+ & f &+ & g & &\\ a^2 &+ & 2ab &+ & 2ac &+ & 2ad &+ & 2ae &+ & 2af &+ & 2ag & &\\ & & &+ & b^2 &+ & 2bc &+ & 2bd &+ & 2be &+ & 2bf & &\\ & & & & & & &+ & c^2 &+ & 2cd &+ & 2ce & &\\ & & & & & & & & & & &+ & d^2 & &\\ & & & & & & & & & & & & & &\\ ab& & & & ad & & bd & & af & & bf & & ah & bb &\\ & & & & bc & & bd & & fe & & ce & & bg & cg &\\ & & & & & & & & ed & & & & ef & df &\\ & & & & & & & & & & & & de & et &\\ & & & & & & & & & & & & & &\\ a& & ac & & b & & ac & & c & & ag & & d & ai &\\ ab& & & & bc & & bd & & cd & & bf & & de & bh &\\ & & & & ad & & & & be & & ce & & cf & cg &\\ & & & & & & & & af & & & & bg & dg &\\ & & & & & & & & & & & & ah & &\\ & & & & & & & & & & & & & &\\ L& &M & &N & &P & &Q & &R & &S & T & \end{array}$
potius inversum:
$\begin{array}{cccccccccccccccccccccccccc} & & & & & & & & & & & &g &+ &f &+ &e &+ &d &+ &c &+ &b &+ &a & \\ & & & & & & & & & & & &g &+ &f &+ &e &+ &d &+ &c &+ &b &+ &a & \\ & & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & &2ga & &2fa & &2ea & &2da & &2ca & &2ba & &aa & \\ & & & & & & & & & &2gb & &2fb & &2eb & &2db & &2cb & &bb & & & & & \\ & & & & & & & &2gc & &2fc & &2ec & &2dc & &cc & & & & & & & & & \\ & & & & & &2gd & &2fd & &2ed & &dd & & & & & & & & & & & & & \\ & & & &2ge & &2fe & &ee & & & & & & & & & & & & & & & & & \\ & &2gf& &ff & & & & & & & & & & & & & & & & & & & & & \\ gg& & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & & & & & & & & & & & & & & & \\ gf& &ge & &gd & &gc & &gb & &ga & &fa & &ea & &da & &ca & &ba & & * & &a & \\ g& & & &fe & &fd & &fc & &fb & &eb & &db & &cb & & & &b & & & & & \\ & & & &f & & & &ed & &ec & &dc & & & &c & & & & & & & & & \\ & & & & & & & &e & & & &d & & & & & & & & & & & & & \\ Z& &Y & &X & &W & &U & &T & &S & &R & &Q & &P & &N & &M & &L & \end{array}$
$a$ aeq. $L$. Si $N$ sit $1$ tum $A$ aeq. $0$.
Quadrati numeri dyadice expressi penultima litera est $0$, seu $M$ aeq. $0$. $ba+b$ aeq. $N$ sed $a$ aeq. $L$. Ergo $bL+b$ aeq. $N$, $1b$ aeq. $N\cdot1+L$ seu $b$ aeq. $N+NL$, si $N$ aeq. $1$ seu $N$ aeq. $ba+b$ vel $0$ si $b$ sit $1$ et $a$ sit $1$, $N$ est $0$ item si $b$ sit $0$ et $a$ sit $0$ hinc ita si $b-a$ aeq. $0$ tunc $b^2-2ba+a^2$ aeq. $1$. Seu $b-2ba+a$ aeq. $1$ tunc $N$ est $1$. Ambo casis habebum tum $b-2ba+a-1$ aeq. $0$ ducas in $b-a$ aeq. $0$ et fiet $b-3ba+3ba-a-b+a$ aeq. $0$ sed ita sumto $\langle$-$\rangle$.