Transcription of Tabula Numerorum Arithmeticæ Replicationis sur Figuratorum interpolata
Tabula Numerorum Arithmeticæ Replicationis sur Figuratorum interpolata
$\int,dx \left(1-x^{\underset{^{.....}}{2:p}}\right)^{\underset{^{............}}{n=m:2}}$ unde in $\int,dx\left(x^{\underset{^{......}}{w:p}}-x^{\underset{^{..................}}{(w:p)+(h:p)}}\right)^{\underset{^{...}}{n}}$, sit $w=0$, $h=2$, $n=m:2$.
Et ex canone generali a me invento, quo summa hæc in casu totali ab $x=0$ ad $x=1$ est $\frac{p, nh, (n-1)h, (n-2)h, \ \textrm{etc usque ad} \ (n-(n-1))h}{p+nw+nh, p+nw+(n-1)h, p+nw+(n-2)h, p + nw + (n-3)h, \ \textrm{etc. usque ad} \ (n-n)h}$ fit $\frac{p, m, m-2, m-4, m-6, \ \textrm{etc usque ad} \ 2}{p+m, p+m-2, p+m-4, p + m-6, p+m-8 \ \textrm{etc. usque ad} \ p+0}$ Ita tam numerator quam denominator exhaurietur si $m$ sit par, seu si $n$ sit integer.
Ponamus jam $p$ esse parem, seu divisibilem per $h$, hoc loco $2$, factores infiniti in numeratore coincidunt factoribus infinitis in denominatore, an ergo se compensabunt ita ut evanescat infinitas ? Exemplum videamus sit $p=6$ et $m=1$, et fiet $\frac{6,1,-1,-3,-5,-7,-9,-11,-13,-15, \ \textrm{etc}}{7,5,3,1,-1,-3,-5,-7,-9,-11,-13,-15, \ \textrm{etc}}$. An ergo $= \frac{6}{7.5.3} = \frac{2}{5.7}$ sed Tabula dat $\frac{1}{\overline{\frac{\frac{3}{2}.\frac{5}{2}.\frac{7}{2}}{1.2.3}}}=\frac{8.1.2.3}{3.5.7} = \frac{16}{5.7}$ Videntur autem $8.2$ (seu $16$) oriri ex totidem $h$ seu $2$ non destructus in numeratore, quot numeri sumantur in ipso denominatore, id ostendi potest restituto $h$, seu $2$, $\frac{6, \frac{1}{2}.2, \left(\frac{1}{2}-1\right)2, \left(\frac{1}{2}-2\right)2, \left(\frac{1}{2}-3\right)2, \left(\frac{1}{2}-4\right)2, \left(\frac{1}{2}-5\right)2, \ \textrm{etc}}{\left(3+\frac{1}{2}\right)2,\left(2+\frac{1}{2}\right)2,\left(1+\frac{1}{2}\right)2,\left(\frac{1}{2}\right)2, \left(-1+\frac{1}{2}\right)2, \left(-2+\frac{1}{2}\right)2, \left(-3+\frac{1}{2}\right)2,\ \textrm{etc}}$
Aliud exemplum : sit $p=8$ et $n=3:2$ et ex canone prodibit $\frac{8 ; \frac{3}{2}.2, \left(\frac{3}{2}-1\right)2, \left(\frac{3}{2}-2\right)2, \left(\frac{3}{2}-3\right)2, \left(\frac{3}{2}-4\right)2, \left(\frac{3}{2}-5\right)2, \ \textrm{etc}}{\left(4+\frac{3}{2}\right)2,\left(3+\frac{3}{2}\right)2,\left(2+\frac{3}{2}\right)2,\left(1+\frac{3}{2}\right)2, \frac{3}{2}.2, \left(-1+\frac{3}{2}\right)2, \left(-2+\frac{3}{2}\right)2,\ \textrm{etc}} = \frac{8}{11.9.7.5}$ sed deberet esse $\frac{1.2.3.4 ; 8}{11.9.7.5}$. itaque canon videtur esse reformandus.
Formemus canonem ex hac Tabula et videamus deinde an prioribus conciliari possit. Sit $p:2=q$ fiat $\int,dx\left(1-x^{\underset{^{.....}}{1:q}}\right)^{\underset{^{.....}}{m:2}} = \frac{q,q-1,q-2, \ \textrm{etc. usque ad} \ 1}{\frac{m+2}{2},\frac{m+2.2}{2}, \frac{m+3.2}{2}, \ \textrm{usque ad} \ \frac{m+(q-1).2}{2}} = \frac{q,q-1,q-2, \ \textrm{etc. usque ad} \ q-(q-1)}{\frac{m+(q-1).2}{2},\frac{m+(q-2).2}{2}, \ \textrm{etc. usque ad} \ \frac{m+(q-q).2}{2}}$ Ita res succedit si $m$ et $q$ sunt integri. Sed si $q$ sit fractus $=p:2$, tamen res succedit modo $m$ sit par ; ex ipsa ratione impar allata, ut arbitror dum scilicet infiniti factores se mutuo destruunt. Quod efficiendum, et deinde pergendum ad $\left(1 - x^{\underset{^{.....}}{1:q}}\right)^{\underset{^{.....}}{m:3}}$, et ad $\left(x^1 - x^{\underset{^{.....}}{1:q}}\right)^{\underset{^{...}}{n}}$ donec perveniatur ad $\left(x^w - x^{\underset{^{.....}}{1:q}}\right)^{\underset{^{.....}}{m:r}}$. Hoc comparetur cum prioribus canonibus, speciatur cum $(x^w -x^p)^n$ in quo etiam fiant interpolationes. Pauci tandem canonem vere universalem obtineamus qui destructionem infinitatis contineat in absolute quadrabilibus infinitam ubi necesse est. Et simul tandem trascendentes quadrabilium ostendat connexiones inter se. Quæ omnia inquam ex Canone habenda sunt, cum XXX in omni Tabula fiat ut diversæ summationes certis in sedibus respondeant inter se, seu ut Tabula sit quasi duplicata. Pes dein promoverimus ad Disectiones figurarum, nam scilicet ultima est non $1$ sed $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, etc, aut $\frac{1}{2}$, $\frac{1}{4}$ et $\frac{3}{4}$, $\frac{1}{8}$ et $\frac{3}{8}$ et $\frac{5}{8}$ et $\frac{7}{8}$. Utrum nunc quoque canones si oris utilitatis excogitari possint. Tandem videndum an non ei ipsi canones derivantur ex nostris seriebus infinitis dum ordinata irrationalis per seriem rationalem exhibetur ; et summatio iri idem, hæc autem applicatur a casus speciales videndum est quomodo proprietates quas Tabula inductione exhibet ex canone deriventur, ut hoc loco quod $A+B=C$.
[marge] Semper duo membra in $A$ et in $B$ perpendiculariter et transverse collineantes in eundem locum $C$ per simplicem saltum : summati componant numerum loci $C$ sic $A, \frac{\frac{5}{2}.\frac{7}{2}.\frac{9}{2}}{1.2.3} + B, \frac{\frac{7}{2}.\frac{9}{2}}{1.2}$ facit $C$ seu $\frac{\frac{7}{2}.\frac{9}{2}.\frac{11}{2}}{1.2.3}$
[Names of columns]
- Recip. $\sqrt{^{}}q$ seu $\fbox{$-1:2$}$
- Null $\fbox{$0:2$}$
- $\sqrt{^{}}q$ $\fbox{$1:2$}$
- Dignitatum simplicium $\fbox{$2:2$}$
- $\sqrt{^{}}q$.cubi $\fbox{$3:2$}$
- Quadrat $\fbox{$4:2$}$
- $\sqrt{^{}}q$.Surdesol. $\fbox{$5:2$}$
- cub $\fbox{$6:2$}$
- $\sqrt{^{}}q$.Surdesolidi secundi $\fbox{$7:2$}$
- Biqudrat $\fbox{$8:2$}$
[Names of lines]
- Subqartanis $1 - x^{1:4}$ $1 - x^{2:8}$
- quadratis subseptimanorum $1 - x^{2:7}$
- Subtertianis $1 - x^{\underset{^{.....}}{1:3}}$ $1 - x^{2:6}$
- quadratis subquintanorum $1 - x^{2:5}$
- Subsecundanis $1 - x^{1:2}$ seu $1 - x^{\underset{^{.....}}{2:4}}$
- quadratis sub3tianorum $1 - x^{\underset{^{.....}}{2:3}}$
- primanis seu subprimanis $1 - x$ seu $1 - x^{2:2}$
- secundanis $1 - x^{2}$ $1 - x^{2:1}$
- nullis $1 - x^{\underset{^{.....}}{\textrm{inf.}}}$ seu $1-0$ $1 - x^{2:0}$
- Recip. qtis $1 - x^{\underset{^{...}}{-2}}$ $1 - x^{2:-1}$
[Title of the table]
Tabulæ Numeri dividentes unitatem exhibent summam figuræ in cujus ordinatis sint æquales multati
[Note upper left]
et residui sunt exaltati ad