<1r>

June 13th 1676.

Dignissime {Vir}

{Quanquam D. Leibnitij, modestia} in excerptis quæ ex Epistola ejus ad me {nuper misisti nostratibus} multum tribuat circa speculationem {quandam infinitarum} serierum de quâ jam cœpit esse rumor: nullus dubito tamen quin ille, non tantum (quod asserit) methodum reducendi quantitates quascunque in ejusmodi series, sed et varia compendia, fortè nostris similia, si non et meliora, adinvenerit. Quoniam tamen ea scire pervelit quæ ab Anglis hâc in re inventa sunt, et ipse ante annos aliquot in hanc speculationem inciderim: ut votis ejus aliqua saltern ex parte satisfacerem iam, nonnulla eorum quæ mihi occurrerunt, ad te transmisi.

Fractiones in infinitas series reduco per divisionem et quantitates radicales per Extractionem radicum, perindè instituendo operationes istas in speciebus ac {in}stitui solent in decimalibus numeris. Hæc sunt fundamenta harum reductionum; sed extractiones radicum multum abbreviantur per hoc Theorema.

P+PQmn=Pmn+mnAQ+mn2nBQ+m2n3nCQ+m3n4nDQ+&c.
Ubi P+PQ significat quantitatem cujus radix vel etiam dimentio quævis vel radix dimensionis investiganda est. P primum terminum quantitatis ejus, Q reliquos terminos divisos per primum, & mn numeralem indicem dimensionis ipsius P+PQ sive dimentio illa integra sit, sive (ut ita loquar) fracta, sive affermativa, sive negativa. Nam sicut Analystæ pro aa aaa &c scribere solent a2, a3, sic ego pro a, a3, c.a5 &c scribo a12, a32, a53, & pro 1a, 1aa, 1a3 scribo a1, a2, a3. Et sic pro aac:a3+bbx scribo aa×a3+bbx13, & pro aab.c:a3+bbx×a3+bbx scribo aab×a3+bbx23: in quo ultimo casu si a3+bbx23 conciapiatur esse P+PQmn in Regulâ; erit P=a3, Q=bbxa3, m=2, & n=3. Denique pro terminis inter operandum inventis in Quoto, usurpo A, B, C, D &c nempe A pro primo termino Pmn, B pro secundo mnAQ & sic deinceps. Cæterum usus Regulæ patebit Exemplis.

Exempl: 1. Est |cc+xx|seucc+xx12=c+xx2cx48c3+x616c55x8128c7 +7x10256a9+&c.. Nam in hoc casu est P=cc, Q=xxcc, m=1, n=2, A=Pmn=cc12=c. B=mnAQ=xx2c. C=mn2nBQ=x48c3, & sic deinceps.

Exempl: 2. Est c5+c4xx5i.e.c5+c4xx515=c+c4xx55c42c8xx+4c4x625c92x1025c9+&c. ut patebit substituendo in allatam Regulam, 1 pro m, 5 pro n, c5 pro {P, & c4xx5c5} pro Q. Potest etiam x5 substitui pro P, & c4x+c5x5 pro Q, et tunc <1v> evadet c5+c4xx5=x{ +c4x+c55x4+2c8xx+4c9x+c1025x9+&c}. Prior modus eligendus est si x valde {parvum sit, posterior si valde} magnum.

Exempl: 3. Est Ny3aay{hoc estN×y3aay13=N×}1y+aa3y3+2a49y5+7a681y7+&c. Nam P=y3. Q=aayy. m=1. n=3. {A=Pmn=y3×13} =y1. hoc est 1y. B=mnAQ=13×1y×aayy=aa3y3&c.

Exempl: 4. Radix cubica ex quadrato-quadrato ipsius d+e (hoc est d+e43 Est d43+4ed133+2ee9d234e381d53+&c. Nam P=d. Q=ed. m=4. n=3. A=Pmn=d43 &c.

Eodem modo simplices etiam potestates eliciuntur. Ut si quadrato-cubus ipsius d+ehoc estd+e5, seud+e51 desideretur: erit juxta Regulam P=d. Q=ed. m=5 & n=1; adeoque A=Pmn=d5, B=mnAQ=5d4e, & sic C=10d3ee, D=10dde3, E=5de4, F=e5, & G=m5n6nFQ=0. Hoc est d+e5=d5+5d4e+10d3ee+10dde3+5de4+e5.

Quinetiam Divisio, sive simplex sit, sive repetita, per eandem Regulam perficitur. Ut si 1d+e, hoc estd+e1sived+e11 in seriem simplicium terminorum resolvendum sit: erit juxta Regulam P=d. Q=ed. m=1. n=1. & A=Pmn=D11=d1 seu 1d. B=mnAQ=1×1d×ed =edd, & sic C=eed3, D=e3d4 &c Hoc est 1d+e=1dedd+eed3e3 d4+&c.

Sic et d+e3 (hoc est unitas ter divisa per d+e vel semel per cubum ejus,) evadit 1d33ed4+6eed510e3d6+&c.

Et N×d+e13 hoc est N divisum per radicem cubicam ipsius d+e evadit N×1d13e3d43+2ee9d7314e381d108+&c

Et N×d+e35 (hoc est N divisum per radicem quadrato-cubicam ex cubo ipsius d+e, sive Nd3+3dde+3dee+e3 evadit N×1d353e5d95+12ee25d13552e3125d185&c.

Per eandem Regulam Genesses Potestatum, Divisiones per potestates aut per quantitates radicales, et extractiones radicum altiorum in numeris etiam commodè instituuntur.

Extractiones Radicum affectarum in speciebus imitantur earum extractiones in numeris, sed methodus Vietæ et Oughtredi nostri huic negotio minus idonea est, quapropter aliam excogitare adactus sum cujus specimen exhibent sequentia Diagrammata ubi dextra columna prodit substituendo in mediâ columnâ valores ipsorum y, p, q, r &c in sinistra columnâ expressos. Prius Diagramma exhibet resolutionem hujus numeralis æquationis y32y5=0; Et hic in supremis numeris pars negativa Radicis subducta de parte affirmativa relinquit absolutam Radicem 2|09455148: et posterius Diagramma exhibet resolutionem hujus liter{ariæ} æquationis y3+axy+aayx32a3=0.

<2r>

_________________________._____________.__________________________________________________ +2,100000000,00544852 +2,09455148 _________________________._____________.__________________________________________________ 2+p=y y3 2y 5 +8+12p+6pp+p3 402p 5 p3summap3 1+10p+6pp+p3 _________________________._____________.__________________________________________________ +0,1+q=p +p3 +6pp +10p 1 +0,001+00,03q+0,3qq+q3 +0,061+01,20q+6,3qq +1,001+10,0q 1 q3summaq3 +0,061+11,23q+6,3qq+q3 _________________________._____________.__________________________________________________ 0,0054+r=q +q3 +6,3qq +11,23q +0,061 0,0000001+00,000r+&cq3 +0,000183700,068q 0,0606420+11,23q +0,061 q3summaq3 0,0005416+11,162r _________________________._____________.__________________________________________________ 0,00004852+s=r ._____________. _ _

_ _________________________._____________.__________________________________________________ ax4+xx64a+131x3512aa+509x416384a3&c _________________________._____________.__________________________________________________ a+p=y y3 +a3+3aap+3app+p3 +axy +aax+axp +aay +a3+aap x3 x3 2a3 2a3 _________________________._____________.__________________________________________________ _________________________._____________.__________________________________________________ 14x+q=p p3 164x3+316xxq&c +3app +316axx32axq+3aqq +axp 14axx+axq +4aap axx+4aaq +aax +aax x3 x3 _________________________._____________.__________________________________________________ _________________________._____________.__________________________________________________ +xx64a+r=q 3aqq +3x44096a&c +316xxq +3x41024a&c 12axq 1128x312axr +4aaq +116axx+4aar x3 x3 6564a3 6564a3 116aax 116aax _________________________._____________.__________________________________________________ _ 000+4aa12ax+131128x315x44096a+131x3512aa+509x416384a3.000

In prior Diagrammate primus terminus valoris ipsorum p, q, r, in prima columnâ invenitur dividendo primum terminum summæ proximè superioris per coefficientem secundi termini ejusdem summæ ut 1 per 10 aut 0,061 per 11,23 et mutando signum quoti.: et idem terminus eodem ferè modo invenitur in secundo diagrammate. Sed Verùm hic præcipua difficultas est in inventione primi termini radicis: methodo um generali em qua id perficitur, {anc} brevitatis gratia prætereo, ut et alia quædam quæ ad concinnandam operationem spectant. Neque enim hic compendia tradere vacat, dicam tantum in genere quod radix cujusvis æquationis semel extracta pro regula resolvendi consimiles æquationes asservari possit; quodque ex pluribus ejusmodi regulis, regulam generaliorem plerumque efformare liceat; & quod radices omnes, sive simplices sint sive affectæ, modis infinitis extrahi {p}ossint, de quorum simplicioribus itaque semper consulendum est.

<2v>

Quomodo ex æquationibus, {sic ad infinitas series reductis, ar}eæ & longitudines curvarum, cont{en}ta et sup{erficies solidorum, vel quorum}libet segmentorum figurarum quarumvis eoru{mque centra gravitatis deter}minantur, & quomodo etiam Curvæ omnes Mechanicæ {ad ejusmodi æquation}es infinitarum serierum reduci possint, indeque Prob{lemata circa ill}as resolvi perinde ac si geometricæ essent, nimis longum foret describere. Sufficiat cerit specimina quædam talium Problematum recensuisse: inque ijs brevitatis gratia literas A, B, C, D &c pro terminis seriei, sicut sub initio, nonnunquam usurpabo.

1. Si ex dato sinu recto vel sinu verso arcus desideretur: sit radius r & sinus rectus x eritque arcus =x+x36rr+3x540r4+5x7112r6+&c. hoc est =x+1×1×xx2×3×rrA+3×3xx4×5rrB+5×5xx6×7rrC+7×7xx8×9rrD+&c. Vel sit d diameter & x sinus versus, et erit arcus =d12x12+x326d12+3x5240d32+5x72112d52+&c hoc est =dxin1+x6+3xx40d+5x3112dd+&c.

2. Si vicissim ex dato arcu desiderentur sinus: sit radius r et arcus z, eritque sinus rectus =zz36rr+z5120r4z75040r6+z936288r8&c, hoc est =zzz2×3rrAzz4×5rrBzz6×7rrC&c; Et sinus versus =zz2rz424r3 +z6720r5z84032r7+&c, hoc est zz1×2rzz3×4rrAzz5×6rrBzz7×8C.

3. Si arcus capiendus sit in ratione datâ ad alium arcum: esto circuli diameter =d, Chorda arcûs dati =x, & arcus quæsitus ad arcum illum datum ut n ad 1; eritque arcûs quæsiti chorda =nx+1nn2×3ddxxA+9nn4×5ddxxB+25nn6×7ddxxC +36nn8×9ddxxD+49nn10×11ddxxE+&c. Ubi nota quod si n est numerus impar, series desinet esse infinita, & evadet eadem quæ prodit per vulgarem Algebram ad multiplicandum datum angulum per istum numerum n.

4. Si in axe alterutro AB ellipseos ADB (cujus Figure centrum C & axis alter DH) detur punctum aliquod E circa quod recta EG occurrens Ellipsi in G motu angulari feratur, & ex datâ area sectoris Ellipticæ BEG quæratur recta GF quæ à puncto G ad axem AB normaliter demittitur: esto BC=q, DC=r, EB=t, ac duplum areæ BEG=z; & erit GF=ztqz36rrt4+10qqqqt120r4t7z5 280q3+504qqt225qtt5040r6t10z7+&c. Sic itaque Astronomicum illud Kepleri Problema resolvi potest.

5. In eâdem Ellipsi si statuatur CD=r, CBqCD=c, & CF=x, erit arcus Ellipticus DG=x+16ccx3+110rc3x5+114rrc4x7+118r3c5x9+122r4c6x11+&c 140c4128rc5 124rrc6122r3 c7 +1112c6+148rc7+388rrc8 51152c85352rc9 +72816c10
Hic numerales coefficientes supremorum terminorum 16.110.114&c sunt in musica progressione, & numerales coefficientes omnium inferiorum in unaquaque columna prodeunt multiplicando continuò <3r> Numeralem coefficientem supremi termini per terminos hujus progressionis 12n12.33n34.54n56.75n78.96n910.&c: ubi n significat numerum dimensionum ipsius c in denominatore istius supremi termini. E:g: ut terminorum infra 122r4c6, numerales coefficientes inveniantur, pono n=6, ducoque 122 (numeralem coefficientem ipsius 122r4c6) in12n12 hoc est in1; et prodit 122 numeralis coefficiens termini proximè inferior{is;} dein duco hunc 122in33n34 sive inn34 hoc est in34 & prodit 388 numeralis coefficiens tertij termini in ista columna. Atque ita 388×54n56 facit 5352 num: coeff: quarti termini & 5352× 75n78 facit 72816 numeralem coefficientem infimi termini. Idem in alijs ad infinitum usque columnis præstari potest, adeoque valor ipsius DG per hanc regulam pro lubitu produci.
Ad hæc si BF dicatur x, sitque r latus rectum Ellipseos & e=rAB; erit arcus Ellipticus
BG=rxin 1+232e} 3r x 2 +3e 58ee } 5rr xx +4 9e +234ee 716e3 } 7r3 x3 10 +30e 1234ee +918e3 45128e4 } 9r4 x4 +&c.
Quare si ambitus totius Ellipseos desideretur: biseca CB in F, & quære arcum DG per prius Theorema & arcum GB per posterius.

6 Si vice versa ex dato arcu Elliptico DG quæratur sinus ejus CF, tum dicto CD=r, CBqCD=c, & arcu illo DG=z erit
CF=z16ccz3110rc3z5114rrc4z7&c. +13120c4+71420rc5 4935040c6
Quæ autem de Ellipsi dicta sunt, omnia facilè accommodantur ad Hyperbolam: mutatis tantum signis ipsorum c & e ubi sunt {ea} imparium dimentionum.

7. Præterea si sit CE Hyperbola cujus Figure Asymptoti AD, AF rectum angulum FAD constituant & ad AD erigantur utcunque perpendicula BC, DE occurrentia Hyperbolæ in C & E, & AB dicatur a, BC b, & area BCED z, erit BD=zb+zz2abb+z36aab3+z424a3b4+z5120a4b5&c: ubi coefficientes denominatorum prodeunt multiplicando terminos hujus arithmeticæ progressionis, 1,2,3,4,5&c in se continuò. Et hinc ex Logarithmo dato potest numerus ei competens inveniri.

8. Esto VDE Quadratrix cujus vertex V, existente Figure A centro & AE semidiametro circuli ad quem aptatur, & angulo, VAE recto. Demissoque ad AE perpendiculo quovis DB & acta Quadratricis tangente DT occurrente axi ejus AV in T: dic AV=a, & AB=x, eritque <3v> BD=axx3ax445a32x6945a5&c. Et VT=xx3a+x415a3+2x6189a5+&c. Et area AVDB=axx39ax5225a32x76615a5&c Et arcus VD=x+2x327aa+14x52025a4+604x7893025a6+&c. Unde vicissim ex dato BD, vel VT, aut areâ AVDB arcuve VD, per resolutionem affectarum æquationum erui potest x seu AB.

9 Esto Denique AEB sphæroides, revolutione Ellipseos AEB Figure circa axem AB genita, & recta planis quatuor, AB per axem transeunte, DC parallelo AB, CDE perpendiculariter bisecante axem, et FC parallelo CE: sitque recta CB=a. CE=c. CF=x. & FG=y; et sphæroideos segmentum CDFG, dictis quatuor planis comprehensum erit.
+2cxyx3cy3x20c3y5x56c5y75x576c7y9&c cx33aax318caax340c3aa5x3336c5aa&c. cx520a4x540ca43x5160c3a4&c. cx756a65x7336ca6&c 5cx9576a7&c. &c.
Ubi numerales coefficientes supremorum terminorum 2,13,120,156,5576&c in infinitum producuntur multiplicando primum coefficientem 2 continuò per terminos hujus progressionis 1×12×3.1×34×5.3×56×7.5×78×9.7×910×11.&c. Et numerales coefficientes terminorum in unaquaque coluna descendentium in infinitum producuntur multiplicando continuò coefficientem supremi termini in prima columna per eandem progressionem, in secunda autem per terminos hujus 1×12×3.3×34×5.5×56×7.7×78×9.9×910×11&c; in tertia per terminos hujus 3×12×3.5×34×5.7×56×7. 9×78×9.&c, in quarta per terminos huius 5×12×3.7×34×5.9×56×7.&c, in quinta per terminos huius 7×12×3.9×34×5.11×56×7.&c Et sic in infinitum. Et eodem modo segmenta aliorum solidorum designari, & valores eorum aliquando commodè per series quasdem numerales in infinitum produci possu{n}t.

Ex his videre est quantum fines Analyseos per hujusmodi infinitas æquationes ampliantur: quippe quæ earum beneficio, ad omnia, penè dixerim, problemata (si numeralia Diophanti et similia excipias) sese extendit Non tamen omninò universalis evadit, nisi per ulteriores quasdem methodos eliciendi series infinitas. Sunt enim quædam Problemata in quibus non liceat ad series infinitas per divisionem vel extractionem radicum simplicium affectarumve pervenire: sed quomodo in istis casibus procedendum sit jam non vacat dicere; ut neque alia quædam tradere quæ circa reductionem infinitarum serierum in finitas, ubi rei natura tulerit, excogitari. Nam parcius scribo, quod hæ speculationes diu mihi fastidio esse cœperunt, adeò ut ab ijsdem jam per quinque ferè annos abstinuerim. Unum tamen addam: quòd postquam Problema aliquod ad infinitam æquationem deducitur, possint indè variæ approximationes in usum Mechanicæ nullo ferè negotio formari, quæ per alias methodos quæsitæ, multo labore temporisque dispendio constare solent. Cujus rei Exemplo esse possunt Tractatus Hugenij aliorumque de Quadraturâ circuli. Nam ut ex datâ arcûs chorda A, & dimidij arcûs chorda D arcum illum proxime assequarim, finge arcum illum esse Z, et circuli radium r; juxtaque superiora erit A (nempe duplum sinûs dimidij z) =z z34×6rr+z54×4×120r4&c. <4r> Et B=12zz32×16×6rr+z52×16×16×120r4&c. Duc jam B in numerum fictitium n & producto aufer A, & residui secundum terminum (nempe nz32×16×6rr+z34×6rr, eo ut evanescat, pone =0, indeque emerget n=8, & erit 8BA=3z3z564×120r4+&c: hoc est 8BA3=z errore tantum existente z57680r4&c in excessu. Quod est Theorema Hugenianum.

Insuper si in arcûs Bb sagittâ AD indefinitè productâ Figure quæratur punctum G à quo actæ rectæ GB, Gb abscindant tangentem Ee quamproximè æqualem arcui isti: esto circuli centrum C diameter AK=d, & sagitta AD=x et erit DB =dxxx =d12x12x322d12x528d32x7216d52&c. Et AE=AB=d12x12+x326d12+3x5240d32+5x72112d52+&c. Et AEDB.ADAE.AG. Quare AG=32d15x12xx175dvel+&c. Finge ergo AG=32d15x, & vicissim erit DG32d65x.DBDA.AEDB. Quare AEDB=2x323d12+x525d32+23x72300d52+&c. Adde AB et prodit AE=d12x12+ x326d12+3x5240d32+17x721200d52+&c. Hoc aufer de valore ipsius AE supra habito et restabit error 16x72525d52+vel&c. Quare in AG cape AH quintam partem AD , & KG=HC; & actæ GBE, Gbe abscindent tangentem Ee quamproximè æqualem arcui Bab errore tantum existente 16x3525d3dx+vel&c; multo minore scilicet quam in Theoremate Hugenij. Quod si fiat 7AK.3AHDH.n, & capiatur KG=CHn erit error adhuc multò minor.

Atque ita si circuli segmentum aliquod BAb per Mechanicam designandum esset: primò reducerem aream istam in infinitam seriem; puta hanc BbA= 43d12x322x525d12x7214d32x9236d52&c; dein quærerem constructiones mechanicas quibus hanc seriem proximè assequere; cujusmodi sunt hæc.

Age rectam AB, & erit segmentum BbA=23AB+BD×45AD proximè, existente scilicet errore tantum x370dddx+&c, in defectu: vel proximiùs erit segmentum illud, (bisecto AD in F et acta recta BF,) =4BF+AB15×4AD, existente errore solummodo x3560dddx+&c. qui semper minor est quàm 11500 totius segmenti, etiamsi segmentum illud ad usque semicirculum augeatur.

Sic et in Ellipsi BAb cujus vertex A, axis alteruter AK, et latus rectum AP, cape PG=12AP+19AK21AP10AK×AP; in Hyperbola verò cape PG=12AP+19AK+21AP10AK×AP: & acta recta GBE abscindet tangentem AE quamproximè æqualem arcui Elliptico vel Hyperbolico AB, dummodo ar{cus} ille non sit nimis magnus. Et pro area segmenti Hyperbolici BbA, in DP cape DM =3ADq4AK, & ad D & M erige perpendicula Dβ, MN occurrentia semicirculo super diametro AP descripto, eritque 4AN+15×4AD=BbA Figure proximè vel proximius propius erit 21AN+475×4AD=BbA, si modoò capiatur DM=5ADq7AK.

© 2017 The Newton Project

Professor Rob Iliffe
Director, AHRC Newton Papers Project

Scott Mandelbrote,
Fellow & Perne librarian, Peterhouse, Cambridge

Faculty of History, George Street, Oxford, OX1 2RL - newtonproject@history.ox.ac.uk

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