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Mrs. William Dinsmore Briggs

STANFORD

UNIVERSITY LIBRARIES

9^

THE WORKS

OF

GEORGE BERKELEY, D.D.

FORMERLY BISHOP OF CLOYNE.

VOL. tit.

a

Sontioii MACMILLAN AND CO.

PVBLISHERa TO THE USlVERSZTr OF

THE WORKS

OF

EORGE BERKELEY, D.D.

FORMERLY BISHOP OF CLOYNE :

INCLUDING MANY OF HIS WRITINGS HITHERTO UNPUBLISHED.

Wit A Prefaces, Annotations, His Life and Letters^ and an Account of his Philosophy ^

BY

ALEXANDER CAMPBELL FRASER, M.A.

PROFESSOR OF LOCIC AND METAPHYSICS IN THE UNIVERSITY OF EDINBURGH.

IN FOUR VOLUMES.

Vol. III.

AT THE CLARENDON PRESS

M.DCCCLXXI i All rights r€9irv9d'\

THE WORKS

OF

iORGE BERKELEY, D.D.

FORMERLY BISHOP OF CLOYNE.

COLLECTED AND ED/TED IV/TN PREFACES AND ANNOTATIONS

BY

ALEXANDER CAMPBELL ERASER, M.A.

PKOFBSSOR OF LOGIC AND METAPHYSICS IN THB UNrV'ERSITY OF RDINBURCH.

AV THREE VOLUMES,

Vol. III.

®ff0rir

AT THE CLARENDON PRESS

M.DCCCLXXl

\AU rights reserved]

7646^13

CONTENTS.

MISCELLANEOUS WORKS.

LATIN.

PACK

CITHM£TICA ABSQUE AlG£BRA AUT EuCUDE DEMONSTRATA. AuC-

lore * * * * Art Bac. Trin. Col. Dub. 1707 3

scellaxea Mathematica : sive Cogitata nonnuUa de Radicibus Surdis, de iEstu Aeris, de Cono -^quilatero et Cylindro eidem Sphaerae circumscriptis, de Ludo Algebraico; et Paraenetica quaedam ad studium Matheseos, praesertim Algebrae. Autore * * * * Art. Bac. Trin. Col. Dub. 1707 41

I MoTV : sive de Motus principio et Natura, et de Causa

communicationis Motuum. 1721 73

ENGLISH.

%SEsn Obedience; or, The Christian Doctrine of not resisting the Supreme Power, proved and vindicated, upon the Principles of the Law of Nature, in a Discourse delivered at the College-Chapel, 1712 103

K.KXS ui THE Guardian. 1713 141

-^ ES5.\Y TOWARDS PREVENTING THE RuiN OF GrEAT BriTAIN. 172I 1 93

PioPosAL for the better supplying of Churches in our Foreign Plantations, and for converting the savage Americans to Christianity, by a College to be erected in the Summer Islands, otherwise called the Isles of Bermuda. 1725 . . 213

OSES on the prospect of planting Arts and Learning in America . 232

^SCELLANEOUS WORKS.

'-'iili

. A T I N

viu CONTENTS.

A Sermon preached before the Incorporated Society for the Propagation of the Gospel in Foreign Parts: at their Anniversary Meeting in the Parish Church of St. Mary- le-Bow, on Friday, February i8, 173I. 1732 ....

The Analyst ; or, A Discourse addressed to an Infidel Mathema- tician. Wherein it is examined whether the Object, Prin - ciples, and Inferences of the Modem Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith. 1734 . • • .

A Defence of Free-thinking in Mathematics. In Answer to a pamphlet of Philalethes Cantabrigiensis, entitled. Geometry no Friend to Infidelity, or a Defence of Sir Isaac Newton, and the British Mathematicians. Also an Ap- pendix concerning Mr. Walton's Vindication of the prin- ciples of Fluxions against the Objections contained in the Analyst. Wherein it is attempted to put this controversy in such a light as that every Reader may be able to judge thereof. 1735

Reasons for not Replying to Mr. Walton's Full Answer, in a Letter to T. P. T. 1735

The Querist, containing several Queries, proposed to the con- sideration of the Public. 1735 — 1737

A Discourse addressed to Magistrates and Men in Authority. Occasioned by the enormous License and Irreligion of the Times. 1736

A Letter to the Roman Cathoucs of the Diocese of Cloyne. Published m the late Rebellion, a.d. 1745

A Word to the Wise : or. An Exhortation to the Roman Catholic

Clergy of Ireland. 1749

Maxims concerning Patriotism. 1750

Three Letters to Thomas Prior, Esq., and a Letter to the Rev. Dr. Hales, on the virtues of Tar-water. 1744 — 1747 . .

Farther Thoughts on Tar- water. 1752

APPENDIX.

A. The First Edition of the ' Querist '

B. First Edition of the ' Maxims concerning Patriotism ' . . . ,

MISCELLANEOUS WORKS.

ERRATA.

Page 9, note i, line 5, for * twenty-three * read * twenty-two.'

Page a 1 5, note i, 11. la, 13, delete Hhen at Bath.'

Page 357, note, 11. 3, 4, for * for the month in which ' read * a few

weeks after.' Page 455, note i, 1. a, delete 'as it seems, in the Dublin Journal'

[See Appendix B.]

BerktUys IVorks, ^oi. III.

VOL. III. /

ARITHMETlCA

ABSQUE

ALGEBRA AUT EUCLIDE

DEMONSTRATA.

AUCTORE * ♦ * * ART. BAG. TRIN. COL. DUB.

1707.

E 2

MAXIMiE SPEI PUERO,

D. GULIELMO PALLISER,

REVERENDISSIMI ARCHIEPISCOPI CASSELENSIS^

FILIO UNICO, INGENIO, SOLERTIA, ERUDITIONS,

ANNOS LONGE PRJEEUNTI,

NUMERISQUE ADEO OMNIBUS AD PRJESTANDUM,

INGENS ALIQUOD SCIENTIIS LUMEN AC INCREMENTUM NATO,

HUNC ARITHMETICJE TRACTATUM,

IN EXIGUUM SUMMI AMORIS PIGNUS,

OFFERT ET DICAT

AUCTOR.

William Palliser, translated to the Arch- Dprick of Cashel in 1694, was previously op of Cloyne. He had been elected a )w of Trinity College, Dublin, in 1668, was tutor to William Molyneux, the d of Locke, father of Samuel Molyneux,

to whom Berkeley dedicated his Miscellanea Matbematica. Palliser was afterwards Pro- fessor of Divinity in Trinity College. He died in 1727. Of the younger Palliser, to whom the Aritbmetica is dedicated, nothing further of interest is recorded.

PR^FATIO.

Plerosqub sciendanim mathematicarum procos in ipso earundem limine csecutientes, sentio simul et doleo. Nimirum cum ea sit, apud nos saltem, mathemata discendi ratio, ut primo arithmetica, deinde geometria, po- stremo algebra addiscatur, Tacqueti* vero Arithmeticam legamus, earn autem nemo probe intelligat, qui algebram non praelibarit, hinc fit ut plerique mathesi operam navantes, dum bene multorum minoris usus theorematum demonstrationes studiose evolvunt, interea operationum arithmeticanim, quarum ea est vis et praestantia, ut non modo cseteris disciplinis mathematicis, verum etiam hominum cujuscunque demum sortis usibus commodissime famulentur, principia ac rationes intactas praetereant. Quod si quis tandem aliquando, post emensum matheseos cursum, oculos in praedictum Tacqueti libnim retorqueat, multa ibi methodo obscura, et quae intellectiun non tam illuminet quam convincat, demonstrata ; multa horrido porismatum et theorematum satellitio stipata inveniet.

Sed nee alius quisquam, quod sciaih, arithmeticam seorsim ab algebra demonstravit. Proinde e re tyroniun futurum ratus, si haec mea qualia- cunque in lucem emitterem, ea postquam, si minus omnia, pleraque certe per integrum fere trienniiun in scriniis delituerint, publici juris facio. Quae cum praeter ipsos operandi modos, eorundem etiam demonstrationes ex propriis et genuinis arithmeticae principiis petitas complectantur, mira- bitiu" fortasse quispiam, quod noster hie tractatus mole vulgares arith- meticorum libros, in quibus praxis tantum tradatur, baud exaequet. Hoc autem exinde provenit, quod cum operationum rh dt&ri explicarem in praeceptis et exemplis, quae vulgus arithmeticorum ad nauseam usque prosequitur, contractior fui; nee eo forsan obscurior. Quippe tametsi

' For Tacqaet, an eminent mathematician Spinoza's letter to De Vries {Epistcla XXVI).

of the serenteenth century, cf. New Theory His Arithmetica Theoria et Praxis^ upon

0/ VisioH, sect. 30, note. He is often re- which Berkeley here remarks, was published

ferred to by contemporary writers. See at Antwerp in 1665.

8 PRiEFATIO.

cseco ad singulos fere gresstis regendos opus sit manuductore, in dara tamen demonstrationum luce versanti sufficit, si quis tenendum tramitem vel strictim exponat. Quamobrem oranes matheseos candidati ad regularum arithmeticse rationes ac fundamenta percipiendum aninios adjungant, summopere velim et exoptem.

Neque id tanti moliminis est, ut plerique fortasse imaginentur. Quas attulimus demonstrationes faciles (ni fallor) sunt et concisae; nee prin- cipia aliunde mutuantur, ex algebra nihil, nihil ex Euclide tanquam notiun supponitur. Ubique malui obvia et familiari aliqua ratione a priori veritatem praxeos comprobare, quam per prolixam demonstra- tionum apagogicarum seriem ad absurdum deducere. Radicum quad- ratarum et cubicanim doctrinam ex ii>sa involutionis arithmeticse natura eniere tentavi. Atque ea, meo quidem judicio, ad numerosam radicum extractionem illustrandum magis accommoda videtur, quam quae ex elemento secundo Euclidis, aut ex analysi potestatum algebraicarum vulgo adferri solent. Regula vulgaris pro alligatione plurium renim non nisi difficulter admodum et per species demonstratur : ejus igitur loco novam, quae vix uUa demonstratione indigeat, e proprio penu sub- stitui. Regulam falsi, utpote mancam et fere inutilem, consulto prae- termisi. Ac, si nihil aliud, novitas fortassis aliqua placebit.

Neminem transcripsi; nullius scrinia expilavi. Nempe id mihi im- primis propositum fuerat, ut numeros tractandi leges ex ipsis principiis, proprii exercitii et recreationis causa, deducerem. Quod et deinceps horis subsecivis prosecutus sum. Nee mihi hoc in loco, absque ingrati animi labc, praeterire liceat Reverendum Virum Johannem Hall*, S.T.D, Academic nostrae Vice-praepositum, ibidemque linguae Hebraicae Professorem dig- nissimum. Cui viro optimo quum me multis nominibus obstringi lubens agnoscam, tmn non id minimum duco, quod illius hortatu ad suavis- simum Matheseos studium incitatus fuerim.

Monstravi porro ad quem collimaverim scopum: quousque ipsum assecutus sim, penes aequos rerum aestimatores esto judicium. Candido quippe horum examini istas studiorum meonim primitias libenter sub- mitto; quicquid interim scioli sentiant et malevoli, panun solicitus.

' John Hall, elected Fellow of Trinity attributes to him his own early enthusiasm

College, Dublin, in 1685, was afterwards for mathematics. In 1713 Hall was

Berkeley's tutor. He was Vice-Provost of appointed to a college living in the diocese

the College, 1697-17 13. Berkeley here ofDerry. He died in 1735.

ARITHMETICS^

PARS PRIMA.

CAP I.

DE NOTATIONE ET ENUNCIATIONE NUMERORUM.

NovEM sunt notas numerales, viz. i, 2, 3, 4, 5, 6, 7, 8, 9, quibus una cum cyfra (o) utuntur arithmetici, ut tantum non infinitos numerorum ordines exprimant. Omne illius rei artificium in eo positum est, quod notarum numeralium loci ratione decupla progrediantur. Series autem numerorum, ea lege quoad locorum valores procedentium, in membra sive periodos enunciationis causa secatur. Rem totam oculis conspiciendam subjecta exhibet Tabella :

* This treatise on * Arithmetic ' is the first proof Berkeley gave to the world of his literary ability. It must have been published early in 1 707, when he was twenty-three years old, for hs took his Master's degree in June of that year, and on the title-page he is designated Bachelor of Arts. From the preceding Preface much of the work seems to have been prepared three years before it was published.

In the original edition, now very rare, the Aritbmetica and Miscellanea Matte- matiea form together a small anonymous volume of 92 pages. Neither of them is contained in the Miscellany of his smaller works which Berkeley published in 1752. They both appear, however, in the various collected editions of his works ; and, as additional proof of their authorship, we have the evidence of their contents, as well as of Berkeley's MS. Common-place-book.

Both the Aritbmetica and the Miscellatiea Matbematiea are marked by the originality, ingenuity, and simplicity which are charac-

teristic of their author. They give internal evidence of the truth of what he says in the Preface to the former — Neminem tran- scripsi ; nullius scrinia expilavi. And they express his juvenile enthusiasm in those mathematical sciences which, nearly thirty years afterwards, involved him in a famous controversy.

The treatise on Aritbmetica not unworthy of study at the present day, is a brief system of the science, unfolded simply and ingeniously from its principles — in three parts. The First Part deduces the elemen- tary rules for the Addition, Subtraction, Multiplication and Division of numbers, with the theory and rules of Squares and Cubes ; the Second treats of Fractions or broken numbers, and the theory of the rules for adding, subtracting, multiplying, dividing, and reducing them ; the Third is concerned with the numerical relations of proportion, alligation, and progression —arithmetical and geometrical.

lO

A rithmetica :

NOTARUM NUMERALIUM SERIES.

Ccnturiae .

Decades

Unitates . . Centurix . .

Decades

Unitates .. Centurix ..

Decades

Unitates .. Centurix ..

Decades

Unitates ., Centurix .. Decades..... Unitates , . Centurix ..

Decades

Unitates .. Centurix ..

Decades

Unitates ") Unesimx J

Decimx

Centesimx Unesimx ., Decimx...., Centesimx Unesimx ., Decimx .... Centesimx Unesimx .

Decimx

Centesimx Unesimx .. Decimx..... Centesimx Unesimx ..

Decimx

Centesimx

• ••• • •••• •

•••••••••«

Quintilionum.

Quatrilionum.

Trilionum.

Bilionum.

Millionum.

Millium.

Integrorum.

Partes.

Millesimarum.

Millionesimarum.

Bilionesimarum.

Trilionesimarum.

Quatrilionesimarum,

qua exponitur notarum numeralium scries, in terniones distrj

Pars Prima. ii

membra autem seu periodi millecupla, loci decupla ratione pro- grediuntur. £. g. Numerus positus in loco unitatum (is per subjectum punctum dignoscitur) denotat septem res integras quas- cunque^ vel saltern ut integras spectatas; numerus ei a dextris proximus, tres partes decimas ejusdem integri^ qui vero locum immediate prsecedentem occupat, indigitat quatuor decadas eorundem integrorum. Eadem proportione decupla locus quilibet sequentem superat, a prsecedente superatur.

Porro, cum infinita unitatum multiplicatione et divisione, notarum series infinite ultra citraque unitatum locum producatur, adeoque innumeri oriantur loci, ut distincti eorum valores expri- mantur, opus est solummodo trium vocum continua repetitione^ modo ternio quivis sive periodus suo insigniatur nomine, uti factimi in Tabella. Nam, progrediendo a loco unitatum versus sinistram, prima periodus numerat simpliciter unitates, sive integra; secunda, millia^ tertia, milliones^ quarta, biliones; atque ita porro. Similiter, servata analogia, in periodis infia unitatem descendentibus, occurrunt primo partes simpliciter, dein mille- simal, millionesimas, bilionesimx, &c atque has quidem partiendae in unesimas, decimas, centesimas; illi vero coUigendi in unitates, decades, centurias.

yt itaque enunciemus numerum quavis e tota serie figura designatum, i% respiciendum est ad valorem notx simplicemj 2% ad valorem loci ; postremo, periodi. JB. g. enuncianda sit 9, in quinta sinistrorsum periodo. Nota simpliciter sumpta valet novem : ratione loci, novem decadas j ratione demum periodi, novem decadas trilionum. Proponatur 5, in tertia periodo: simpliciter sumpta dicit quinque; ratione loci, quinque unitates; ratione periodi, quinque unitates millionum, seu quinque milliones. In secunda infra unitatem periodo, detur 8 : simplex notae valor est octo J ratione loci, octo centesimae ; ratione periodi, octo cente- simal millesimarum.

Quod si numerus enunciandus non habeat adscripta vocabula valores pcriodorum locorumque indigitantia, is punctatione a loco unitatum dextrorsum sinistrorsumque instituta in terniones dis- tinguatur; deinde, cuique loco et periodo assignato nomine, proferatur. Sit, f. g. numerus propositus 73'48o*i95. Notis in periodos distinctis, primum quxro quinam sint valores figurx ad sinistram primx ; qux, quoniam collocatur in secundo loco tertix

12

Arithmetica :

period!, valet septem decadas millionum : quia vero numeri ratione decupla prc^ediuntur, intellecto notx primx valore, cseterarum valores ordine sequuntur. Sic ergo enunciabimus numenun propositum; septem decades et tres unitates millionum, quatuor centurias et octo decades millium, una centuria, novem decades ^ et quinque unitates ; vel contractius, septuaginta tres millione^ quadringenta octaginta millia, centum nonaginta quinque. Hind ^ cernimus quod cyfra, licet per se nil valeat, necessario tameD' scribatur, ut unicuique notx debitum assignemus locum.

Facillimum erit numeros quantumvis magnos scribere et enun^ ciare, modo qux dicta sunt perpendantur^ quorum etiam scientia in sequentibus maximi erit momenti : siquidem qua ratione open- tiones arithmeticx in digitis perficiantur ipsa docet natura}. arte vero opus est ad easdem in numeris grandioribus accurate i cxercendas, qux sane omnis in eo versatur, ut quod opus simul et uno quasi ictu peragi non sinit humanx mentis angustia, id in plures partiamur opellas sigillatim inquirentes digitonim ] aggregata, dlfFerentias, producta, &c. dein hxc ita componamus | ut exhibeant summam, residuum, aut productum, &c. totale; cuJQi ' rei ratio omnis et artificium petitur ex simplici locorum progres- - sione, et in ea ultimo fundatur.

N.B. Non me latet arithmeticos nonnullos numerorum seriein alitor ac a nobis fiactum est partiri j sc. in senarios (composita denominatione) loco ternionum. Cum vero methodum quam tradimus sequantur etiam' alii, visum est et nobis eam (utpote simpliciorem) retinere.

• [v. g. CI. WalHsius in Matbes. Univen., et le P&e Lamy dans ses EUtnens des MaUbmatiquet^ — Author. Wallis* Matbesis Universalis is the first article in his Opera Matbematica (Oxford, 1695). See ch. V, * Numerorum Procreatio,' for the opinion to which Berkeley refers. Bernard Lamy (or Lami), priest of the Oratory, a Cartesian,

was author of various works in matheautkt and theology. One of these, TVaiti de la Grandeur en giniraly qui eomprend fAridh tnitique, FAlgibre^ I'Analyse, &c., was published at Paris in 1 680. The second edition of this book appeared in 1691, under the title EUmens des Matb^mtUiptM,

Pars Prima.

13

CAP. II.

DE ADDITIONE.

Additione qiMcritUT duorum pluriumve numerorum aggre- lltmn; quod ut obtineatur, numeri aggregandi sub invicem xribantur ea lege, ut unitates unitatibus, decades decadibus, partes decimx decimis, &c. respondeant. Quamobrem, ubi adnexae faeiint partes decimaies, oportet unitatis locum adjecto commate insigiiire. Deinde, sumpto a dextris initio, notx in primo loco icairrentes una addantur; decades autem siqux proveniant, idjectis punctulis notatas sequenti loco annumerandas sunt, cujus itideni numeris (reservatis interim decadibus, qux ad locum fequentem pertinent) in unam summam aggregati infra scribantur. Atqoe ita porro.

JB. ^. In primo, infta-scriptorum exemplo, 9 et 5 feciunt 14; decadcm punctatam servo, cum 4 progredior; 4 et 8 sunt 12, pQDCtata igitur decade, a subscribo ; ad secundum locum accedens, reperio 6, quibus addo 2, scil. decadas in primo punctatas, 8 et 2 faciunt decadem, quam notatam servans, qux sola superest i subscribo. Et sic deinceps.

Addend.	20 I 8- 8-2 2-5- 4369	523,9702 8 ',35 6o,2cx)5	£ s. d 7 8 9 3 "• 5 'o.:.7 2
Sum.	14612	665,5207	II 8 4

Quod si proponantur colligendx res diversarum specierum, simili prorsus methodo opcrandum, dummodo habeatur ratio proportionis, juxta quam progrediuntur diversa rerum genera. E. g, Quoniam Lib. Sol. et Den. non ratione decupla ut numeri progrediuntur; adeoque non 10 denarii sed 12 constituant solidum; non 10 solidi sed 20, libram. Propterea in hisce speciebus addendis, loco dccadis, numerus quilibet in denariis, duodenarius, in solidis^ ncenarius, sequenti loco adscribendus est.

H

A rithnetica :

CAP. III.

DE SUBDUCTIONE.

SUBDUCTIONE quseritur duorum numerorum diflbrentia, sivc quodnam superfuerit residuum sublato uno ex altero^ cujus obti- nendi causa, numeri minoris nota quxlibet notae majoris ejusdem loci subscribatur j deinde subducendi prima dextrorsum nota ex nota suprascripta auferatur, residuumque infra notetur^ atque ita porro, usque dum perficiatur subductio totius.

Si vero accidat numerum aliquem minorem esse quam ut ex eo nota subscripta auferri possit, is decade augeatur, mutuata sdL unitate a loco sequente.

Detur J 189 subtrahendus ex 32034. Numeris ut in exemplo subjecto scriptis, aggredior subductionem notac primx 9 ex supra- posita 4 ; verum cum 4 ne semel quidem contineat 9, adjecta de- cade, fiat 14^ ex 14 subductis 9, restant 5. Dein versus sinistram pergens, reperio 8, a a (loco 3, habita nimirum ratione mutuatx decadis) subducenda, quod quoniam fieri nequit, aufero 8 a la, et restant 4. Proxima subducendi nota est i, quae quia a nihilo, sive o, non potest subtrahi, loco cyfras o, substituo 9, (9 inquam, quo- niam, mutuata decas unitate numero prascedenti jam ante adjecta truncatur) ablata demum 1 ab i, restat nihil. Porro peracta sub- ductione restant 3, quae itidem subscribo.

Haud dissimili ratione subductio specierum diversarum perfi- citur: modo advertamus non semper decadem, sed numerum qui dicit quotuplus locus quilibet sit praecedentis, in supplementum defectus notae alicujus mutuandum esse.

Subdue.	32034 1189	73*9><545 3042,100	ji s. d. 4 8 3 265
Resid.	30845	4287,545	2 I 10

N.B, £x dictis liquet arithmetical (quam hactenus tradidimus)

Pars Prima,

15

arti&cium consistere in perficiendo per partes id quod una vice fieri nequeat; rationem vero in additione, reservandi, in subductione, iwituandi decadas, a decupla locorum progressione omnino peten- dam esse.

CAP. IV.

DE MULTIPLICATIONE.

MULTIPLICATIONE toties ponitur multiplicandus quoties jubet moltiplicans^ seu quaeritur numenis qui eandem habeat rationem jd multiplicandum, quam multiplicans ad unitatem. Humerus aotem iste appellatur productum sive rectangulum^ cujus latera SOI £actores dicuntur uterque tum multiplicandus, turn numerus per quem multiplicatur.

Ut productum duorum numerorum inveniamus, scripto numero nultiplicante sub multiplicando, hie multiplicetur per quamlibet aotam illius, incipiendo a dextris^ cujusque autem producti nota prima directe sutKScribatur notse multiplicanti, reliquae versus Ise- fam ordine sequantur.

Peracta multiplicatione, producta particularia in unam coUigan- tur summam, ut habeatur productum totale, in quo tot loci partibus sunt assignandi, quot sunt in utroque factore.

Proponatur 30,94 ducendus in (sive multiplicandus per) %6^^. Quinquies 4 dant 20, cujus primam figuram o subscribo notae mul- tiplicanti (5), reliquam a servo ^ porro 5 in 9 dant 45 ; 5 cum % scrvatis faciunt 7, quae subscribo, 4 sequenti loco ponenda servans j et sic deinceps.

	3O594 26,5	53886 24	6000 56
1 1	15470 18564 6188
1 1 1 1 1	211544 105772	3<5 30
Prod. tot.	819,910	I 269264	336000

1 6 A rithmetica :

Quoniam numeri cujusque duplex est valor, ut multiplicati^ recte instituatur oportet utriusque rationem haberij adeo ut nottj qusevis multiplicetur juxta valorem cum simplicem turn localeil^ figuras multiplicantis. Hinc nota prima cujusque particulariH product! scribitur sub nota multiplicante. E.g. in secundi ex^ empli multiplicatore, nota a valet duas (non unitates sed) dc^ cadas; ergo in 6 (primam multiplicandi notam) ducta produccfi! duodecim (non quidem unitates, verum) decadas. Proinde pri-!. mam producti notam in loco decadum t.e. directe sub nota mol* tiplicante 2, poni oportet.

Ob eandem rationem, ubi in factoribus occumint partes, num0> rus ex prima multiplicandi nota in primam multiplicantis ducta genitus, tot locis detrudendus est infra notam multiplicatam, quol multiplicans dextrorsum ab unitate distat; adeoque tot loci ill producto totali partibus seponendi sunt, quot fuerant in utroque factore.

N.B. Si factori utrique aut alterutri a dextris accedant cyfm non interruptae, multiplicatione in reliquis notis instituta omit- tantur istac mox producto totali adjiciendae : quippe cum lod pro* portione decupla progrediantur liquet numerum decuplum, centch plum, millecuplum, &c. suiipsius evadere, si modo uno, duobus aut tribus locis promoveatur.

CAP. V.

DE DIVISIONE.

DiviSlO opponitur multiplicationi ; nempe productum quod haec conficit, ilia sibi dissolvendum sive dividendum proponit. Nume- rus in divisione inventus, dicitur ^jiotiens : siquidem dicit quoties dividendus continet divisorem vel (quod idem est) rationem divi- dend! ad divisorem 4 seu denique, partem dividend! a divisore denominatam.

In divisione, scriptis dividendo et divisore, sicut in exemplorum subjectonim primo, captoque initio a sinistris, pars dividend! divi- sor! aequalis, vel eum proxime superans (intelligo valorem tantum simplicem) interposito puncto seponatur. Quaerendum dein quoties divisor in membro isto contineatur, numerusque proveniens erit

Pars Prima. 17

prima quotientis nota; porro divisor ducatur in notam inventam, pscbctoque a membfx> dividendo ablato, residuum infra notetur, QU adscripta sequente dividendi nota, confit novum membrum

: Afidcnduin, unde eruatur nota secunda quotientis, mox in divi- mcm ducenda, ut producto ex membro proxime diviso ablato,

\ jcnduum una cum sequente dividendi nota, praebeat novum mem- bnm ; atque ita porro, usque dum absoluta fiierit operatio. Sub- dncds demum locis decimalibus divisoris ab iis qui sunt in divi- dendo, residuum indicabit quot loci partibus assignandi sunt in fiotiente; quod si nequeat fieri subductio, adjiciantur dividendo loC qrfrx decimales quot opus est.

Feiacta divisione, si quid superfuerit, adjectis cyfris decimalibus onntinnari poterit divisio, donee vei nihil restet, vel id tam exi- gaum sit, ut tuto negligi possit \ aut etiam quotienti apponantur DOtac residuse subscripto iisdem divisore.

Si uterque, dividendus nempe et divisor, desinat in cyfras, hae squall numero utrinque rescindantur; si vero divisor solus cyfris tenninetur, ese omnes inter operandum negligantur^ totidemque pQstrenue dividendi notas abscissae, sub finem operationis restitu- antur, scripto infra lineolam divisore.

Proponatur 4583a, dividendus per 67. Quoniam divisor major est quam 45, adjecta nota sequente fiat 458, membrum primo dividendum ; hoc interposito puncto a reliquis dividendi notis

I scccmo. 6 in 45 continetur septies, et superest 3 ; veruntamen quoniam 7 non itidem septies in 28 reperitur, ideo minuendus est quotiens. Sumatur 6 ; 6 in 45 invenitur sexies, atque insupcr 9, quin et 98 continet 7 sexies, est igitur 6 nota prima quo- tientis. Haec in divisorem ducta procreat subducendum 402, quo sublato a 458, restant 56^ his adscribo 3, proximam dividendi

IDOtam, unde confit novum membrum, nimirum 563, quod sicuti prius dividens, invenio 8 pro nota secunda quotientis: 8 in 67 cfat 536, hunc subduco a membro 563, residuoque 27 adjiciens rtlk]uam dividendi notam, vi%. 2, habeo 272 pro novo dividendo, qucxl divisum dat 4, qua primo in quotiente scripta, dein in divisorem ducta, productoque ex 272 ablato, restant 4 quotienti, scripto infra lineolam divisore, adjicienda.

Expeditior est operatio, ubi subductio cujusque notae multipli- ationem immediate sequitur^ ipsa autem multiplicatio a sinistra dextrorsum instituilur, E. g. Sit 12 199980 dividendus per 156

VOL. III. c

1 8 ArUhnteiica :

(yide exempl. 3) sub 1219 primo dividendi membro scripto din sore, constat hunc in illo septies continent quamobrem 7 scr in quotiente. Septies i est 7, quibus subductis ex 12, deleo t notam multiplicatam i turn 12 partem membri unde aufcreba productum, residuum 5 supra notans ; dein accedo ad proxim divisoris notam 5; 7 in 5 dat 55; 35 ex 51 ablatis, restaot 1 quae supra scribo, deletis 51 et 5. Deindc autem 7 in 6 dm productoque 42 ex 69 subtracto, supersunt 27, quse proindc no! deletis interim tum 69 turn 6, ultima dividendi figura. Pono dia sorem jam integre deletum, denuo versus dcxtram uno lot promotum scribo, perque iilum membnim suprascriptum (qud quidem fit ex residuo membri proxime divisi sequente nota audi quemadmodum pracedens dlvldo. Eodem modo divisor 1 promoveatur quoad dividendum totum percurrerit,-''

401 534	200)8200 2) «2 (41 8	smxH
368	3	IIT
	00
004

Jam vero prxceptorum ratio dabiturj et primum quidem liqi cur quotientem per partes investigemus.

a. Qujeri potest, cur v. g. in exemplo supra allato habeatur] pro quotiente membri primi per divisorem divisi, nam 67 in ^ centuriis (pro centuriis nimirum habendae sunt cum duobus lot, sinistrorsum ab unitate distent) non scxies, sed scxcenties i tinetur? Respondeo, revera non simpliciter 6, sed 600 scribi { quotiente; duae enim notx postmodum inventx istam sequuntl| atque ita quidem quotient! debitus semper conservatur valor; unicuique notfe tot loci in quotiente, quot membro unde eruebata in dividendo postponuntur.

d luliini rejciceil ii

>fp<rfm.

Pars Prima. 19

5. Quandoquidem nota quselibet quotlentis indicat quoties id, X quo eruebatur, dividendi membrum divisorem contineat sequum Bt at ex divisore, in notam proxime inventam ducto, confletur ■bdocenduin : tunc nempe aufertur divisor toties ad amussim ooties in dividendo continetur, nisi forsan aequo major aut minor it mimerus ultimo in quotiente scriptus. De illo quidem errore oostabit, si productum tam magnum fuerit, ut subduci nequeat; le hoCy si e contra productum oriatur tam exiguum ut peracta obdiicticHie residuum divisore majus sit vel ei sequale.

4. Ratio cur tot loci partibus seponantur in quotiente, quot nm iis qui sunt in divisore sequentur locis decimalibus dividendi, X CO cemitur, quod numerus dividendus sit productum, cujus actoies sunt divisor et quotus, adeoque ille tot habeat locos Icdmales quot hi ambo, id quod demonstravimus de multipli catione agentes.

5. Fatet cyfras decimales ad calcem dividendi adjectas ipsius valorem non immutare. Nam integros quod attinet, ii dummodo eodem intervallo supra unitates ascendant, eundem sortiuntur valorem; decimales vero non nisi praepositis cyfris in inferiorem gradum deprimuntur.

6. Quoniam quotiens exponit seu denominat rationem dividendi ad divisorem, patet proportione ilia sive ratione existente eadem, eundem fore quotientem; sed abjectis cyfris communibus, ratio seu numerorum ad invicem habitudo minime mutatur. Sic v. g. 200 est ad 100, vel (quod idem est) aoo toties continet 100, fioties 2 continet i, quod sane per se manifestum est.

CAP. VI.

DE COMPOSITIONE ET RESOLUTIONE QUADRATI.

Productum ex numero in seipsum ducto, dicitur numerus jfrnUrstttr. Numerus autem ex cujus multiplicatione oritur quad- atus, nuncupatur iatus sive radix quadratai et operatio qua numeri Propositi radicem investigamus, dicitur extractio radich quadratjey njus intelligendx causa juvabit genesin ipsius quadrati, partesque rz quibus componitur, earumque ordinem situmque contemplari. /eruntamen quoniam in inquirenda rerum cognitione consultius

c 2

Ariiknietka :

est a simplicissimis et taciliimis ordiri, a contemplatione gwM quadrati, ex radice binomia oriundi, initium capiamus.

Attentius itaque intuendum est, quid ftat ubi numerus d( notis constans in seipsum ducatur. Et primo quidem maniffl est, primam a dextra radicis notam in notam supra posi seipsam nempc, duci ; unde oritur quadratum minoris mei Dcindc vero, eadem nota in sequentem multiplicandi, ». *. alt radicis notam ducta, provenire rcctangulum ab utroque n membro conflatum constat. Porro pwracta multiplicatione t multiplicandi per primam radicis notam, ad secundam accedj qua in primam multiplicandi notam ducta, oritur jam d rcctangulum duarum radicis binomije notarum ; deinde set multiplicandi nota, »'. e. eadem per eandem, multiplicata secundi membri radicis binomia quadratum.

Hinc ergo colligimus, quadratum quodvis a radice bio

procreatum constare primo ex quadrato membri minorisj sec

duplici rectangulu mfmbronim ; tertio quadrato membri maja

Proponatur radix binomia, v. g. 23 quadranda, juxta ea

cap. 4. traduntur ; primo duco 3 in 3, unde producit

^■^ quadratum membri minoris. Secundo duco 3 in a, alt

^3 radicis notam; prodit 6, rectangulum utriusque 1

g Tertio, ex 2 in 3 ducto oritur jam secunda vice rt

g gulum membrorum. Quarto, 3 in a gigiut 4, i

ratum membri majoris.

Progrediamur ad gencsin quadrati a radice trimembri. i

hie, primo quidem, prima radicis nota in intcgram radtcem <

procreat, primo, primi membri quadratum ; secundo, rcctang

membrorum primi ac secundi; tertio, rectangulum membn

primi ac tertii. Secundo, secunda radicis nota multipl

radicem dat, primo, rectangulum membrorum primi ac sea

secundo, quadratum membri secundi; tertio, rectangulum I

brorum secundi ac tertii. Tertio, ex tertia radicis nota in rad

ducta oritur, primo, rectangulum membrorum secundi ac t)

secundo, rectangulum membrorum secundi ac tertii ; tertio, <

ratum tertii membri radicis.

Hinc porro colJigimus quadratum quodvis a radice trini genitum cumplecti, primo, quadratum notx radicis primx ; secq) duplex rectangulum noise prim^ in duas reliquas ductx; tc) quadratum duarum rcliquarum, i. t. bina singularum quadrati

Pars Prima, 21

amndem duplex cectangulum, qux quidem constituere quadratum taanim notanim jam ante ostendimus.

Simili methodo ostendi potest quadratum 4, 5, quotlibet notarum iwitinrrc, primo quadratum notas infimae^ secundo, duplex rect- taguhun ex infima in sequentes omnes ducta genitum^ tertio, indratum notamm omnium sequentium^ quod ipsum (uti ex jnemissis manifestum est) continet quadratum notas a dextris iwnndar^ duplex rectangulum ejusdem in omnes sequentes ductas, |BHbatum notarum omnium sequentium; quod pariter continet fudratum notx tertiae, bina rectangula illius et sequentium haramque quadratum, atque ita porro, usque quoad ventum sit ad ^fadratum altissimae radicis notas.

Inventis tandem partibus ex quibus componitur quadratum, Rstat ut circa earum ordinem situmque dispiciamus. Si itaque fBdratum indpiendo a dextris in biniones partiamur, ex genesi fBin supra tradidimus constabit, primum (a sinistris) membrum oocupari a quadrato notas primas sive altissimx, simul ac ab ea A^ds rectanguli ex notis prima et secunda in invicem ductis fiooflati portione, quae extra primum sequentis binionis locum itdandat: secundi locum primum continere dictum doplex rectangulum, atque insuper quicquid quadrat! ooCx secunda^ excurrat ; secundum capere quadratum Dolae secundas, et quod redundat duplicis rectanguli doarum priorum notarum in tertiam ductarum (quoad iiotam infimam) ad locum primum tertii binionis per- tiaentis, et sic deinceps; v.g. in exemplo apposito, membrum primum 10 continet 9 quadratum notas primae 3, simul ac i qua la (duplex rectangulum notae 10*3^*4^ 3 in sequentem 2 ductal) locum primum secundi membri transcendit. Primus locus secundi binionis capit % (duplicis rectanguli notarum 3 et !2 reliquum), atque etiam id (yaod extra locum proxime sequentem redundat, &c.

Perspecta jam compositione quadrati, ad ejusdem analysin aocedamus. Proponatur itaque numerus quivis (f. g. 103041), node clicienda sit radix quadrata. Hunc incipiens a dextris, in biniones (si par sit locorum numerus, alioqui membrum ultimum ex unica constabit nota) distingue. Quaero dein quadratum maximum in (10) membro versus Isevam primo contentum, cujus radix (3) est nota prima radicis indagandac, ipsum aulcm quad-

321 321

321 64 a 963

22 Arithmetica:

ratum (9) a membro (10) subduca Ex residtio (1)

ioaoj.if^2i ^^J^^^ (3) ^'^^ prima sequentis mcmbri confit o dividendus (13), quern divido per notam inventam

duplicatam (/. e. 6), quotiens (2) erit nota radicalis

6)13*^ secunda; qua primo in divisorem, deinde in

'H seipsam ducta, productisque in unam summam ool-

"~I lectis, ita tamen ut posterius uno loco dextix»sum

^^ ^ promoveatur {e. g.^ 4) habeo numenim sobdii-

641 cendum (124), hunc aufero ex dividendo (13) aucto

(o) nota reliqua secundi membri : residuo (6)

000 (^^^ notam primam tertii binionis, ut fiat novas

dividendus (64), qui divisus per (64) dupimii radicis hactenus inventae dat (i) notam tertiam radids in* dagandse; hac tum in divisorem tum in seipsam ducta, fao* tisque ut supra simul aggregatis, summam (641) subduco a din* dendo (64) aucto accessione notas alterius membri tertii : eadem plane methodo pergendum quantumvis producatur operatio.

Si quid post ultimam subductionem superfiierit, id tibi indicio sit, numerum propositum non fuisse quadratum; verumtamen adjectis resolvendo cyfris decimalibus operatio extendi poterit quousque lubet.

Humerus locorum decimalium, si qui fuerint, in resolvendo bipartitus indicabit, quot ponendi sunt in radice. Cujus ratio cemitur ex cap. 4.

Ratio operandi abunde patet ex praemissis. Nam e. g. adhibiii (6) duplum notas inventae pro divisore, propterea quod ex traditt quadrati compositione, duplex rectangulum notae illius (3) in sequentem (2) ductae dividendum complecti rescissem, eoque adeo diviso per duplum factoris unius (3) confactorem ejus (a) i. i- notam proximam radicis innotescere. Similiter, subducendum conflavi ex duplici rectangulo quotientis et divisoris, simul ac quotientis quadrato in unum, ea qua dictum est ratione, coUectis; quia bina ilia rectangula et quadratum eo ordine in residuo et membro sequente, ex quibus fiebat subductio, contineri depre- henderam, atque ita quidem potestatis resolutio ex ipsius com- positione facili admodum negotio deducitur.

Pars Prima. 23

CAP. VII.

DE COMPOSITIONE ET RESOLUTIONE CUBI.

Radix in quadratum ducta procreat cubum. Ut sternamus

fiam ad analysin cubi, a compositione potestatis (quemadmodum

in capite praecedenti Return) sumendum est initium. In pro-

doctione igitur cubi a radice binomia primum radicis membrum

offimdit, primo, suiipsius quadratum, unde cubus notas primae;

duplex rectangulum membrorum, unde duplex solidum

i notae primae in alteram ducti \ tertio, quadratum membri

aherias^ unde solidum ex nota prima et quadrato secundae genitum.

Suniiiter, facta multiplicatione per membrum secundum, oritur

frimo^ solidum notae secundae et quadrat! primas ; secundo, duplex

loiidum notx primx et quadrati secundas; tertio, cubus membri

tecundi.

Cootinet ergo cubus a radice binomia procreatus singulorum aanbforum cubos et 6 solida, nimirum 3 facta ex quadrato oembri utriusvis in alterum ducto.

Hinc ratiocinio ad analogiam capitis prascedentis protracto, ooostabit, si (ut quadratum in biniones, ita) cubus a quantavis radice genitus, in temiones distribuatur, ternionem seu membrum a sinistris primum continere cubum notae sinistrorsum primae, simul ac redundantiam (si quae sit) 3 solidorum quadrati ejusdem in secundam ducti; locum primum secundi caperc dicta solida et redundantiam 3 solidorum quadrati notae secundae in primam, locum secundum eadem 3 solida et redundantiam cubi notae se- ; tertium occupari a dicto cubo, simul ac redundantia 3 ex quadrato notarum prxcedentium in tertiam ducto gmitorum : locum primum tertii membri solida ultimo memorata obcinere, et sic deinceps. Hinc facile derivabimus methodum eliciendx radicis cubicx, quae est ut sequitur.

Incipiendo a dextris, resolvendum (806:21568) in temiones (prarter membrum postremum quod minus esse potest) punctis ioterpositis distribuo. Dein cubum maximum (64) in (80) primo versus sinistram membro contentum subduco, scriptaque illius radice (4) in notam primam radicis quaesitae, residuo (16) abscribo (6) notam proximam resolvendi, unde confit dividendum (166)

24 Ariihmetica.

quod divide per (48) triplum quadrati notac inventae : quotiens (3) est nota secunda radicis : hanc duco, primo in divisorem ; secundo, ipsius quadratum in triplum notx primas; postremo, ipsam in seipsam bis. Producta ea lege aggregata, ut secundum a primo,

}I44 ) 108 >subduco

a dividendo aucto accessione duarum notarum reliquarum

membri secundi. Ad eundem modum, 80*62 1. 568(432 utut prolixa sit operatio, numerum divi- <$4 dendum semper prsestat residuum, adjuncta

prima sequentis membri nota: divisorem

^ ' ' vero, triplum quadrati notarum radicis hac-

^^ ' g tenus inventarum : et subducendum, nota

5547; ,rt ultimo reperta in divisorem ducta, ejusdem

^ quadratum in triplum notarum praecedentium:

ocxxxxx) postremo illius cubus, ea qua diximus ratione

aggregati constituent. Si numerus resolvendus non sit cubus, quod superest, adjectis locis decimalibus, in infinitum exhauriri potest.

Radici assignanda est pars tertia locorum decimalium resol- vendi.

N. B. Operationes syntheticae examinari possunt per analyticas, et vicissim analyticae per syntheticas : adeoque si numero alterutro ex summa duorum subducto, restet alter, recte peracta est additio; et vice versa, extra dubium ponitur subductio, quoties aggregatum subducti et residui xquatur numero majori dato. Similiter, si quotiens in divisorem, aut radix in seipsam ducta procreet divi- dendum, aut resolvendum, id tibi indicio sit, in divisionem aut resolutionem nullum repsisse vitium.

ARITHMETICiE

PARS SECUNDA.

CAP. I.

QUID SINT FRACTIONES ?

ScuPTO divisore infra dividendum^ ductaque llnea intermedia, dirisionem utcunque designari, jam ante * monuimus. Hujusmodi autem quotientes dicuntur numeri fracti seu fractioneSy propterea quod numerus superior, qui dicitur etiam numerator, dividitur seu frangitur in partes ab inferiore dcnominatas, qui proinde didtur denominator: e. g. in hac fractione f 2 est dividendus seu numerator, 4 divisor seu denominator^ ipsa autem fractio indicat quotientem qui oritur ex divisis 2 per 4, h. e. quadrantem duanim rerum quarumvis, vel duos quadrantes uniusj nempe idem sonant.

N. B. Patet numeros qui partes decimales denotant, quique vulgo fractiones decimales audiunt, subscripto nominatore^ per modum fractionum vulgarium exprimi posse. £. g. ,25 valent

tV\f> j004 valent tthht ^^* ^^ S^^ faciamus oportet, aut sAltem factum intelligamus, quotiescunque ex in fractiones vulgares aut vicissim hx in illas reducendx sint, aut aliam quamvis opera- tionem, utrosque fractos, vulgares et decimales ex xquo respi- cientem, fieri contingat.

* [Cap, V. p. I.] — Author.

26

Aritknutica :

CAP. II.

DE ADDITIONE ET SUBDUCTIONE FRACTIONUM.

1. Si fractiones, quanim summa aut di£Ferentia quxritur, eun- dem habent nominatorem, sumatur summa aut diS«rentia nume- ratx)rum, cui subscriptus communis nominator quxsitum dabit.

2. Si non sunt ejusdem nominis, ad idem reducantur. Nomina- tores dati in se invicem ducti dabunt novum nominatorem; co- jusque autem firactionis numerator, in nominatores reliquanim ductus, dabit numeratorem novx fractionis datx aequalis. Dein cum novis fractionibus operandum ut supra.

3. Si int^er fraction! addendus sit, aut ab ea subducendus, vel vice versa, is ad ^actionem datx cognominem reducatur ; nempe illi in nominatorem datum ducto idem nominator subscribendus est.

Additio	•^ ad f sum. \
Subductio	•^ a f resid. \
Additio	1 ad 1, i.e. A ^ iV S"™* ii
Subductio f a f , /. ^. A ^^ A '^sid. ^
Additio	3 ad 1, i.e. V ad | sum. V
Subductio	1 ex. 3 1. e. V resid. V

Primo, Dicendum est, cur fractiones, antequam operemur, ad idem nomen reducamus : atque id quidem propterea fit, quod nu- meri res heterogeneas numerantes in unum colligi, aut ab invicem subduci nequeant. E.g. Si velim addere tres denarios duobus

Pars Secunda. 27

iolidi% summa non erit 5 soL aut 5 den. neque enim ilia prius haberi potest quam res numeratas ad idem genus reducam, adhi- bendo looo duonim solidorum 24 denarios, quibus si addam 3 den. oritur aggregatum 27 den. pari ratione 2 partes tertias et 3 quartas una colligens, non scribo 5 partes, tertias aut quartas ; sed earum locx> usurpo 8 duodecimas et 9 duodedmas, quarum summa est 17

Secundo, Ostendam quod fractiones post reductionem idem Takant ac prius, e.g. quod f xquentur ^\ siquidem uterque nominator et numerator per eundem numerum (v. g. 4) multipli- cantur; omnis autem fractio exprimit rationem numeratoris, seu dindendi, ad nominatorem, seu divisorem; proinde dummodo ratio ilia eadem manet, fractio eundem retinet valorem; sed ducto ■troque rationis termino in unum eundemque numerum, certum

m

est rationem non mutari : e. g. si dimidium rei cujusvis sit dimidii aherius rei duplum, erit et totum illud totius hujus duplum; quod qoidem tarn liquido patet, ut demonstratione non indigeat.

Tertio, Integer ad fractionem reductus non mutat valorem : nam si 2 numerorum rectangulum per unum eorundem dividatur, quo- dens erit alter; sed in reductione integri ad fractum is in nomi- oatorem datum ducitur, et per eundem dividitur : igitur quotiens, b.e. fractio valet integrum primo datum.

N.R, Utile nonnunquam erit, fractionem ad datum nomen re- ducere; e.g. ^ sad alteram, cujus nominator sit 9: quod quidem fit per regulam trium (de qua vide par. 3. cap. i.) inveniendo nu- merum, ad quem nominator datus ita se habeat ac fractionis datx oominator ad ejusdem numeratorem; is erit numerator fracti cujus datum est nomen, valor autem idem qui prioris ; quippe inter fractionis terminos eadem est utrobique ratio.

CAP. III.

DE MULTIPLICATIONE FRACTIONUM.

f. Si ducenda sit fractio in fractionem, datarum fractionum numeratores in se invicem ducti, dabunt numeratorem product!; dati item nominatores procreabunt ejusdem nominatorem.

28 A rithmetica :

2. Si multiplicanda sit fractio per integrum, ducatur integer datus in numeratorem fractionis, eodem manente nominatore.

3. Si in fectore altenitro, vel utroque occurrant int^ri, aut fractiones heterogeneae, ei claritatis causa una coUigi poterunt.

EXEMPLA MULTIPLICATIONIS.

Multiplic I f per I pro. f^ | | per 2 prod. |

Multiplic. I a & I per I & f /. ^. V per i

Manifestum est quotientem eadem proportione augeri, qua divi- dendum: E.g. si 2 continetur ter in 6, continebitur bis ter in bis 6; liquet insuper eundem eadem proportione minui, qua cresdt divisor. E.g. si numerus 3 continetur quater in 12, continebitur bis 3 duntaxat bis in 12: igitur cum ut multiplicem 1 per |, augenda sit fractio f ratione quintupla, quoniam per 5, et minu- enda ratione octupla, quoniam non simpliciter per 5, sed solum- modo ejus partem octavam multiplicaturj duco dividendum 2 in 5, et divisorem 3 in 8.

2. Qucxl ad regulam secundam, constat bis 4 res quasvis aequari 8 rebus ejusdem denominationis, qusecunque demum sit ilia.

CAP. IV.

DE DIVISIONE FRACTIONUM.

1. Fractio per integrum dividitur, ducendo integrum datum in nominatorem fractionis datas.

2. Si fractio per fiactionem dividenda sit, numerator divisoris ductus in nominatorem dividendi dabit nominatorem quotientis; et ejusdem nominator ductus in numeratorem dividendi dabit numeratorem quotientis.

3. Quotioscunque admiscentur integri aut fractiones diversi no- minis, fecilius operabere si membra utriusque, tum dividendi turn divisoris, in binas summas colligantur.

Pars Secunda, 29

EXEMPLA DIVISIONIS.

Div.	1	f per 3, quot. f 1
Div.	Iper	i quot. ff
Div.	2i + ^	per 3f , ;. *. V per	1 1

i""* Quantum ad primain regulam, ex capite pnecedenti constat,

bctionem eadem proportione minui seu dividi, qua multiplicatur

oominator.

3^ Postquam dividens ^actionem unam per aliam, t.g, ^ per f,

I duxi nominatorem 9 in 2, fractio -^ dicit tantum quoties 2 con-

tiiittur in dividendo^ illius vero quintuplum indicabit quotie3

pEtfs quinta numeri % ibidem continetur; quapropter quotientem

primum -^ duco in 5, inde fit ff .

N^ Si fiactiones datas sunt homogeneas, brevius est et concin- nius dividere numeratorem dividendi per numeratorem divisoris, quotiescunque ilium hie metitur. Sic divisis f per f quotiens erit %y quascunque enim numerantur 6 bis continent 3.

2. Si extrahenda sit radix e fractione data, radix nominatoris radici numeratoris subscripta constituet fractionem quae erit radix quaesita. E. g, f est radix quadrata fractionis |, et cubica fractionis •V \ nam ex iis quae de multiplicatione diximus patet, f in f pro- ducere | et f in | dare ^.

CAP. V.

DE REDUCTIONE FRACTIONUM AD MINIMOS TERMINOS.

I. QuoNiAM fractionum quas ex minimis terminis constant valor clarius agnoscitur, utile est fractionis terminos, quoties id fieri potest, per communem aliquam mensuram dividere. Quanto autem major fuerit communis iste divisor, tanto minores erunt (juotientes seu termini fractionis datae aequalis. Oportet itaque.

30

Arithvietica :

datis duobus numeris, ictelligere methodum invcniendi mzxim^M eorum communem mensuram, >. e. divisorem maiciinum qui datc^ dividat absque residuo. Qui est ut sequitur :

2. Divide majorem e datis per minorem, et divisorem per divisionis residuum, ct si quod denuo supersit residuum, per illud residuum prius, i.e. ultimum divisorem dividas; atque - ita poiTO, donee veneris ad divisorem qui dividendum suiun exhauriat sivc nietiatur; is est maxima datorum communti = mensura.

E. g. Proponantur 9 et 15. Divjdo 15 per 9, restant 6. Divido 9 per 6, restant 3 : porro divisis 6 per 3, restat nihil. Ergo j est maxima communis mensura datorum numerorum 9 ct 15: quod sic ostcndo.

(i) 3 metitur 6, at (i) 6 metitur 9 demptis 3 ; igitur 3 metitur 9 demptis 3; sed 3 metitur seipsum, metitur ergo integrum 9: atqui (() 9 metitur 15 demptis 6, ergo 3 metitur 15 demptis (5, metitur vero 6; igitur metitur integrum numerum 15. Hinc patet 3 esse propositorum 9et 15 communem mensuram, Superest ut ostendam eandem esse maiimam. Si negas, esto alia quxpiam major, puta 5 ; jam quoniam (</) 5 metitur 9, {e) g vero metitur ij demptis 6, liquet 5 metiri ij demptis 6; sed et integrum 15 (ex hypothesi) metitur, igitur metitur 6; 6 autem metitur 9 demptis 3, ergo 5 metitur 9 demptis 3, Quoniam igitur 5 metitur et integrum 9, et 9 demptis 3, metietur ipsum 3, i. e. (/) numerum minorem; quod est absurdum.

Inventa maxima communi mensura, patet fractionem ^ deprimi posse ad hanc |, quam priori xqualem esse sic ostendo. Omnis fractio denotat quotientem numeratoris divisi per nominatorem ; in divisione autem, quotiens dicit rationem dividendi ad divi- sorem, dum igitur ratio eadem manet, erit et quotiens seu fractio eadem. Porro rationem non mutari, terminis ejus pariter divisis, liquido constat; e.g. si res quaelibet sit afterius rei dupla, vel tripla, erit et dimidium ilHus, dimidii hujus, duplum vcl triplum, 6cc.

[■»Qui fractiones per integros dividere et multiplicare novit, si in fractionibus (ut vocant) fractionum ad simplices reducendis

(a) pa coniL (t) pa toot. («) per conn, (if) p«r hjp. («) pn conil. (/) per hyp.

' Not in (be 170; edition. J

Pars Secunda. 31

im difScultatem ezperietur. Nam v. g. hacc fractio fractionis \ ecquid aliud est quam pars quarta fractionis f triplicata, ^ ducta in integrum 3 ? Similiter, ductis in invicem tam eratoribus quam nominatoribus, fractio fractionis fractionis, ad integrum reducitur. Hsec cum tam clara sint et per se ifesta, mirum profecto per quantas ambages, quam operosam rematum citationem, et specierum supellectilem a nonnullis onstrantur, dicam, an obscurantur ?]

i. "3 "T

s v.-

J123, propositis

Liiur terdum qi

■sciini vcl pond

^ dec inter p

ondas (18)

^^cx cubicus

:? jmdasj in qi

- - -- ■ -r- r - -^.jr^riii ii^citi pondus

^ -.•^:: r -^ i. 'jsz^ si capiimus - T . -:!>■:': ..jr :^:l j^mLiioS dare p

:x l:-.-— :m^-.r. Quin patet

^ ' • .\-^ r-- rr rr mss^neiiio 16

^- v^^ V. , . r,' 5- issKmij qui ]

►. z --^-i-t* ic 3uzisniur a

^ • .: .:.:.:: "naxris sunw

.-X > •. -.^in^rLTcn. aizmcra

\ •

^■,.c^.-. ^cJ rrriicata

m • •• • -

Pars Tertia. 33

! bctoribus, quotientem. esse alterum. Igitur, si dividam pro iKtuin duorum terminonim intermediorum (6 et 4) per primum jty^quotiens (12) exhibebit quartum proportionalem quxsitum.

j^m^esti0 I . Viator tribus horis conficit quindecim milliaria ; quot gDnfidet novem horarum spatio? Resf. 45. Patet enim ex quse- Hkme, ut 3 ad 15, ita 9 esse ad quacsitum : i. ^^ 3 : 15 : '• 9 : ergo 135, productum ex 9 in 15, di visum per 3, dabit qusesitum, viz, 45.

Sji^st. 2. Si 2 operarii 4 diebus merentur 2s, 5 quantam mer-> eedem merebuntur 7 diebus ? i». f . ut 2 in 4 ad 2, ita 5 in 7 ad ifBssitum : sive 8 : 2 : : 35 ? Unde invenitur qusesita merces, viz.

Sjt^est* 3- Tres mercatores, inita societate, lucrifaciunt 100/. xipendebat autem primus 5/. secundus 8/. tertius 10/. Quxritur fiantum lucri singulis seorsim contigit? summa impensarum Gt 23/. Die itaque, ut 23 ad 5, ita 100 ad qusesitum: numerus |iO¥eniens indicabit quantum primo de communi lucro debetur^ quum nempe est, ut quam proportionem habet cujusque impensa adsummam impensarum, eandem habeat ipsius lucrum ad summam hcrorum. Porro ad eundem modum dicendo 23 : 8 : : 100? et jj : 10 : : 100 : ? cxterorum lucra innotescent.

[•Proportio composita inversa in simplices facillime resolvitur.

lib. lib.

y. g, 2 homines expendunt, 5, 6 diebus : 30 quot diebus expendent 8 homines.^ Die primo 2 : 5 : : 8 ? inveniens 20; die igitur denuo 20 : 6 : 30 : ? et habebis qusesitum. Qua vero ratione terminus qua^itus simul et semel per regulam satis intricatam innotescat, cxplieare superfluum duco.]

Qj^Mst. 4. Quatuor fistulx implent eisternam 12 horis j quot horis implebitur ilia, ab 8 ejusdem magnitudinis ? Dieendum 8:4:: 12? Proinde 4 in 12, h, e. 48, divisa per 8, exhibent quarsitum, viz. 6. Neque in hoe easu, ubi invertitur proportio ulJa est nova diffieultas^ nam terminis rite dispositis, semper habcbimus bina aequalia rectangula, quorum unius notum est iftnimque latus, alterum vero conflatur ex noto termino in ignotum ducto: quare dividendo productum illud prius per notum latus, scu factorem hujus, provenict terminus ignotus. Quo autem ordine disponendi sint termini, ex ipsa quaestione palam fict.

* Not in the 1707 edition. VOL. 111. D

34

Arithntetica :

CAP. 11.

DE ALLIGATIONE.

Regula alltgatiams ssmflkis dicitur, qua, propositis duaboi rebus diversi pretii aut ponderis, &c. invenitur tertium quoddam genus, ex datis ita compositum, ut illius pretium vel pondus, Aac; xquetur dato cuidam pretio vel ponderi, &c. inter proposita intermedio. E. g. Pollex cubicus auri pendit uncias (18), poUa cubicus argent! uncias (12). Quxritur pollex cubicus metalli cujusdam ex utroque mixti qui pendat 16 uncias; in quo pio blemate, pondus intermedium 16 superat argenti pondus per 4, et superatur ab auri pondere per 2. Jam, si capiamus ^ cobi argentei, et ^ cubi aurei, patet eas una conflatas dare pollicem cubicum; quippe |. et ^ xquantur unitati. Quia patet etian metalli hujusce mixti pondus xquari dato intermedio 16; nam argenti, quod levius est per 4, accepimus 2 partes; igitur defectus est 2 in 4; auri vero, quod gravius est per 2, accepimus 4 partes: adeoque excessus est 4 in 2, /• e. xqualis defectui ; qui proinde se mutuo tollunt.

Hinc oritur regula pro alligatione rerum duarum : Fractio quse nominatur a summa di£Ferentiarum, et numeratur a defectn minoris infra medium indicat quantitatem majoris sumendam; et vicissim quse eundem habens nominatorem^ numeratur ab excessu majoris supra medium, indicat quantitatem minoris sumendam.

^ijfdest. Sunt duo genera argenti, uncia purioris valet 7, vilioris 4, quxruntur 3 uncix argenti, qux valeant singulx 5? ResoL constat ex regula, si accipiam f uncix vilioris, et \ imdx purioris ' argenti, haberi unam unciam mixti quxsiti ; hxc triplicata quxstionem.

Quod si res alligandx sint plures duabus, dicitur alligati^ poslta. E,g. sunt quinque vini genera, vis massici est j, diii 5^ falerni 5, cxcubi 7, corcyrxi 9 : volo mixtum cujus vis sit 4. Mixti xqualiter ex chio et massico, vis erit 2: nimirum dimidium sum- mx datarum i et 3, uti per se patet. Similiter, mixti acqualiter ex falerno cxcubo et corcyrxo, vis erit 7, i,e, ^ numeri ai, sci summx virium misturam hancce componentium. 2 et 7 alligc cum vi intermedia data, viz, 4, defectus est 2, excessus 3, summa

Pars Tertia, 35

jdiflFerentianim 5: igitiir sumendx sunt | misturas prioris, \ pos- Skrioris; poiro divisis 4 per 2, quotiens indicat quantum singu- lontm, chii et massici, accipiendum sit. Similiter \ divisx per dicent quantum falerni, &c. mixturas quassitas inesse debet. Ic ^ massici, ^ chii, ^ felerni, ^ caccubi, ^ corqrrxi qiutsitum. Ifinc cernimus, quomodo alligatio composita ad simplicem Niminim pondera, pretia, magnitudines, aut quxcun- demum sunt alliganda, in binas coUigantur summas, qux sunt, utraque, per numerum terminorum qui ipsam mstitinmt: quotientes juxta regulam alligationis simplids alli- cum termino intermedio: qux proveniunt fractiones, di- singulx per numerum rerum, mixturam sive summam ad spectant ingredientium, indigitabunt quantitatem ex sin- fBlis capiendam. Demonstratio patet ex dictis.

K.Bn \cL alligatione plurium rerum quxstio quxvis innumeras

idniittit solutiones, idque ob duplicem rationem: nam primo

tennini deficientes cum excedentibus diversimode colligi pos-

ittnt^ unde varii prodibunt quotientes, cum dato termino inter-

I iMdio alligandi. Cavendum tamen est ne dicti quotientes sint

simul majores, aut simul minores medio ; quod si eveniat, patet

'• fBsesitum esse impossibile. Secundo, unum eundemque terminum

Gcct sxpius repetere ; unde illius portio augebitur, reliquorum vero

portiones minuentur.

Libel in studiosorum gratiam hie exhibere solutionem Celebris illius problematis, ad Archimedem ab Hierone propositi.

j^jutft. Ex conflatis auro et argento fit corona : quxritur quan- tum ei insit auri, quantum argenti ? coronam interim violari non sinit tyrannus. Respon. Parentur binx massx, una auri, altera argenti, quarum utraque sit ejusdem ponderis ac corona. Quibus paratis, patet problema, alia forma, sic proponi posse : datis v. g. libra auri, et libra argenti, invenire libram metalli ex utroque compositi, qux sit datx intermedix molis : igitur inquirendx sunt massarum et coronx magnitudines. Quoniam vero coronx soli- ditas geometrice determinari nequeat, opus est stratagemate. Sin- gulx ergo vasi aqua pleno seorsim immergantur; mensuretur autem quantitas aqux ad cujusque immersionem profluentis quam immersae moli magnitudine xqualem esse constat: immerso uti-

D 2

36 A rithnietica :

que auro, aqua exundans sit 5, argento 9, corona 6. Hue igitur redit quasstioj datis libra auri cujus magnitudo est 5, et libn argenti cujus magnitudo est 9, quxritur quantum ex singulis Or pere oporteat, ut habeamus libram metalli cujus magnitudo sit 6: proinde alligatis 9 et 5 cum magnitudine intermedia 6^ innotcsod quantitas auri, vit,, | lib. et \ lib. quantitas argenti, coroiis immisti.

Hinc patet, quam non difficile sit problema, ob cujus solutioncm notum illud tUpriKa ingeminavit olim Archimedes.

CAP. III.

DE PROGRESSIONE ARITHMETICA ET GEOMETRICA, ET DE

LOGARITHMIS.

Progressio Arithmetica dicitur series numerorum, eadem com- muni diflFerentia crescentium vel decrescentium. E.g. In hac serie i. 4. 7. 10. 13. 16. 19. 32. 25, 3 est communis excessus, quo terminus secundus excedit primum, tcrtius secundum, quartus ter- tium, et sic deinceps: et in hac altera decrescentium' serie, 15. 13^ II. 9. 7. 5. 3. I, 2 est communis defectus, quo terminus quilibet a praecedenti deficit.

Jam ex ipso serierum harumce intuitu et quam prsemisimos definitione, manifestum est, unumquemque terminum contineie minorem extremum, simul ac communem diflFerentiam, multipfr catam per numerum locorum quibus ab eodem distat. E.g. In prima serie terminus quintus 13 constat ex minore extremo i,ct communi differentia 3, ducta in 4, /. e. numerum iocorum quibai a minimo extremo distat. Hinc dato minore extremo, ct axDr muni differentia, terminus quivis, e. g. a minimo undecimus ex- clusive, facile inveniri potest, ducendo difFerentiam 3 in 11, et productum 33 minori extremo i addendo. Idem invenitur, datis majore extremo, differentia communi, et numero locorum quibus terminus qusesitus a maximo sejungitur, ducendo communem dif- Ferentiam in numerum locorum datum, et productum e majore extremo auferendo. Patet etiam qua ratione datis termino quo- libet, ejusdem indice, et communi differentia, terminus primus

Pars Tertia. 37

assignetur ^ et quomodo ex datis termino quovis, illius indice, et minore extremo, communis diflFerentia itemque ex datis termino, differentia, et minore extremo, termini index eruatur. Quin et illud etiam patet, v/». dimidium summx duorum terminorum acquari medio proportionali arithmetico. JS.^. 7 et 13 feciunt 20, cujus dimidium 10 est terminus inter datos medius {ytde seriem frimam). H«c et alia bene multa theoremata ac problemata, corumque solutiones, ex ipsa progressionis arithmeticx natura hcilc quisquam deduxerit, prsesertim si logistica speciosa utatur. Quapropter ea exercitii causa tyronibus relinquo.

Fr9gresno Geomefrica vocatur series numerorum, eadem con- tinua ratione crescentium vel decrescentium. E.g. 3. 6. \%. 24. 48. 96. sunt in progressione geometrica, cujus ratio communis est dupla, nimirum terminus iquisque duplus est prxcedentis. Simili- ter numeri hujus decrescentis seriei, 81. 27. 9. 3. i. progrediuntur ratione subtripla, i.e. terminus quilibet prxcedentis subtriplus est sive \.

Ubi observandum est, terminum quemvis conflari ex potestate communis rationis, ipsi cognomine, in terminum primum ducta. E.g. In serie prima, 48, terminus exclusive quartus, producitur ex 16, potestate quarta numeri 2 (/. e. qux generatur ex 2 ter in seip- sum ducto, siquidem ipsa radix dicitur potestas prima) per ter- minum primum 3 multiplicata. Quamobrem ea qux de prc^res- sione arithmetica diximus etiam hie locum habent, si pro additione ct subductione multiplicationem et divisionem, pro multiplica- tione et divisione involutionem et evolutionem, sive radicum*' extractionem adhibeamus. E.g. Quemadmodum in progressione arithmetica summa extremorum bisecta dat medium arithmeticum, ita in progressione geometrica medius proportionalis est radix producti extremorum. Adeoque theoremata et problemata quod ^ spectat, iis, cum ilia ex nuda serierum contemplatione facillime eruantur, ulterius deducendis non immorabimur.

At vero unum est progressionis geomctricx theorema, ex quo olim derivata fuit, et etiamnum dependet nobilis logarithmorum scientia, quodque adeo hie visum est explicare.

In progressione geometrica cujus principium est unitas, rectan-

' [N. B. Quomodo potestatum quanimvis et cubo eorunique radicibus agentes, in- ndiccs extrahantur, lector diiigens, juxta vcstigarc potcrit.] — Author. ■etbodum quam secuti sumus de quadrato

38 A rithmetica :

gulum duorum quorumlibet terminonun aequatur termino ejusdeoi progressionis, qui pro indice habet summam indicum factonmu B*g.

Si sequentis seriei j i- 2. 4- 8. 16. 32. 64. I du^amus tcnniiiiim ^ I o. 1 . 2. 3* 4. 5* 6. J

secundum 2 in quartum 8, productum 16 est terminus quintus,

cujus index 4 aequatur indicibus secundi et quarti una cx>llectis.

Ratio manifesta est: nam quaelibet potestas, in aliam quam- cunque ejusdem radicis ducta, procreat tertiam, cujus dimensiooes tot sunt, quot fuere in utraque potestate generante. Sed in pro- gressione geometrica, cujus terminus primus sit unitas, patet reli- quos omnes subsequentes esse potestates ex communi ratione genitas, quarum singular tot habeant dimensioned quot locis ab unitate distant.

Igitur si infinitse progressioni geometrical adscriberetur indicum series itidem infinita, ad obtinendum duorum terminonim rectan- gulum baud necesse foret unum per alterum multiplicare ; opor- teret solummodo, indicibus una coUectis, quaerere indicem qui aggregato asquetur, is sibi adscriptum ostenderet rectangulum quassitum. Similiter, si dividendus sit unus terminus per alium, differentia indicum, si extrahenda sit radix quadrata aut cubica, i aut ^ indicis, quaesitum quotum, vel radicem, indigitaret.

Hinc patet, difficiliores arithmetical operationes insigni com- pendio exerceri posse, si conderentur tabular, in quibus numeri naturali ordine collocati habeant singuli indicem a latere respon- dentem: tunc quippe multiplicatio, sola additione; divisio, sub- ductione; extractio radicum, bisectione vel trisectione indicum, peragerentur. Sed indices illos, sive logarithmos, numeris accom- modare, hoc ofus, hie labor est ; in quo exantlando plurimi desuda- runt mathematici.

Primi ^ quidem tabularum conditores hac fere methodo usi sunt Numeris i. 10. 100. 1000, &c. in progressione decupla existenti- bus, logarithmos assignarunt 0.0000000. 1.0000000. 2.0000000. 3.0000000, &C. Deinde ut numeri alicujus, v. g. 4, inter i et 10 intermedii, logarithmum invenirent, adjectis utrique septem cy- fris, inter i. 0000000, et 10.0000000, medium proportionalem qux- siere^ qui si minor esset quam 4, inter ipsum et 10.0000000, si vero major, inter eum et 1.0000000, medius proportionalis in- dagandus erat : porro inter hunc (si minor esset quam 4) et jx-ox-

' Not Napier. He took fourth proportionals, not mean.

Pars Tertia. 39

ine majorem, sin major, et proxime minorem, denuo quxrebant aMdimn proportionalem^ et sic deinceps, usque dum ventum ftiis- JRt ad numenim, non nisi insensibili particula, puta looo^toooi la poposito 4 difierentenu Hujus autem logarithmus obtinebatur, jawmimdo medium arithmeticum inter logarithmos numerorum J ct \Oy et alium inter ipsum et logarithmum denarii, &c. Jam ■ bipartiatur logarithmus numeri 4, liabebitur logarithmus binarii, dnplicatus dat logarithmum numeri 16^ et si logarithmo

ioois addatur logarithmus binarii, summa erit logarithmus Simili methodo, ex uno logarithmo numerii 4 alii in-

inTcniri possunt. Ad candem modum, cum caeteris numeris inter unitatem et decadem intermediis aptati essent logarithmi, alios quamplurimos conun gummar, di£Ferentiae, &c suppeditarunt. Sed de his satis^ ■eqiie cnim omnia qux ad logarithmos spectant tradere statuimus : y dnstazat propositum fuit, eorum naturam, usum, et inven- tkmein quadantenus ezponere.

MISCELLANEA MATHEMATICA:

SITE

COGITATA NONNULLA

DE

RADICIBUS SURDIS, DE JESTU AERIS. DE CONO ^QUILATERO ET CYLINDRO EIDEM SPHJEKJE CIRCUMSCRIPTIS,

DE LUDO ALGEBRAICO; •

ET

PARjENETICA QUiEDAM AD STUDIUM MATHESEOS,

PR.«SERTIM ALGEBRA.

AUTORE » * * ♦ ART. BAG. TRIN. COL. DUB.

1707.

EGREGIO ADOLESCENTI

D. SAMUELI MOLYNEUXS

:aoemia dubliniensi sociorum commensali, filio viri clarissihi gulieljii moltneux^y faucis ab hinc annis acsrbo, tam fatrls quam rsi litsrari, fato denati.

EGREGIE ADOLESCENS,

JTTA fiiit patris tui, dum viveret, apud eruditos existimado, ut me Lis pergratam factunim arbitrer, si filium, sui acuminis ac solertiae iem, ipsmn reliquisse palam faciam. Fatendum quidem est, patnium , vinim doctrina jiixta ac humanitate insigni, tale aliquid jam m * fecisse. Viderat niminim vir clarissimus, eam esse tui necdum scentis indolem, ut te olim patema pressurum vestigia verisimile aret Cujus tanti viri auctoritas apud me usque eo valuit, ut eps magnam de te spem conceperim. Nunc autem, cum ipse orum tuorum conscius, te saniori philosophise et mathesi operam

unnel Molyncax, to whom the MiS' ta Maibtmaiiea are addressed, was iQ of WiUiam Moljrnetix (the friend >rrespondent of Locke), by whom the

OK Human UmUntOHding was in- xd into Trinity College soon after its oblication. Cf. New Tbtory of Vision^ ot's Preface/ and sect. 13a. The |er Molyneuz was bom in 1689, at cr, where his family had retreated for : from the tyranny of Lord Tyrconnel's unent. He was trained by his father lireat care, according to the method of :'s tract on Education^ and afterwards, his father died (in October, 1698), by de Dr. Thomas Molyneux. Samuel lenx was Berkeley's pupil at Trinity ;e, DubliiL In the early part of his

life he wis secretary at Hanover to the : of Wales, afterwards George II. He need his former tutor to the Prince and •ss, which was the first occasion of

Berkeley's being known to Queen Caroline. Mr. Molyneux lived much at Kew. He devoted himself to optics and astronomy, from which pursuits an appointment in the Admiralty in a great measure withdrew him. He died in 1728. The interesting corre- spondence of the elder Molyneux and also of the uncle with Locke, from July 169 a till January 1699, should be studied in con- nection wi^ the introduction of the Lockian and Newtonian philosophy into Dublin. (See Locke's Works, vol. IX. pp. 389-4 7a.) • [Vide epistolam Thomac Molyneux, M.D. ad Episcopum Clogherensem. PbUo' BOpb.Transact.'So. iSi.] — Author, Thomas Molyneux, younger brother of William, was Professor of Medicine in the University of Dublin, and Physician-General to the Army. He attained high repute, and was made a baronet in 1730. He died in 1733. He was F.R.S., and contributed many Essays to the Tranuctions.

44 D E D I C A T I O.

strenue navantem cemam ; quum spinas quibus obsepta videtur mal quseque alios quamplurimos ab ejus studio deterrere solent, te e c ad alacrius pergendum stimulare ; quum denique ad industriam iU sciendi ardorem prseclaram ingenii vim sentiam accedere ; exunds nequeo cohibere Isetitiam quin in orbem literatum effluat, teque ex cipuis (si modo Deus vitam largiatur et salutem) ineuntis saeculi mentis fore, certissimo sane augurio prsenuntiem. Proinde, sequei quantuliscunque ad te delatis, ansam hancce tecum publice coUoq arripere gestiebam; cum ut ipse proprio cedam afifectui, tum i expectatione de te coorta, tanquam vinculo quodam, alioqui non in| illi reram pulcherrimanim studio devinciare.

SCELLANEA MATHEMATICAL

DE RADICIBUS SURDISl

mihi olim in mentcm venit, ut putarem praxin algebraicam n iri nonnihil feciliorem, si ablegate signo radicali, alia am cxcogitaretur potestalum imperfectarum radices com- di methodus, qux ab usitata in reliquis operationum forma > abhorrcret. Nimirum, quemadmodum in arithmctica longe is tractantur fractiones a vulgaribus ad decimales rcductx^ :unc notx cujusque loco nominatoris vicem obeunte, altera rte truncantur, similique forma ac integri descriptx, eandem- um iis sericm constituentes, iisdem itidem legibus subji- jr; sic si ex logistica etiam speciosa ablegaretur nota ista lHs [ V'] qux, ut nominator inter fractiones et integros, ionum diversitatem inter radices surdas ac rationales in- praxis proculdubio minus intricata evadcret. idni itaque radices quascunque surdas, perinde ac rationales, udas duntaxat literas designemus, v. g. pro ^h substituto i? Quippe surdis ad hunc modum designatis, nihil intererit eas ac potestatum perfectarum radices j additio, subductio, plicatio, &c. ad eundem modum utrobique peragentur. Scd

hesc * Mathematical Miscellanies,' d along with the ' Arithmetic * in -•ntain some ingenious operations and ions in Algebra, as well as a specu- >n the cause of the Atmospheric They conclude with an ardent re to the study of Mathematics, y Algebra, to which Berkeley was thusiastically devoted. He adduces -a) Sir William Temple, Bacon, Pes Malebranche, and Locke as autho- favuur of Mathematics, in particular . as a mental discipline ; and he ends by

lamenting that other studies, dry and jejune enough, were at that stage in his life- time superseding charming Mathematics, to which he hoped soon to return. It seems that the diversion of his attention lasted long. Cf. Analyst^ sect. 50, published nearly thirty years after, when he next appeared in mathematical literature, and with a dif- ferent purpose.

' This essay on Surds docs not carry much weight. The suggestion with which it commences has not met with favour, and is apt to produce confusion.

46 Miscellanea McUkemoHca:

objiccrc in promptu est, vcl magis quam signum radicale, sfcaa hac rationc multiplicatas calculum dlvexare. Siquidem com nulla sit affinitas seu conncxio inter * et f , adeoqus una ci altera agnosci nequeat, videtur illius radix aptius designari per V*, cujus statim ac cernitur innotesdt significatio. Rtsfmin^ huic malo mederi posse, si v. g, Grxcum alphabetum ad desig- nandas radices introducamus, scribendo fi pro V*, h pro V^ &c. Quo pacto non tarn ipsx literae quam charactercs variap buntur, et nota quxvis substituta in tantum rcferet primitivaii^ ut scrupulo non sit locus.

Quantitatis ex aliarum multiplicatione aut divisione conflatx radix designabitur per earundem radices similiter multiplicatas

seu divisas. E.g. ^hc^fitt^ et / — = — -.

Si vero proponatur quantitas multinomia, seu constans ex pluribus membris (in quibus nulla sit quantitas ignota) signis+ aut— inter se connexis; designetur horum aggregatum (quod ct alias quidem sxpe fit) per unicam aliquam literam. B. g. fiat a 4 t^e =g cujus radix est y.

Quxris autem quid fiat ubi ignotx quantitates notis connec- tantur J sit v. g. potestas imperfecta /+x; nam si utamur ^ et ( partium nempe potestatis radicibus, ex iis nequit determinari radix totius? Quidni igitur exsequemus potestatem datam im- perfectam alteri cuidam perfectx, w«. /+*=#+ i^+f6 vd ^+3/f + 3/« + fft, &c.? Tunc enim erit /+ f = -y/+ x vel ^/+*, &c.

Sed illud prxtermissum est, qua ratione radicis genus dignos- catur ; utrum scilicet sit quadratica, aut cubica, aut biquadratica. Num itaque quadraticis linquendi sunt characteres Graeci, rcli- quisque deinceps alii itidem assignandi? An potius manente eodem charactere, puncto supra notato radicem quadratam, binis cubicam, tribus biquadraticam, atque ita porro indigitemus: e. g, a significet radicem quadraticam quantitatis per a designatae,

a radicem cubicum, a biquadraticam, &c.? quo quidem modo fluxiones primx, secundx, tertix, &c. designantur. Seu denique id satis ducamus quod per retrogressum innotescat radicis deno- minatio? Quippe inter operandum nihil interest cujus generis sit radix aliqua, quandoquidem omnes absque signo radicali notatx, iisdem subsint legibus, et ad eundem mocum tractentur.

De Radtcibus Surdis.- 47

Cnida quidem sunt hxc et imperfecta, quamque nullius sint

^Rtii ut a me pioponuntur, sat cemo. Tu autem, clarhsime

flMncnUy cui nee otium deest nee ingenium, ex hoece sterquilinio

pboni aliquid fbrtasse extraxeris. Caeterum haud sdo, an ea quae

imus tyronibus (reliquos ista flocci facturos scio) quadan-

usui esse possint^ eorumque ope disquisitionis analyticae

-flum Donnunquam enodetur eliminatis, eum ipso signo radicali,

operationibus quae illud eomitantur heterogeneis. Utut id sit,

Buhi visas sum iis ex parte adhibitis, vulgarem de surMs doctrinam,

bevius et elarius quam ab uUo quod sciam factum est, posse

ciplicare. Proinde rem ipsam aggredion

Radices surdx dicuntur esse conimensurabiles, cum earum ad invieem ratio per numeros rationales exprimi possit; quod si fieri nequeat, inconmiensurabiles appellantur. Porro si propositis duabos radidbus surdis, quxrere oporteat, utrum sint commen- sorabiles necne; inveniatur exponens rationis existentis inter potcstates quibus praefigitur signum radicale: hie si sit potcstas perfecta, habens eundem indicem ae radices propositx, erunt iibc commensurabiles : sin minus, incommensurabiles censendx sunt. E. g. Sint radices propositx ^24 et ^54. ^ fr actio quad- rata exponit rationem potestatis unius 24 ad alteram 54 ^ adeoque radices sunt commensurabiles, viz. 4/^4-- \/ 5^ • • ^ • 3» Propo- oatur denuo ^/s^o et .,y 135 : ratio numeri 320 ad 135 exponitur per ff, cubum nempe perfectum, cujus radix ^ indicat rationem radios unius ^^320 ad reliquam y/12^. Demonstratio manifesta est, siquidem norunt omnes radices quadratas esse in ratione subduplicata, cubica in subtriplicata, biquadraticas in subquad- ruplicata, et sic deinceps potestatum respectivarum.

QiKxl si radices sint heterc^enex quarum exploranda est ratio, ad idem genus reducantur, involvendo numeros signo radicali affixes, singulos juxta indicem radios alterius^ quibus sic involutis prxfigenda erit nota radicalis cum indice ex indicibus primo datis in se mutuo ductis conflato. E. g. Sint radices surdx hetero- genex ij ^ et y/\\. Cubatis 5, et quadratis 11, proveniunt 125 et 111 : his prxfixum signum radicale cum indice 6 prxstat radices homogeneas ^^125 et ^121. Hujus operationis ut cematur ratio, designemus 4/5 per speciem quamvis simpliccm, puta *, et ^11 per Ci eritquc ^SSz=^^y et ^ccc=.^\i^

48 Miscellanea Matkematica :

et !^ MUt =1 ^ 1%^ et l^cceccczsi!^ 121. Ubi poiro patct qi

Additionem quod attinet radicum surdarum, ilia, si sint mensurabiles, fit praefigendo summam terminorum rationis radicali, cui suffigendus est communis divisor cujus ope rationis termini innotuerunt. E.g. ^/a4 + 4/54= 5^/6. Na ex antedictis, et iis quse sequuntur de multiplicatione, ^s % ^6j et \/54=3 n/^' Ad eundem modum fit subductio^ quod differentia terminorum exponentis signo radical! pi Si addendx sunt aut subducendx radices surdac incoi biles, mcdiantibus signis + aut — connectantur. £. g*^6 V3 et Js/6 — >v/3 sunt summa et differentia radicum num< 6 et 3 ; quo quidem modo surdis adduntur aut subducuntur numeri rationales.

Si radix surda per aliam homogeneam multiplicanda sit; angulo potestatum prxponatur nota radicalis, simulque communis. E.g. /^3x^7=v^ai et \/gx^x=z^pe. cujus praxeos demonstrationem, designentur radices rum 3 et 7 per S et ^, ut sit ^^=3 et ^=7, et liquido stabit, quod ^^S dd-=zhd^ i. e. radix quadrata producti aequatiff] producto radicum quadratarum. Idem ad eundem modum ost< potest de aliis quibuscunque radicibus, cubicis, biquadratids, && Radices heterogenese, priusquam multiplicentur, ad homogeneas ft-^ ducendae sunt. Si numerus rationalis in surdum ducendus sit^eleveCur ^ ille ad potestatem datx imperfcctx cognominem, cui pnefigatur' nota radicalis, unaque ejusdem potestatis index. Cxtera ut prius. -E.^. 5><>/4=x/ia5XN/4 = N/5co. Vel brevius sic, 5 ,^4; et generaliter h x ijc^ijb^c vel h ^c.

Divisionem quod attinet, quoties dividendus et divisor sunt ambo radices surdx, ablata (si qua sit) heterogeneitate, nota radicalis cum proprio indice quotienti potestatum praefixa, quotum quxsitum cxhibebit. E.g. v/7-^>/3=x/i = N/H' Si vero ex duobus alteruter duntaxat numerus seu species signo radicali atficitur^ reliquus, juxta indiccm radicis datx involutus^ notx radicali suffigatur : dcinde ut prius. E. g. y/^6 -4- 4=

4/96 ^ 4/64 = Vfr = n/ i • Vel sine prxparatione >^-?--. Et generaliter ^r -*- ^ =^>^ vel ^ .--. Hxc, velut prxcedentia, facillime dcmonstrantur.

De y£stu Aeris.

49

DE ^STU AERISl

^ ita pridem incidi in librum cui titulus, De Imferlo Soils et m Corfmra humana^ authore viro cl. M.D. et S.R.S*. Qui uantus sit, et quantulus sim ipse, non ignoro. Sed ut libere

quod sentio, sententiam ejus De jEstu Aeris^ quam ibidem itam dat, utpote celcberrimi Newtoni principiis innixam, us ulnis amplexus sum. Verumtamen baud scio, an author osus [diacnomenon quorundam isthuc pertinentium causas !Cte assecutus sit. Quam vero justa sit dubitandi ratio, tu, >erspectum habeo acumen, optimc judicabis. buit vir cl. altiorem aeris circa xquinoctia tumorem figurx ndali terrae : differentiam insuper inter xris intumescentiam,

liina meridionali, et illam qux a luna (ut ita loquar) anti- ionali in sphxra obliqua excitatur^ eidem causas acceptam

Ego vero neutrius istorum phxnomenon explicationem lata sphxroide petendam duco. Propterea quod^ primo, is sententia qux massam aerco-terrestrem ea esse figura idit, rationibus tam physicis quam mathematicis compro-

ct nonnuUis item phxnomenis pulchre respondeat j non

apud omnes usque adeo obtinet, ut nulli veteris, vel oppositx sententix fautores, iique non minimx notx viri, rcpcriantur. Et sane memini, D. Chardellou, astronomix simum, abhinc plus minus sesquianno, mihi indicasse^ sibi ;crvationibus astronomicis axem terrx diametro xquatoris rtum esse longiorem: adeoque terram esse quidem sphx- 1, scd qualem vult Burnetius', ad polos assurgentem, prope

V speculation on the * Atmospheric poses some absurd errors, but it is see what its intrinsic value is. To hexnatidan it seems to involve a

appreciation of what constitutes atical proof. : author here referred to was Dr.

Mead, bom 1673, an eminent physician, author of various works 3ne and natural philosophy, whose ttssed through many editions. His IT Imptrio SolU et Luna was first d in London in 1704, and editions

. III.

afterwards appeared in Leyden, Naples, Amsterdam, and Frankfort. It was trans- lated into English in 1 708.

* This reference is to a curious book by Dr. Burnet — Telluris Tbeoria Sacra : orhu nostri originem, et mutationes geturales quas aut jam^subit, aut olim subiturus eit, com- plectens. London, 1681. The opinion re- ferred to is thus stated : ' Manifcstum est partes polares altiores fuisse zquinoctialibus, sive remotiores a ccntro : unde aquae ceci- derunt versus polos, in medias tcrrse partes defluere deberunt, et totam fere telluris

E

50 Miscellanea Mathematica :

aequatorem vero humiliorem. Attamen quod ad me mallem quidem viri clarissimi observationes potius in vocare^ quam aigumentis quae terram esse oblatam demc obviam ire. Nihilominus^ quoniam sententia ista non ( aeque arridet, illam tanquam principium ad phxnomenoi explicandum adhiberi nollem, nisi res aliter commode < nequeat. Sed secundo, tantum abest quod supradictonu tuum explicatio sj^xroidalem terras iiguram necessario p( vix uUam inde lucis particulam mutuari videatur : id quo sitis qux in banc rem scribit vir clarissimus, ostendere \ ^^Altius (inquit) solito se attollit aer circa duo aeqi quoniam cum aequinoctialis linea illi globi terrestris cin versa respondeat qui diametrum babet maximam, utrumq dum in ilia versatur terras est vicinius." Dt Imp, SoL et L Jam vero, utrum vicinior iste luminarium situs par sit at aeri in cumulum solito sensibiliter altiorem, merito ambig Etenim tantilla est differentia inter axem transversum jugatum ellipseos, cujus volutione gignitur sphasrois terrc

ilia ad sphasram quamproz cedat. Verum ut accural prosequamur, designet a c . tionem per polos massas aere tris, in qua sit d c axis a 1 1 asquatoris. Jam inito calcuk hendi vim lunae attractricc vel a non esse 7^^ sui part rem quam foret in c vel a polo alterutri directe immii proinde differentiunculam eflFectui ulli sensibili edendo imparem omnino esse, i randum etiam, lunam ab aequatore nunquam tertia par ^ d distare, dictamque proinde quantulamcunque diffi adhuc valde minuendam esse. Quod autem de luna id de sole, cum multis vicibus longius absit, adhw

constabit.

Verum quidem est D. Mead alias insuper causas acsti

luperficiem irrigarc' (Lib. II. cap. 5.) Estay, in a series of tracts (i Burnet wrote several tracts in defence of his afterwards collected as a review theory. He also wrote Remarks on Locke's

De y£stu Aeris. 51

xquinoctia altioris attulisse; viz. ^ agitationem fluidi sphxroidis in majori orbe se revolventls majorem, prxterea vim centrifiigam eficctum habentem eo lod longe maximum V Quod ad primam, etsi ilia prima fronte nonnihil prse se ferre visa sit, fatendum tamen est, me non omnino percipere, quomodo aliquid inde ad distinctam rei propositus explicationem fia,ciens colligi possit. Quod ad secundam, constat sane vim centrifiigam prope sequa- torem esse longe n^ximam, et propterea massam aereo-terrestrem figuram oblatse sphxroidis induisse : quid vero aliud hinc sequatur non intelligo.

Venmi etiamsi concedamus aerem, propter causas a clarissimo viro allatas, drca aequinoctia ad aequatorem supra modum tume- fieri; non tamen inde apparet, quamobrem apud nos, qui tam procul ab xquatore degimus, tum temporis altius solito attollatur : quinimo contrarium scqui videtur. Sequenti pagina sic scribit D. Mead^ ^Ut finem tandem feciam, in iisdem parallelis ubi liiDX dedinatio est, ilium coeli polum versus qui altissimus in- Siirgit, validissima est attractio, cum ilia ad ejus loci meridianum vierticem accedit, minima vero, ubi pervenit ad meridianum loci oppositi ; quod contra contingit in parallelis his adversis. Causa est in sphxroide terrx artherisque figura." Ergo vero causam non esse in terrx et ambicntis xtheris figura propterea puto, quod posita terra vel perfecte spherica, vel etiam oblonga, idem certe eveniret, uti infra patebit.

Restat ut harum rerum explicationem ipse aggrediar, siquidem CO prxsertim nomine suspecta mihi fuit ratio a sphxroidali terrx figura deducta, quod, nulla ipsius habita ratione, res tota darissime simul ac facillime exponi posse videbatur.

Newtonus, Operis sui Physico-Mathematici, lib. iii. prop. 24. obi xstuum marinorum phenomena explicate hxc habet : ^< Pendet etiam effectus utriusque luminaris ex ipsius declinatione seu distantia ab aequatore. Nam si luminare in polo constitueretur, traheret illud singulas aqux partes constanter, absque actionis intensione et remissione, adeoque nuUam motus reciprocationem cieret. Igitur luminaria recedendo ab xquatore polum versus eflfectus suos gradatim amittent, et propterea minores ciebunt

* Di Imptrio Soils ei Luna, p. 7. £ 2

52 Miscellanea MathemaHca :

sestus in syzygiis solstitialibus quam in sequinoctialibus." non alia causa videtur quxrcnda ullius phaenomeni sestus quam qux ad similem dFectum in xstu marino czcitandun ciat. Sed ut id quod a viro per totum orbem longe celeb breviter adeoque subobscure traditum est, uberius exponai in priore figura a J c i meridianus, et s i axis massse terrestris; sol autem et luna in polo constitui concdpi Manitestum est, quamvis massz aeiae partem, puta Jy d circumvolutione diurna, candem semper distantiam a lumin tueri, adeoque vi ubique xquali in eonim corpora trahi. Pi aer non uno tempore attollitur, alio deprimitur, sed per diem in eadem hau'et altitudine. Venun secuodo, in i figura repraesentet a e i J an^uatorem aut parallelura qui luminaria interim in piano sequinoctiali existant; quo te manifestum est, turn ipsum sequatorem, turn singulos paia ellipticam induere figuram. Manifestum etiam est, aerei nunc «, aptcem axis transversi, obtinet, adeoque altissimi surgit, post sex horas, t, extremum axis conjugati, ubi I limus deprimetur, occupatum ire, maximamque proinde reciprocationem cieri. Ut igitur rem omnem simul ab« gibbos sphxroidis xstuosse trtplici ratione locari concipi vel in polis, vel in sequatore, vci in locis intermediis. In casu, esset planum rotationis diumsE ad axem sphscroldis per culare, adeoque circulus^ unde nullus fbret zstus: in set esset ad eundem parallclum, adeoque ellipsis, inter cuju: maxima sit differentia ^ unde maximt forent zstus : in tctti magis ad situm perpendkularem accederet, eo circulo vi esset, adeoque minores forent Kstus.

Reliquum est ut dcmonstrem, differentiam qu* est in s obliqua inter xstum quemvis ct subsequentem, ubi luna extra iequatorem vagatur, terra posita vel oblata, vel ad amussim sphxrica, vel etiam oblonga, perinde causatum iri. Sit « i axis mundi, g d xquator, k locus quivus, / * loci parallcJus, h I axis sphaeroidis xstuosx ob actionem, potissimum, lun«

De Cono. 53

tiimentis. Luna autem prope / constituatur. Demon- csX c k altitudinum aeris^ luna prope loci meridianum ite, majorem esse cf^ aeris altitudine, ubi luna meridianum oppositi transient. Ducatur f s parallelus priori ex adverso ct producantur c k^c fad f ct s. Per constructionem / i xqualis est arcui k I ,• ergo arcus / h major est arcu tf I ergo propter ellipsin recta / / minor est recta kpyttfc minor

E CONO ^QUILATERO ET CYLINDRO, EIDEM SPH^RiE CIRCUMSCRIPTIS^

LEMMA.

Latus trianguli aequilateri est ad diametrum inscripti circuli, «l ^ 3 ad I ; et perpendicularis ex angulo quovis ad latus oppo- stum demissa, est ad eandem, ut 3 ad 2.

Haec cuivis, algebram et geometriam utcunque callenti, facile coQstabunt.

PROBLEMA.

Invenire rationem qux existit inter Cylindrum et Conum sequi- btenim eidem Sphserx circumscriptos.

Ponamus diametrum et peripheriam basis cylindri esse singulas vnitatem. Eruntque, per Lemma, diameter basis coni cjusdemque peripheria singulx ^/^. Proinde i x i = i = bas. cylindri ; et \ = summx basium. Et ^^ x i V3 = J = bas, coni, et super- kies cylindri seu quadruplum baseos =1. Et superficies simplex

coni = J = — - X V 6. Nam >v/f {^» ^. media proportionalis inter

4 s^i latus coni, et basis radium seu \/J) est radius circuli aequalis

superficiei conicx. Et per prxcedentia i + 1 = f = sup. tot.

cylindri, et | -f | = J = sup. tot. coni. Porro per hypothesin

^ Thb matter of the Cone is at best ao ingenious conceit.

54 Miscellanea Mathematica :

et lemma, axis cylindri est i, et coni f . Soliditas autem qrli = i X I = i, et soliditas coni = | x ^ = f . Hinc, comp tis inter se homogeneis, eruitur sequens

THEOREMA.

Inter Conum sequilaterum et Cylindrum eidem Sphacrae ciro scriptos, eadem obtinet ratio sesquialtera, quoad superficies tc superficies simplices, soliditates, altitudines, et bases.

Duobus abhinc annis^ Theorema illud non sine admiratione qua inveni. Nee tamen propriam ingenii vim aut sagacita ullam, quippe in re tam facili, sed quod Tacquetus^, notissL matheseos Professor, tantopere gloriatus sit, de invento cui in non sit tyro, id demum admiratus sum. Nempe is inven partem aliquam Theorematis prxfati,^^. quod ^^ conus sequilat sit cylindri, eidem sphxrac circumscripti, soliditate et superi tota sesquialter ; quodque adeo continuata csset ratio'* inter coi aequilaterum, cylindrum, et sphseram. Hasc est ipsa ilia pn sitio, ad quam spectat schema, quod praefati authoris tractatus Theorematis ex Ar chime de selectis^ in ipsa fronte, una cum graphe insci iptum praefert. Quin etiam videas qux dicat Jesui in prsefatione, in scholio ad prop. 32, et sub finem propositi^ 44t« ejusdem tractatus. Ubi Theorema hocce tanquam illustre quod inventum, et Archimedscorum aemulum ostentat. Idem c Tacquetus, etiam CI. Wallisius" in additionibus et emendation ad cap. Ixxxi. algebrae suae, aD. Caswello^'^ ope Arithmetices In torum demonstratum exhibet. Qucxi ipsum, quoad alteram partem, fecit D.Dechalcsi'' in librosuo de indivisibilium Metb prop. 20. Sed tam ipsa indivisibilium Methodus, quam qux i: fundatur Arithmetica infinitorum, a nonnuUis minus Geomet censentur.

Integrum autem Theorema a nemine, quod sciam, ant<

* i. e. in 1 705. " Des Chales, a native of ChamtM

* Cf. p. 7. Savoy, was professor of mathematit '* i. e. Tacquet. Qeremont, and afterwards in Turin. *^ Wallis, the eminent mathematician and edition of Euclid was long a popular

logician, died in 1 703. book. His works were published at L

" John Caswell, an Oxford mathematician, in four folios, under the title of M

author of A Brief Account of On Doctrim Matbematicut. He died in 1678. 0/ Trigononutry (1689) and other works.

De Ludo Algebraico,

55

Ltum fiiit. Attamen si verum est quod opinatur Tac- : * Iddrco Archimedi inter alia tarn multa et praeclara in- ilhid quo cylindrum inscriptx sphxrae soliditate et superficie Lltenim esse demonstrate prae reliquis placuisse : quod corpo- et superficierum corpora ipsa continentium, eadem esset una rationalis proportio :' si, inquam, hoc in causa fuit, cur tcjliadrum sphxrx circumscriptum tumulo insculptum voluit; tandem faceret senex ille Siculus^ si unam eandemque ratio- proportionem bina corpora quintuplici respectu intercedere ^Ddisset? Illud tamen quam facile ex ejus inventis pro- mode vidimus.

[■^Simili fere methodo ac nos illud omnia Tacqueti Theoremata ;is subjuncta, adde et centum istiusmodi alia si cui pretium videbitur, difficile erit invenire et demonstrare.]

DE LUDO ALGEBRAICO^^

Sob idem tempus quo Theorema illud, Ludum etiam Algebraicum

unreni. Qujppe cum vidissem e familiaribus meis nonnulios, per

dimidios fcrme dies, Scacchorum^^iudo gnaviter incumbentes, acre

corum studium in re nihili admiratus, rogavi quidnam esset quod

tantopere laborarent ? lUi porro pergratum animi exercitium re-

nantiant. Hoc ego mecum reputans, mirabar quamobrem tarn

pauci ad mathesin, utilissimam sane scientiam eandemque jucun-

dissimam, animum applicarent. An quod difficilis sit ? Sed multi

et ingenio valent, nee laborem in nugis fastidiunt ullum. An

podus, quod gratissimum animi exercitium non sit ? Sed quxnam,

fueso, est ilia ars, aut disciplina, aut quodcunque demum opus,

quod omnem animi facultatem, solertiam, acumen, sagacitatem

pulchrius cxerceat? Sed ludus est mathesis? Nihilo secius

jucunda: co tamen si venisset nomine, tunc forsan lepidi isti

** Not cootained in the 1707 edition.

* This carioas Game, contrived as a sub- ititste for Chest, and at the same time as a piasng and oseful exercise in Algebra, is ftrj dbaracteristic of Berkeley. Portions of vfaat foQows, especially the formube for the poMMe wiationt and combinations which the cocMSitions of the Game admit of that are cnotained in the Appendix, are given in MS.

in his Common-place-book. The Game itself is a sort of lottery — not to solve but to draw a set of simple equations. It is worth little, save as showing the bent of Berkeley's mind towards the practical side even of a game of chance. In reading it, he supposes himself, like a spider, in the centre of the Tabula. »* Chess.

5«

Miscellanea Mathemah'ca :

homunciones, qui tempus ludendo terunt, ad ejus studiu) protinus accingerent. Subiit adl«cc sapientissimi viri Johannil Lockii"', in re non multiim absimili, consilium. Sequentem f inde lusum ad praxJn algebra exercendam, rudi feteor Minerva, excogitavi, scd qualis adolesccnti, aliis praesertim studiis occupata facile spero condonabitur,

Problcmata algcbraica immediate constituunt ajquationes data qua in quajstionibus determinatis quantitates quasitas numen exsequant, Quaclibet autem jequatio duobus constat mcmbris x litatis signo connexis, in quorum utroque consideranda veniuntj primo, species, utrum scilicet quantitates datas aut quxsitas desij nent ; deinde, sjgna quibus conncctuntur. Efficere itaque ut h« omnia ad constitucndas questioncs sorte obveniant, ludumqaf tam ex quacstionum formatione, quam ex earundem resolution^ concinnare operam damus.

In asserculoj qualis ad dominarum aut scaccborum lusum v adhibcri solet, depingatur circulus quadrato inscriptus, reliquaque omnia quK in appositoSchemate'" continentur; nisi quod loco cir^ ccllomm nigrantium facienda sinl foramina. Quibus peractit habcbimus Tabulam lusoriam. Parandus iDsuper est stylus tenui e ligno, qui alicui ex dictis foraminibus infigatur. Reliquum t ut horum usum exponamus.

Ut vidcs, operationum logisticarum Symbola ad latera et anga los Quadrati scribunlur: porro latera prioribus, anguli vero pM terioribus, sequationum membris signa impcrtiunt. Circulus autO inscriptus a sedecim cuspidibus in totidcm partes aequalcs dispq citur, ita ut tres cuspides ad latus et angulum quemvis spcclenf sed alise directe, alix oblique: qux oblique latus aliquod aut an- gulum respiciunt, eae angulo et lateri communes sunt; qu* vero directe latus aliquod intuentur, etc ad angulum nullum pertinent, sed ad utrosque adjaccntes pariter rcferuntur. Et vicissim qux angulum aliquem directe intuentur, eae ad latus nullum pertinent, sed ad utraquc adjaccntia pariter referri censenda sunt.

In fbrmanda itaque qusestionc, primo obscrvanda est cuspis quam stylus respicit, latusque et angulum ad quos pcrtineat; horum signa notentur, quippe qua;, ut diximus, species utrius<jiif

" Sur Enay w lb4 Comdna a/ iht Undtr- " In thr origin»l edition

ilanding, t ;. Thii ii ihe finl tliuiion to Lusaria otcupies in enlugci Lnckc in BRkrlr}r'i workt, facO ihii tecliDD.

De Ludo Algebraico.

TABULA LUSORIA.

cojuslibet aequationis membri connectent. Dein, stylo liters ad ptacdictam cuspidem scrtptse imposito, numera i, eoque inde juxta rectae linese ductum translate (ut faciunt astrologi, nominum qui- bus fcrix appellantur rationem assignantcs) ad literam oppositam, oumcia 2. Tunc ad alteram linex, tanquam continuata esset per annulum intermedium extremitatem pergens, numera 3J et sic deinceps, donee litera primse cuspidi adjacens recurrat Hinc recta dcscendens ad cuspidem in convexitate interioris circuli tcnninatam, foramini alterutro adjacent! intigc stylum.

Numenis ultimo numeratus indicabit, quot quantitates quEsitEc, Tcl (quod idem est] quot aequationes datx tiierint in quxstione. Hanim membra priora quantitates ignotae altcrnatim sumptw et ligoo lateral] connexx, posteriora quantitates cognitx vel incog- nitae (prout determinarit litera ad cuspidem internam scripta) quacsitis signo angulari alligatx, constituent. Porro d adhibendas

I

58 Miscellanea Mathematica :

quantitatum cognitarum species diversas, s unam solummodOy / figuras numerates 2, 3, 4, &c. x quantitates quaesitas repetendas esse indicat. Notandum autem, in cujusque aequationis membro posteriore non alias poni quantitates ignotas, quam qux in primo membro sequentis sequationis reperiantur. Dicta exemplis clarcsccnt.

Ponamus itaque stylum occupare foramen stellula insignituin, cuspisque quam respicit pertinebit ad latus cujus signum est +, et ad angulum cujus signum est x, qux signa in charta noto, laterale a sinistris sive primum deinde angulare. Porro e ad cuspidem scribitur, ad quam numero i ; inde (liberum autem est e duabus lincis utriusvis ductum sequi) sinisttx)rsum pergens ofFendo a^ ad quam numero 2; hinc transiens ad £ numero 3; inde autem transversim eunti denuo obversatur e, litera prinuc cuspidi apposita, ad quam numerans 4, recta descendo ad cuspidem intcriorem litera d insignitam. Erunt igitur quatuor quantitates quxsitx in quasstione, qux signo lateral! +, altematim connezse, constituent prima xquationum datarum membra. Posteriora veio ficnt ex quantitatibus ignotis et notis (propter d) dlversis per signum angulare, nimirum x , conjunctis ; ad hunc modum :

y-^z=zad y=?

z-^a=z ef zz=?

Quod si ponamus stylum foramini prxcedenti infixum esse, quo pacto + laterale directe intucbitur, lineasque sinistrx ductum scquamur, provenient tres quantitates investigandae, et cuspis interior habcbit literam/. Unde numerus xquationum datarum et primorum carundem membrorum signa, itemque posteriorum species detcrminantur, Sed quoniam in hoc casu cuspis indi£Fe- rentcr se habet respectu duorum angulorum adjacentium, idcirco eorum signa per vices usurpanda sunt : secundum quas conditiones hujusmodi struatur quxstio.

y^a=4e y=?

Posito autem stylum sequenti foramini infigi, cuspis stylaris inxangulare dirigetur, signaque lateralia+et— pariter respiciet.

De Ludo Algebraico. 59

Proindc, si fert animus dextram inire semitam, juxta leges prae missas sequens prodibit quxstio :

e—y-ssiay f = ?

y-^-a^iae y^?

19 Notandum autem primo, quod varietatem aliquam in signo- nim et specierum combinationibus prxscriptse leges admittant. Unde fit, quod cuspide semitaque determinatis, diversas oriantur quaestiones.

Secundo, quod etsi ad primx literse recursum sistendum esse supra statuimus^ lex tamen ilia pro cujusvis arbitrio mutari possit ; ita ut progrediamur donee singulx, a^ e, «, x, obversentur, vel aliqua ex iis bis, vel ad aliam quamcunque metam. Sed ad lusum properamus.

Primum itaque e lusoribus aliquis ad mcthodum jam traditam qaaestionem sibi formet. Quod et cxteris deinceps iisdem legibus fiudendiun est. Porro fbrmatis singulorum quasstionibus, ad ejus quae sorte obtigit solutionem se quisque accingat. Faciat dcin unusquisque fractionem, cujus numerator sit numerus quantitatum in suo problemate quaesitarum, et nominator, numerus graduum sive aequationum quas^ dum solveretur quxstio^ chartis mandabat. Penes quern maxima sit fractio, is vincat.

Proinde, siquando fugitivx quantitates inhiantem eluserint alge- bristam, is omni victorix spe excidisse censendus est. Neque id prorsus injuria, siquidem potius eligentis culpa quam infortunio accidat quxstionem esse indeterminatam.

[«>Quotiescunque inter ludendum deveniatur ad xquationem aflFec- tam supra ordinem quadraticum, nihil opus erit exegesi numerosa aut constructione per parabolam, sulfficit si radix incognita mutata specie pro cognita habeatur.]*

Peractis omnium quxstionum solutionibus, quisque proximi opus percurrat ; ad quod Pellii margines conducant,

Qux pignora et mulctas spectant, quisquam ad libitum com- miniscatur : hxc enim aliis permitto.

Problemata quod spectat, ilia quidem difficilia non sunt, alioqui inepta forent ad lusum ; sed ea tamen, quorum solutio in ingens lusorum commodum cesserit, dum rectum tramitem inire student,

*• [Vide Appendicem.]-' Author. * Not in the 1707 edition.

6o Miscellanea Mathcmatica :

dum longos consequentiarum nexus animo recolunt, integramquc analyseos seriem brevissimo conceptu claudere laborant.

Permitte jam, adolescens optime ut alios paulisper alloquar; tibi enim, quern ipsa trahit difficultas, nihil opus hortatore. Vos, adolescentes academici, compello, quibus inest sagadtas, men- tisque vigor et acumen j tristem vero in musseo solitudinem, duramque eorum qui vulgo audiunt Pumps^ vitam aversamini, satius inter congerrones, per jocum et lusum, ingenium prodere ducentcs Videtis quam merus lusus sit algebra, et sors locum habet, et scientia: quidni igitur ad tabulam lusoriam accedatis? Neque enim, quod in chartis, scacchis, dominis^ &c. usu venit, ut dun alii ludunt, alii oscitanter adstent, hie etiam metuatis. Nam quotcunque ludendi incesserit libido, iis omnibus ludere simul ac studere, adde et nonnullis, lucelli aliquid corradere hs est. Ast aliquem audire mihi videor in hujusmodi verba erum- pentem : Itane vero nos decipi posse putas ? Non ii sumus, quos ad difficillimam artem sudore multo addiscendum, oblata lusus specie, inescare liceat. Respondeo, algebram eatenus esse di£S- cilem quantum ad lusum requiritur: quod si tollas omnem diffi- cultatem, tollitur simul recreatio omnis ac voluptas. Siquidem ludi omnes totidem sunt artes et scientiae^ nee aliud est inter cscteros et hunc nostrum discrimen, quam quod illi prxsens solum- modo oblectamentum spectent^ ex hoc vero, practer jucundissi- mum laborem, alii etiam iique uberrimi fructus percipiantur. Tan- tum autem abest quod hoc in lusus detrimentum cedat, ut is id- circo omnibus numeris absolutus jure habeatur, juxta tritum illud poetac,

' Omne tulit punctum qui miscuit utile dulci.*

Sed quinam sunt illi quos praedicas fructus ? Hos ut enumerem, universa, quaqua patet, mathesis, artesque omnes ac scientiae, quas rem militarem, civilem, et philosophicam promoventes complec- titur, perlustrandae forent. Quippe per hasce omnes di£Funditur mirifica algebrx vis. Eadem apud omnes ars magna, mirabilis, supremus cognitionis humanae apex, universal matheseos nucleus et clavis, imo apud nonneminem scientiarum omnium fundamen- tum audit. £t sane quam difficile esset algebras limites assignare, cum philosophiam etiam naturalem et medicinam jamdudum in- vasit, inque dies dissitissima quaeque argumenta aggreditur. Ut alia taceam, in Actis Philosoph. N^ 257, de certitudine testimo-

De Ludo Algebraico,

6i

niorum et traditionum humanarum algebraica extant theoremata. £t pro certo statuendum est, ubicunque datur magis ac minus, ubicunque ratio aliqua aut proportio invenitur, ibi locum habere algebram.

Verum dixerit fortasse aliquis, se nee mathesin ipsam, nee res mathematice tractatas morari. Ut lubet: demus hoc voluntati cujuspiam, demus ignorantise : nimirum ex ignorantia rerum prx- darissimarum, qu^ue vos a barbaris Mstsnguunt^^y contemptum pro- fidsci afHrmare ausim. £stne vero quisquam qui ingenium sagax, intellectum capacem, judicium acre parvi faciat? Siquis usque adeo rationis expers inveniatur, is demum mathesin spernat, qux quanti sit momenti ad optimos quosque mentis habitus comparan- dos, apud omnes in confesso est.

Verulamius alicubi, in iis quae de Augmentis Scientiarum con- scripsit'^, analogiam quandam inter pilse palmariae hisum et mathe- sin notat. Nempe quemadmodum per ilium, ultra voluptatem qux primum intenditur, alia eaque potiora consequamur, v$%. cor* poris agilitatem et robur, promptumque oculorum motum : sic dis- dplinx mathematics, prseter fines ac usus singulis proprios^ illud etiam coUaterale habent, quod mentem a sensibus abstrahant^ ingeniumque acuant et figant. Idem hoc tam olim veteres, quam hodie e modernis cordatiores quique agnoscunt. Quod vero recen- tiorum algebra ad ingenium formandum imprimis conducat, inter alios ostendunt Cartesius^^^et prolixeMalbranchius'^* Delnquirenda

* [Vide Tentamen Anglicum de Hortis Efkmi, a Gulielmo Temple, Equite Aurato coQscnptum.] — Author. The reference to Sir William Temple is contained in the foDowiog sentence : — * More than this, I know DO advantage mankind has gained by the progress of natural philosophy, during 10 many ages it has had vogue in the world, excepting alwajrs and very justly what we owe to the Mathenutics, which is in a manner all that seems valuable among the dviHscd nations, more than those we call barbarians, whether they are so or no, or Dorc so than ourselves.' — Essay upon the Gardens o/Epieunu (1686). See Temple's Consded Works, voU I. p. 172 (ed. 1720). Of. Guttrdian, No. 130, in which the above pascan is referred to in a similar manner.

^ The passage alluded to is contained in the AdpatuemetU of Learning, the earlier vork (1605), and is not reproduced in the traoilation, in the corresponding passage of

the Dt Augmentis (1623). The words are these : — * For if the wit be too dull, they (Pure Mathematics) sharpen it ; if too wandering, they fix it; if too inherent in sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures : so in the Mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended.' Advancement of Learning, B. II. But Bacon repeats his recommendation of Mathematics, especially as an education of the power of attention, in the De Augmentis, VI. 4, and in the Essay on Studies in 1625.

" See Discours de la M6tbode, pp. 143- 146, in Cousin's edition of the works of Des Cartes. In another passage in the same work, Des Cartes speaks rather in disparage- ment of Algebra.

^ This paragraph contains Berkeley's

62

Miscellanea Mathematica,

Veritate, lib. vi. part i. cap. 5. et part. 2. cap. 8. alibique passim. £t regulas quidem quas hie in qua^tionum solutione observandas tradit, lib. vi. part. 2. cap. i. quasque tarn sunt eximias, ut meliores angelum non fuisse daturum credat auctor quidam ingeniosus: illae, inquam, regulas angelicas ex algebra desumi videntur. At quid alios memorem, cum vir omni laude major, Jdiannes Lockius, qui singulos intellectus humani defectus, eorumque remedia, siquis alius, optime callebat, cum universae matheseos, tum praesertim algebras studium, omnibus supra plebem positis, tanquam rem infiniti usus vehcmenter commendat? Vide inter Opera ejus Posthuma^ % pag. 30, 3 1 , 32, &c. Tractatus de Regimine Intellectus : opus exiguum quidem illud et imperfectum, sed quod vastis et elaboratis aliorum voluminibus jure quisquam praetulerit. At vcro auctor magni nominis ad disciplinas mathematicas acrem nimis meditationem, quasque homini generoso et voluptatibus studenti minus conveniat, requiri putat. Respondeo, suadente Lockio^ frustra opponi dissidentis Santevremontii'^^ judicium. Deinde hie ineptus matheseos judex merito habeatur, quippe qui, uti ex ejus vita et scriptis plusquam verisimile est, eam vix a limine salut^raL Si vero cortex durus videatur et exsuccus, quid minim ? Sed ut dicam quod res est ^ prasstat singulos rem ipsam expertos propria sequi judicia. Nee est cur quis ingentes difficultates sibi fingat, eo quod vox algebra nescio quid asperum sonat et horrificum; artem enim, quantum ad ludum nostrum requiritur, intra breve unius mensis spatium facile quisquam perdiscat.

Exposita demum lusus et consilii nostri ratione, lectorem ma- thematicum, ut tenues istas studiorum meorum primitias candide accipiat, rogo, potiora forsan posthac daturus. Imprassentiarum autem me alia distinent studia quas, arida satis et jejuna, suavissi- mam mathesin exceperunt. Tu interim, Clarissime Adolescens, hanc nugarum rhapsodiam, tanquam aliquod mei erga te amoris symbolum, cape, et vale.

earliest published allusions to Bacon, Des Cartes, and Malebranche, here adduced as authorities in fayour of the study of Mathe- matics. It may be added that Malebranche, in his Reebtrcbet Lir. VI. p. ii. ch. 8, alludes to the commendation of Algebra in Des Cartes' Discours (U la Mitbod*.

^ The reference is to the Es$ay on tbt

Conduct of the Understanding, § 7, eulogized in the following sentence.

** Saint Evremond, a famous French wit of the seyenteenth century, who came to England in the reign of Charles II, and died thert in 1703. He was connected with Bayle, Des MarzeauiL, and other men of letters of that time.

APPENDIX.

Ut mentem nostram quilibet plenissime assequatur, visum est, sequentibus paginis, omnem in qua^tionibus Combinationum et Specierum varietatem quam praefatas ludendi conditiones patian- tur oculis subjicere.

Notandum autem : Primo, qucxl sequentes formulas, quoad modos combinandi et quantitatum species, non item omnes quoad nume- nun aequationum datarum, ad Cuspides respectivas pertinent : saspe enim plures quam tres quantitates investigandae enint.

Secundo, quod ut omnes quasstionum formulas haberi possint, metx diversas, prout fieri posse supra monuimus, statuendas sunt : alioqui duas tantum ex quatuor classibus ad Cuspidem quamcunque pertinebunt.

Primam dico Cupisdem quae in + laterale dirigitur, secundam huic a dextris proximam, atque ita porro.

AD LECTOREM.

IsTA adolescentiae nostras, obiter tantum proprioque marte ad quantulamcunque matheseos scientiam olim enitentis, conainiii3' in lucem protrusisse sero aliquoties poenituit. Quin et poeniteret etiamnum, nisi quod hinc nobile par Ingeniorum, in spem nascen- tis saeculi succrescentium, una propalandi enascatur occasio. Nequc enim nos aliunde Rempublicam Literariam demereri gloriamur. Atque hsec quidem ad temeritatis, &c. censuram, ut et invidiam, si quam mihi forte conflaverim, amoliendum dicta intelligantur.

Appendix. 65

Cusfis frima,

a-^e=z^xe e^b by,y y^b exb b^e yxb b^y

e-\-y^b'~y y xb b^aaxb y—^b b xy ^^b bxA

s y^a-ssibx^a—b bx^ e^b axb b^ae xb b^e

a-{-e=zbxe e—bb xy y—b exb b-^e yxb b^y

e+y=c-^yy xc c—aaxc y — c c xy a^c c xa

d y'\'a^^dxaa-^ddx^ ^ — da xd d^a e xd d~~e

a-^e=2xee—22 xyy—2 ex22—eyx 22— y

/^ -I- tf =4 X <i i»— 44 X ^ ^— 4^ X 4 4— <i ^ X 44— .^

a-^-e^iexy e-^yexyy — e

e+y^y — ayxaa^yaxy xy-^-aszaxea—eax^e—a

Cusfis secunda.

a-^-e^bxe bxy

e-\-y=:bxy bxa

s y -\-a=bxa bxe

a-\-eT=bxe bxy

e-i-y^zcxy cxa

dy '\-az=zd xadx e

if + ^=2 xe 2xy ^+7^3x73Xtf

fy-Yaz=i^xa^X e

a-\-e = e xy

e-\-y=:yxa

X y^a=zaxe

Cuspis tertia.

a-\-e a''ez=exbyxb se-^ye-k-y^yxbaxb y-\-a y^a^a xb exb VOL. III. F

66 Miscellanea Mathematical

d e—y e-\-y^yxc axe y-\^ay—a^axd exd

a-\^e a — e=zex2yx 2

fe^y e+y=yx3ax3

y-\-ay^a=zax4 ex 4

a-^-e a—ezr^exy

X e^y e-\-yz=yxa

y^a y—a^axe

Cusph quart a.

a — e:=zbxe bxy

5 e—y=bxy bxa

y^az=^bxa bxe

a—e^bxe bxy

d e-^y^c xycxa

y^a=.dxad xe

tf— f = 2 xe2 xy

X e—y-szyxa y-^a=^axe

Cuspis quinta.

a-'ez=.exb b^e yxb b^y bxe e-r-b bxy y-r-b

s e ^y=y~i~b b xy a^b bxa b-r-y yxb b-^a axb

y — a-z^a xb b-r-a exb b-^e b xa a-^b bxe e-i-b

a^e^exbb-r-e yxb b^y bxe e~--b bxy y-^b

d e—y=.y^c cxy a^c c-~a c-:r'y yxc c-^^a axe

y^ a=za xdd-r-a e xd d-^ed xa a-^^dd xe e-r-d

Appendix. 67

i»— *=ex2 2-5-^/ X a 2-*-/ %y,e>e-r'b zxyy-i-^ f ^-y^y-^i 3 ^y ^-*-3 3 X ^ 3-^7 J' X r 3-^tf tf X 3

y^az^a x 4 4-4-<t e x 4 4-r-^ 4 X aa^d^ x e ^-*-4 a—e^exy e-*-/ ^xy y-^e

X e-^y^zy^a y X a a^y a xy y-^a^zaxe a-r-eax^ e-i-a

Cuspis sexta.

a^e=6~i~e h-^y e^^h y^^h

y — a^h-^a h-^e a-^h e-r-lf

a^e^h-^e h-^y e-^h y-^h

d e — y=zc~i~y c-^ay-i-c a-^c

y -^a^d-^a d-if-e a-i-d e-jjrd

tf — ^=2-?-^ 2-4-/^-5-2/ -^- 2 /^-/ = 3-e-/3-4-^/H-3tf-r-3 7 — tf=4-i-tf4-T-^tf^ 4^-5-4

a^e^-e-r-y y-r-e X e—y—y-r-aa^y

/ — if = if -a-f e-r- tf

Cuspis septtma,

tf — e iix* = ^-4-^ b^e y^b b^y

sexy e—y^y^b b^y a^b b-r-a

y-^ay xa^a-i-b b-i-a e-i-b b-^e

a — e axez=e^b b-i-e y-r-b b^y

d exy e^y=zy^c c^y a-r-c c-^a

y — ay xar=^a-^dd.^a e-^d d-r-e

a—e ax e=^-!-2 2-4-^7-j-2 2-T-7

f e \ y e--y=y^^^^ya^7,^^a

J —ay xi3r=i»-r-4 4-T-i»<*~4 4-»-r

K 2

68 Miscellanea Mathematica.

X exy e-~y=y-^a a-^y

Cusfis octava. a X esse-i^S h^t y-^^h h^y

y X a=za^h b-^a e-^b b^e

a X e==e^b b-^e y^b b^y

d e xy^y-^c c^y a^c c^a

y X a^a^r-dd^i-a e-i~dd^i^e

fexy=^y-^33^ya^33-^a

y X a=.a^^^^a ^-5-44-*-^

axez=ie~rry y-^e

X exy^y-i-aa-i-y

yxa^a^ee-i-a

Cuspis nana.

axe=ib^e e^b b-\-y y-^b e-^b b-t-ey-^-b b^y

s exy^b-^-y y-\-b b^aM-\'b y^b b-^-y s^b b-^-M

yxa = b'\'aa-^b b-^e e^ba-^b b^m e-]-bb^e

axe=zb-\-e e^b b-^y y-i^b e-^-b b^e y^b b^y

d exy^e-i-y y-\-c c^a a-\-c y-^c c-^-y a~^c c-^-a

yxa^zd-^aa^i-ddi-e e-^d a-^-d d~i~ae'\'dd~i~e

tf X^ = 2 + ^^-^-2 2 +7 7-*-2 ^-1-2 2h-^/+2 2^y

;fXi»=4 + tftf-4-44+f *-i-4if + 44-i-i»*-|-4 4-T-^

axes^e-^y e-^y e-\-y y^e

X exy^y-^^ay^aa^ya-^y

y X ^=a -}- e a-r-e a -j- e e-^a

Appendix, 69

Cuspts decima.

s e x/=j' + ^# + *

J exy=y'\-c a-\-c yxMz=m-\-de'\'d

axe=ze-\-2y-\.2

yxa^a^^e + 4,

axe=ze+y

X exy-^y-^a

yxa=zM-\-e

Cusfis undecima*

axe a^esze-^-^y-^lf

s e^y exy^y-i-^ a + ^

yxay^a = a-f^ e-\-h

axe a^ez=e^t y'\-h

d e^y exy=y'\-ca+c

yxay'r'a:=a-{-de'^d

axe a-r-e = e+ 2 y-\- 2

fe^y exy=y + 3a-\-3

yxay^a=a-\-4, e + 4

axe a~i~e=e'\'y

X e^y exy=y'\'a

yxay-r-a^za-^-e

Cuspts duodecima.

s e^r'y^b-\-y b-^ a y^a^h-^-a b-\- e

70 Miscellanea Mathematical

Je^yz=c+y c-^a y-^az=.d-^a #/+ e

a^e = e-\-y

X e^y=y+a

y-^az^a-^-e

Cuff is decima tertia.

a~r'^=ze + ^ i—ey-^-i t—y S + e e-^t t-\-y y-^b

s e-^yzizy^t t^y a—h h-^-ah^y y-\-b b-^a m^b

y-i-azsa-^-b b^a e-^ b b^e b -{-MS^b b-\-e e^b

a^i-e^e-^-b b^e y-\^b b^y b-^-e e^b b'\-y y^b

d e-r-y^y—c c -^y a^c c + a c^y y^c c^m <«+ c

y-i^a^rza-^-dd^a e-^-d d^e d-\-a a—dd-^e e^d

a-^e^ze^a 2— e^ + a 2—^2 + ^ f — 2 2+77—2

*^^J'=J'~3 3+/^-3 3+^ 3-/7 + 3 3-^^+3 7-4-i»=i» + 44— iff + 44-^4 + ^4— 44+^ f— 4

a^e^ze-^-y e^y e-\-y y^e

X e^yz=iy'-ay-\.aa'-y a^y

y-i-aziza+e a—e a-\-e e^a

Cuspss decima quarta,

a-r-ez^b-^e b—y e — by-^b

s e^y^zb—y b—ay—b a^b

y-r-azzib-^a b^e a-^be — b

a-^ezzb^e b—y e—b y-^b d e^^yzzic—y c-^a y^c a — c y-r-azzd—a d—e a-^d e-^d

Appefuiix, 7 1

7-T-i» = 4— tf 4 — ^ ii— 4 ^—4

Cuspis decima qusnta,

a^e a-^re^e^h y-^b b-^e b^y

5 e-^y e~r-y=zy — ba — b b^y b-^a

y-T-ay + a^a^b e—b b — ab — e

a^e a-\-e^e — b y^b b^e b—y ^ ^-\-y e^i^yzsiy'^c a^c c^y c—a

a^e <2 + ^ = ^ — 2/— 2 2 — f 2—/

/''+7 '-^7=7-3^-33-/3-^ /-f-tf^ 4-^=^—4 ^—44—^4-/1

X f +^ e-i-yzrzy^aa^y

Cuspis decima sexta,

a-\-e^=ze^b y — b b^e b—y

s e-\-yr=zy — b a — b b—y b—a

y-\-a = a — b e — b b — a b— e

a-^e =:e — b y-^b b — e b—y d e -^-y-^y — c a—c c—y c—a y-\-a=.a — de — dd'^ad— e

a-\-e=ze — 2y — 2 2—e 2—^

fe^y=y-^a-i^-y 3-a

^ + tf = tf — 4 f — 44 — 4r4--f

72 Miscellanea Mathematica.

X 9-\-yz=,y'-.Ma^y y^a^^a-^e e^m

N,B. Est et alia varietas in prioribus aequationum membri ubi signum analyticum reperitur, viz. si Species transponamu

E.g. in cuspide quarta adhibitis "^J^ — 'X ^^ duodecima -{/-»-^

duplicabuntur quaestiones.

[I Ne quis forte putet quaestiones omnes in ludo nostro possibite a Tabulis exhiberi, notandum est illas revera esse innumeras. Nan metae infinities variari poterunt: ex his vero pendet numeruf quantitatum in quovis problemate quassitanim, qui proinde pre metarum diversitate erit infinite variabilis^ unde quaestiones ori- cntur innumeras, in quarum tamen singulis non alias senrands sunt methodi pro signis^ combinationibus^ et speciebus determi- nandis, quam quas in solis quasstionibus imparls cujusvis pnetei unitatem numeri quantitatum quaesitarum atque adeo in Tabulis quas apposuimus exhibeantur.]

* Not contained in the 1 707 edition.

DE

MOTU:

SIVE

DE MOTUS PRINCIPIO ET NATURA, ET DE CAUSA COMMUNICATIONIS MOTUUM.

1721.

DE MOTU\

\d veritatem inveniendam prxcipuum est cavisse ne voces intellectx'^ nobis officiant: quod omnes fere monent philo- pauci observant. Quanquam id quidem baud adeo difficile r, in rebus praesertim physicis tractandis^ ubi locum habent , experientia, et ratiocinium geometricum. Seposito igitur, im licet, omni prasjudicio, tarn a loquendi consuetudine

5 curious and yaloable Tract is 's first appearance as an author s famous Dialogues on Matter, 1 in 1 713. It is said to have been It Lyons in 1720, on his way home ly, and to have been sent to the cademy of Sciences at Paris. The was proposed by that Society in ir a prize. The Memoir which he prize was written by Crousaz, - of Philosophy and Mathematics inne, and author of the well-known of Logic. That Berkeley should led in the competition need not us, when we consider the boldness ch, in this tract, he attacks received and assails ontological assumptions 'e still powerful. It was published >n in 1721, soon after its author's ■) England, and republished by him iscellany in 1752.

De Motu contains a Theory of n — efficient and physical. Abstrac- j misuse of language are accused of Qg our notion of Power. The b referred to our consciousness nal activity or volition — the only in our experience of a Cause so called. Throughout this Tract, reiterates and applies the favourite of his Principles and Dialogues — ivity belongs to minds or persons riy ; and that the supposed relation lion among sensible things is really >ign and Signification, not of Cause .'ct. The movements of the ma- orld are thus a natural language, ng in, and inspired by, Supreme

Tract opens with two introductory

sections, on the abuse of abstract terms in ancient and modem philosophy— especially those which refer to Motion, and Power or Force. What follows may be subdivided into three parts : —

(i) Sect. 3 — 4a investigate the Prindple or Efficient Cause (prineipium) of motion, and offer proof that this is and must be Mind. The vulgar hypothesis, that power is in- herent in body, is attributed to the confusion introduced into our conceptions by meta- physical abstractions. When we duly attend to our experience, we find that sensible things are essentially passive, and that terms significant of causality have no meaning as applied to them. Spirit is the sole origin of motion. * Af<m5 agitat moUm.* The evidence of experience is alleged for this.

(2) The Nature or Definition of motion is the theme of sect. 43 — 66. Motion itself is something clearly enough perceived by the senses. It has been obscured by the con- jectures of philosophers. Some examples of obscure definitions of motion are given. Power or force, the cause of motion, it is argued, on the other hand, transcends the senses and appertains to Mind. Motions themselves, the effects of this spiritual ef- ficiency, are alone objects of sense, and their laws form physical science. Motion is simply the successive appearances of sensible things in a variety of places. And place is relative, not absolute. Absolute place and absolute space are alike absurd. They are both due to confusion of thought, generated by abstraction. All place and all space must be relative, and given in the concrete experience of the senses.

(3) The cause of the Communication of

76

De Motu.

quam a phiJosophorum auctoritate nato, ipsa renim natura dili- genter inspicienda. Neque enim cujusquam auctoritatem usque adeo valere oportet, ut verba ejus et voces in pretio sint, dummodo nihil clari et certi iis subesse comperiatur.

2. Motus contemplatio mire torsit veterum^ philo60{Aoniin mentes, unde natae sunt varias opiniones supra modum difiEidks^ ne dicam absurdas, quas, qjuum jam fere in desuetudinem abieiint, baud merentur ut iis discutiendis nimio studio immoremur. Apod recentiores autem et saniores hujus acvi ^ philosof^os, ubi de Nfeta agitur, vocabula baud pauca abstracts nimium et obscurac signifi- cationis occurrunt, cujusmodi sunt soUcttatso p'dvifatisy coMsSm^ vires mortu^j &c. quas scriptis alioqui doctissimis tenebras offmi- dunt, sententiisque non minus a vero quam a sensu hominom communi abhorrentibus ortum prxbent. Haec vero necesse est ut, veritatis gratia, non alios refellendi studio, accurate disoH tiantur.

3. Solicitatio et nisus, sive conatus, rebus solummodo animatis^ revera competunt. Cum aliis rebus tribuuntur, sensu metaj^iorioo accipiantur necesse est. A metaphoris autem abstinendum philo- sopho. Porro, seclusa omni tam animae affectione quam corporis motione, nihil clari ac distincti iis vocibus significari^ cuilibet constabit qui modo rem serio perpenderit.

motion among bodies is considered in the remaining sections of the Essay. The only true or doicient cause of motion and of its difiiision, as was argued in the preceding sections, is the Intelligence to which all cor- poreal change is due. But the rules according to which It takes place — the Laws of motion — may be termed its nuebanieal or apparent cause. If we remember that physical causation means only the order according to which sensible dianges take place — that it is only Natural Language — we may speak of motions transmitted by mechanical causes. But Causation properly so called belongs to metaphysics or primary philosophy, and active causes can be drawn forth from the shades in which they are inyolred only by contemplation and rea- soning.

The De Motu should be compared, on the one hand, with Berkeley's Principlis and Dialogues on Matter, and, on the other hand, with SiriSt being itself intermediate

between his earlier and later phikMopliy. See also Ahipbron, Dial.- VII. sect. 4.

* * voces male intellects.' Cf. Prme^ of Human KnowUdg€, ' Intiodiictioo«' sect 6, 33—25.

' 'veterum.' The history of •ndenl speculations about motion, from the pan- dozes of Zeno downwards, is, in some tort, a history of ancient metaphjrsict. The problem involves the nature of Space, Timc^ and Sensible Things, and their ultimate le- lation to Mind.

* * hujus sBvi.' See Bacon, De Mohtj and the questions raised by Newton, Letboitx, and others, alluded to in the foUowiog sections.

* * animatis.' Sect. 3^ — ^2 refer the various terms in Unguage which signify the cause of motion exclusively to animated beins;, mind, or spirit, and interpiet mett- phoncally their application to •«^*- things.

De Motu.

n

4« Quamdiu corpora gravia a nobis sustinentur, sentimus in aobismet ipsis nisum, fatigationem, et molestiam. Percipimus tdam in gravibus cadentibus motum acceleratum versus centrum telluris; ope sensuum prxterea nihil. Ratione tamen colligitur caosam esse aliquam vel principiuih horum phaenomenon ^ illud antem grmtas vulgo nuncupatur. Quoniam vero causa descensus gravium caeca sit et incognita, gravitas ea acceptione proprie dici oequit qualitas sensibilis^ est igitur qualitas occulta. Sed vix, et ne vix quidem, concipere licet quid sit qualitas occulta, aut qua latioQe qualitas ulla agere aut operari quidquam possit. Melius itaque fbret, si, missa qualitate occulta, homines attenderent sdummodo ad effectus sensibiles^ vocibusque abstractis (quan- tumvisillae ad disserendum utiles sint) in meditatione omissis, mens in particulahbus et concretis, hoc est in ipsis rebus, defigeretur.

5. Vis^ similiter corporibus tribuitur: usurpatur autem vocabulum illud, tanquam significaret qualitatem cognitam, distinctamque tarn a motu, figura, omnique alia re sensibili, quam ab omni animali afiectione: id vero nihil aliud esse quam qualitatem occultam, rem acrius rimanti constabit. Nisus animalis et motus corporeus vulgo spectantur tanquam symptomata et mensurae hujus qualitatis occulta^.

6. Patet igitur gravitatem aut vim frustra poni pro principio^ motus : nunquid enim principium illud clarius cognosci potest ex eo quod dicatur qualitas occulta? Qucxi ipsum occultum est, nihil explicat : ut omittamus causam agentem incognitam rectius did posse substantiam quam qualitatem. Porro w, gravitas^ et istiusmodi voces, saepius, nee inepte, in concrete usurpantur, ita ut connotent corpus motum, difficultatem resistendi, &c. Ubi vero a philosophis adhibentur ad significandas naturas quasdam ab hisce onmibus praecisas et abstractas, qux nee sensibus subji- ciuntur, nee ulla mentis vi intelligi nee imaginatione effingi* possunt, tum demum errores et confusionem pariunt.

• •ris.* This tnpposed power or force in matter is the great example offered of the abase of abstractions. It seems easily con- odrable, tmt reflection dissolves the apparent meaning. Cf. Sirit, sect. 848.

"* * principio ' — the ultimate explanation or effidcnt cause. Cf. sect. 36. Mere metaphors, due to metaphysical abstraction, are accepted for explanations, it is argued,

when • gravity/ * bodily power/ or * force/ are taken as causes of motion. To call these * occult causes' is to say nothing. The sensible effects and their laws are all we know through sense. Mere physicists are still deluded by these metaphors.

" Cf. sect. 53, where sen$e, inielligence, and imagination are distinguished from one another.

78

De Motu.

7. Multos autem in errorem ducit, quod voces generales ct abstractas in disserendo utiles ^ esse videant, nee tamen eamm vim satis capiant. Partim vero a consuetudine vulgari inventc sunt illae ad sermonem abbreviandum, partim a philosophis ad docendum excogitatas ; non quod ad naturas rerum aceomniodatB sint, quae quidem singulares et concretae existunt, sed quod idones ad tradendas disciplinas, propterea quod faciant notiones, vel saltern propositiones, universales.

8. Vim corpoream esse aliquid conceptu facile plenimqu^ existimamus. li tamen qui rem accuratius inspexerunt in divefsa sunt opinione, uti apparet ex mira verborum obscuritate qua laborant ubi illam explicare conantur. Torricellius ait vim ct impetum esse res quasdam abstractas subtilesque et quintessentias^ quae includuntur in substantia corporea, tanquam in vase magioo Circes 10. Leibnitius item in naturae vi explicanda haec habet^ Vis activuy frtmitiva^ qua est Ivr^Xiy^ia tsp&rri^ anima vel forma ii(- stantiali respmdet. Vide Acta EruMt. Ups. Usque adeo necesse est ut vel summi viri, quamdiu abstractionibus indulgent, voces nulla certa significatione praeditas, et meras scholasticorum umbras sectentur. Alia ex neotericorum scriptis, nee pauca quidem ea, producere liceret, quibus abunde, constaret, metaphysicas abstrac- tiones non usquequaque cessisse mechanicae et experimentis, sed negotium inane philosophis etiamnum facessere.

9. Ex illo fonte derivantur varia absurda, cujus generis est illud— vim percussionisy utcunque exigua^ esse infinite magnam, Qupd sane supponit, gravitatem esse qualitatem quendam realem ab aliis omnibus diversam ^ et gravitationem esse quasi actum hujus quali- tatis a motu realiter distinctum: minima autem percussio pro- ducit efFectum majorem quam maxima gravitatio sine motu; ilia scilicet motum aliquem edit, haec nullum. Unde sequitur, vim percussionis ratione infinita excedcre vim gravitation is, hoc est, esse infinite magnam^^ Videantur experimenta Galilaei, et qu« de definita vi percussionis scripserunt Torricellius, Borellus, et alii.

• • utiles.* Cf. Alcipbnm, Dial. VII. sect. 8.17.

^^ [La Materia altro non e che on vaso di Circe incantato, il quale serve per ricettacolo della forza et de momenti dell' impeto. La forza I'impeti sono astratti tauto sottili, sono quintessenze tanto spiritose, che in altre ampolle non si possono rachiudere, fuor che

neir intima corpulenza de solidi naturtlt. Vide Lezioni Aeademicbe,'] — ^Author. This comparison of Matter to the enchanted vise of Circe, illustrates a mass of crude ph]rsical speculation. With this and the six follow- ing sections cf. Alciphron^ Dial. VII.

" See Borellus, Di Vi Pereussioms, cap. XXIV. prop. 88, and cap. XX^'11.

De Motu.

79

' 10. Veruntamen fatendum est vim nullam per se immediate sentiii, neque alitcr quam per eflPectum^^ cognosci et mensurari. Sed vis mortux, seu gravitationis simplicis, in corpore quiescente sobjecto, nulla facta mutatione, efFectus nullus est; percussionis autem, efiectus aliquis. Quoniam, ergo, vires sunt e£Fectibus prop(»tionales, concludere licet vim mortuam ^^ esse nullanu Neque tamen propterea vim percussionis esse infinitam : non enim oportet quantitatem uUam positivam habere pro infinita, propterea quod ratione infinita superet quantitatem nullam sive nihil.

II. Vis gravitationis a momento secemi nequit; momentum

autem sine celeritate nullum est, quum sit moles in celeritatem

ducta: porro celeritas sine motu intelligi non potest; ergo nee

vis gravitationis. Deinde vis nulla nisi per actionem innotescit,

et per eandem mensuratur; actionem autem corporis a motu

praescindere non possumus; ergo quamdiu corpus grave plumbi

subjecti vel chordae figuram mutat, tamdiu movetur; ubi vero

quiesdt, nihil agit, vel, quod idem est, agere prohibetur. Breviter,

▼oces istx vis mortua et gravitatioy etsi per abstractionem meta-

physicam aliquid significare supponuntur diversum a movente,

moto, motu et quiete, revera tamen id totum nihil est.

12. Siquis diceret pondus appensum vel impositum agere in diordam, quoniam impedit quominus se restituat vi elastica : dico, pari ratione corpus quodvis inferum agere in superius incumbens, quoniam illud descendere prohibet : dici vero non potest actio cor- poris, quod prohibeat aliud corpus existere in eo loco quem occupat.

13. Pressionem corporis gravitantis quandoque sentimus. Verum sensio ista molesta oritur ex motu corporis istius gravis fibris ner- visque nostri corporis communicatio, et eorundem situm immu- tantc ; adeoque percussioni accepta referri debet. In hisce rebus myitis et gravibas prsejudiciis laboramus, sed ilia acri atque iterata meditatione subigenda sunt '^5 vel potius pcnitus averruncanda.

^ « per effectam* i. e. by its sensible effects — power or force not being, in itself, 1 dead object of the senses, but the living ^Hfitaal efficacy of which we are conscious. Of. Sins, ted, 350.

" • vim mortuam/ The only power recognised in this philosophy is the living power of Mind, perpetually active in the uiiverse, in itself imperceptible through the tenses, and revealed to them only in its sensible effects, * Dead power* and • gravi-

tation' are distinguished from the sensible effects called * rest' and ' motion' only through an illusion due to metaphysical abstraction. There is no physical power intermediate be- tween spiritual agency, on the one hand, and the order we observe in the sensible effects of that agency, on the other. Cf. sect. II.

** • meditatione subigenda sunt.* Cf. Theory of Vision Vindicated, sect. 35, 70.

8o

De Motu.

14. Quo probetur quantitatem ullam esse infinitam, ostendi oportet partem aliquam finitam homogeneam in ea infinities coa- tineri. Sed vis mortua se habet ad vim percussionis^ non ut pais ad totum, sed ut punctum ad lineam, juxta ipsos vis infinitie per- cussionis auctores. Multa in hanc rem adjicere liceret^ sed vexeor ne prolixus sim.

[5. Ex principiis prsemissis lites insignes solvi possunt, qme viros doctos multum exercuerunt. Hujus rei exemplum sit oon- troversia ilia de proportione virium. Una pars dum concedit, momenta, motus, impetus, data mole, esse simpliciter ut veloci- tates, afiirmat vires esse ut quadrata velocitatum. Hanc autem sententiam supponere vim corporis distingui >^ a momento, motu, ct impetu ; eaque suppositione sublata corruere, nemo non videt

16. Quo clarius adhuc appareat, confusionem quandam miram per abstractiones metaphysicas in doctrinam de motu introductam esse, videamus quantum intersit inter notiones virorum celebrium de vi et impetu. Leibnitius impetum cum motu confundit. Juxta Newtonum'6 impetus revera idem est cum vi inertias. Borellus'^ asserit impetum non aliud esse quam gradum velodtatis. Alii impetum et conatum inter se difierre, alii non difierre volunt Plerique vim motricem motui proportionalem intelligunt. Non- nulli aliam aliquam vim prxter motricem, et diversimode mcn- surandam, utpote per quadrata velocitatum in moles, intelligere prsB se ferunt. Sed infinitum esset haec prosequi.

17. Vis^ gravitas^ attractto^ et hujusmodi voces, utiles >» sunt ad ratiocinia et computationes de motu et corporibus motis ; sed non ad intelligendam simplicem ipsius motus naturam, vel ad quali- tates totidem distinctas designandas. Attractionem certe quod attinet, patet illam ab Newtono adhiberi, non tanquam qualitatem veram et physicam, sed solummodo ut hypothesin mathematicam^'.

" • distingui.' It is here argued that so- called power in the material world is in- dittinguishable from the sensible pheno- mena or effects — in short, that there is no real power in matter. To the inconceivable supposition that there is, he attributes the differences among the learned mentioned in the following section. The province of natural philosophy, according to Berkeley, is to inquire what the general rules are according to which sensible effects are manifested. Of. Siri$^ sect. 236, 247, 249.

»• Principia Matb, Def. III.

*' De Vi Pereusiionit^ cap. I.

" 'utiles.' Berkeley's doctrine of the function of common terms as useful instru- ments for calculation, even when not signi* ficant of what he calls id»<u, is illustrated in Alciphrofif Dial. VII. So here such words as • force,' * power,* * gravity,* * attractioo,' arc allowed to be convenient in physical reasonings about the laws of motion, but worthless as philosophical expressions of the cau$€ of motion, which transcends Cf. Sirii, sect. 234, 235.

" Cf. sect. 67.

De Motu, 8 1

Qmnetiam Leibnitius, nisum elementarem seu solicitationem ab imprtu distinguens, fatetur ilia entia non re ipsa inveniri in rerum natiara, sed abstractlone facienda esse.

i8. Similis ratio est compositionis et resolutionis virium qua-» nimcunque directarum in quascunque obliquas, per diagonaleno et latera parallelogrammi. Hxc mechanicas et computationi in* serviunt : sed aliuJ est computationi et demonstrationibus mathej- inaticis inservire, aliud rerum naturam exhibere.

19. Ex recentioribus multi sunt in ea opinione, ut putent motum neque destrui nee de novo gigni, sed eandem^o semper motiis quantitatem permanere. Aristoteles etiam dubium illud olim proposuit — utrum motus factus sit et corruptus, an vero ab xtemo? Fhys. lib. xiii. Quod vero motus sensibilis pereat, patet scDsibus: illi autem eundem, impetum, nisum, aut summam virium eandem manere velle videntur. Unde affirmat Borellus, ▼im in percussione non imminui, sed expandij impetus etiam contrarios suscipi et retineri in eodem corpore. Item Leibnitius nisum ubique et semper esse in materia, et ubi non patet sensibus, ratione intslligi contendit. — Hacc autem nimis abstracta esse et obscura, ejusdemque fere generis cum formis substantialibus et cntelechiis, fatendum.

20. Quotquot, ad explicandum motus causam atque originen^ vel principio hylarchico, vel naturae indigentia, vel appetitu, aut denique instinctu naturali utuntur, dixisse aliquid potius quam cogitasse censendi sunt. Neque ab hisce multum absunt qui sup- posuerint^', partes terrje esse se movent es^ aut etiam spirit us iis iwtplantatos ad instar form e^ ut assignent causam acceleration is gravium cadentium: aut qui dixerit--, "in corpore praeter solidam cxtensioncm debere etiam poni aliquid unde virium consideratip oriatur." Siquidem hi omnes vel nihil particulare et determi- natum cnuntiant; vel, si quid sit, tam difficile erit illud explicare, quam id ipsum cujus explicandi causa adducitur^^.

21. Frustra ad naturam illustrandum abhibentur ea quae nee

• 'einclefiu' So in recent discussions on a real causality in the material world afc

the coosenration of force. merely a cover for mcaninglessiiess. Only

■ [Borellos.]— Author. See De Vi Per- through our experience of personal activity

cap XXIII. does meaning enter into this portion of

• [Lcibatttas.] — Author. language. This is argued in detail in lect.

■ On Berkeley's reasoning in the text, ai — 35. all tcnnt which hivolve the assumption of

VOL. III. G

De Moiu.

sensibus patent, oec ratione intelligi possunt, Videndum ergo quid sensus, quid experientia, quid demum ratio iis innixa siiadcat. Duo sunt summa renim genera — corpus ct an'ima. Rem extcnsam, solidam, mobilem, figuratam, aliisque qualitatibus quje sensibus occurrunt prKditam, ope sensuum ; rem vero sentieotem, perci- pientem, intelligentem, conscicntia quadam interna cognoviraus, Porro, res istas pSane inter se diversas'* esse, longequc hetero- geneas''^, cernimus. Loquor autem de rebus cognitis: dc in- cognitis enim disserere nil juvat.

aa. Totum id quodnovimus, cui nomen torptis indidimus, nihil) in se continet quod motus principium seu causa efficiens

possit ; — etcnim impenctrabilitas, cxtensio, figura nullum includunt vel connotant |X)tt;ntiam producendi motum ; quinimo e contrario non modo illas, verum etiam alias, quotquot sint, corporis, quali- tateS sigillatim percurrentcs, vidcbimus omnes esse rcvera passivas, nihilque iis activum inesse, quod ullo modo intelligi possit tan- quam fons et principium motus. Gravitatem quod altinet, voce ilia nihil cognitum et ab ipso efFectu sensibili, cujus causa quxritur, diversum significari jam ante ostendimus. Et sane quando corpus grave dicimus, nihil aliud intelligimus, nisi quod feratur deorsi de causa hujus effectus sensibilis nihil omnino cogitantes,

23. De corpore itaque audacter pronunciare licet, utpoti comperta, quod non sit principium motus. Quod si quisquan^ prater solidam extensioncm cjusque modificationcs, vocem torpus qualitatem etiam occultam^ virtutem, formam, esscnliam com- plecti sua significationc contcndat; licet quidem illi inutili ncgotio sine idcis disputare, et nominibus nihil distincte exprimcntibus abuti. Cieterum sanior philosophandi ratio videtur ab notionibus abstractis et generaiibus (si modo notiones dici debent quse tclligi nequeunt) quantum fieri potest abstinuisse.

lihUia \ essff

deorsun^ :>te de j9

■divtrut — ' helefogcriMS,' Cf. Prin- I 0/ Human Snaalidgi, lect, 1, a, id h 0111 knowledge ii ciitsiGed by rcfci-

1

nMntial pikaWity of ideas giveo id the mitenil world, is niaiiiuincd 11 1 nr- diiul principle nfthe Berkeleian philoioph; — on the positive ground of our eupciisKcof whit Miiiible Ihingt are. CF. Sirit. tect IjJ. 391. On the other hand, to ipcak of 'the origin of motion tt lamilbiHg in maiUr' ii. he irjoa {lert. 14), merely to oy thi! we know nothing ibont the origin of motiou. Cf. wcl. iR. 19. infn.

J

De Motu. 83

24. Quicquid continetur in idea corporis novimus; quod vero Dovimus in corpore, id non esse principium motus constat. Qui praeterea aliquid incognitum in corpore, cujus ideam nullam habent, comminiscuntur, quod motus principium dicant, ii revera nihil aliud quam principium motus esse incognitum dicunt. Sed hojusmodi subtilitatibus diutius immorari piget.

25. Prarter res corporeas, alterum est genus rerum eogttantium'^\ In iis autem potentiam inesse corpora movendi, propria expe- rientia didicimus^ quandoquidem anima nostra pro lubitu possit dere et sistere membrorum motus, quacunque tandem ratione id fiat. Hoc certc constat, corpora moveri ad nutum animx, eamque pioinde baud inepte dici posse principium motus; particulare quidem et subordinatum, quodque ipsum dependeat a primo et aniversali Principio^^,

26. Corpora gravia feruntur deorsum, etsi nullo impulsu

apparente agitata; non tamen existimandum propterea in iis

cnntineri principium motus: cujus rei banc rationcm assignat

Aristoteles'^s : ^^ Gravia et levia (inqiiit), non moventur a seipsis;

id enim vitale esset, et se sistere possent/' Gravia omnia una

cademque certa et constanti lege centrum telluris petunt, neque

in ipsis animadvertitur principium vel facultas ulla motum istum

sistendi, minuendi, vel, nisi pro rata proportionc, augendi, aut

denique uUo modo immutandi: habent adeo se passive. Porro

idem, stricte et accurate loquendo, dicendum de corporibus per-

cussivis. Corpora ista quamdiu moventur, ut et in ipso percus-

sionis momento, si gerunt passive, perinde scilicet atque cum

quiescunt. Corpus iners tam agit quam corpus motum^ si res ad

venim exigatur : id quod agnoscit Newtonus^^^ ubi ait, vim inertias

ease eandem cum impetu. Corpus autem iners et quietum nihil

agit, ergo nee motum.

27. Revera corpus seque perseverat in utrovis statu, vel motus vel quietis. Ista vero perseverantia non magis dicenda est actio corporis, quam existentia ejusdem actio diceretur. Perseverantia

* 'genos renun cogitantium.* Cf. Prin- * *t primo ct uniyewali Prindpio* — I.e.

dflm 9f Human Knoudedge, sect. 1, in the Divine Spirit, on whom human and

vliich & Ego^ or Self it distinguished from other finite spirits depend, in a way which

'ideas* or tbe mm'Ego, For Berkeley's Berkeley does not attempt to rationalize.

tefefocc oi all the power in the universe * Cf. Sim, sect. 280.

czdutively to spirits, cf. Prineipin, sect. * PHncip. Matb. Def. HI. 16—33 ; alto sect. 30 infra.

G 2

De Motu.

nihil aliud est quam continuatio in eodem modo existendi, quae propric dici actio non potest. Caterum resistenliam, quam ex- periniur in sistendo corpore moto, ejus actionem esse fingimus vana specie delusi. Revera enim ista resistentia'" quam sentimus, passio est in nobis, neque arguit corpus agere, sed nos pati: con- stat utiquc nos idem passuros fuiss^, sivc corpus illud a se move- atur, sive ab alio principio impellatur,

28. Actio ct reactio dicuntur esse in corporibus: nee incom- mode i' ad dcmonstrationes mechanicas. Sed cavendum, ne prop- terea supponamus virtutem aliquam realem, qua; motus causa sive principium sit, esse in iis. Etcnim voces illae eodem modo intelli- genda: sunt ac vox attractio; et qucmadmodum haec est hypothesis solummodo mathematica >», non autem qualitas physica: idem etiam de illis intelligi debet, et ob eandem rationem. Nam sicut Veritas et usus theorematum de mutua corporum attractione in philosopSiia mechanica stabiles manent, utpote unicc fiindati in motu cor- porum, sive motus iste causari supponatur per action?m corporum se mutuo attrahentium, sive per actionem agentis alicujus a corporibus diversi impcikntis et moderantis corpora; pari rationc, quaecunque tradita sunt de regu'is et legibus motuum, simul ae theoremata indc deducta, manent inconcussa, dum modo conce- dantur cft^'ectus sensibilis, et ratiocinia Jis innixa; sive supponamus actionem ipsam, aut vim horiim effectuum causatricem, esse in corpore, sive in agente incorporeo.

29. Auferantur ex idea corporis cxtcnsio, suliditas, figura, remanebit, nihil". Sed qualitatcs ist« sunt ad motum iodiff-'ercntcs, nee in se quidquam habenl quod motus principium dici possit. Hoc ex ipsis ideis nostris pcrspicuum est. Si igitur voce corfmt significatur id quod concipimus, plane constat inde non peti posse principium motus; pars scilicet nulla aut attribulum illius causa cfficicns vera est, quae motum producal. Vocem autem proferre, ct nihil conciperc, id demum indignum esset philosopho.

■he Printifiti and DitJeguB

ancc i> vulgarlj' oHeied >i in ilidf eridence or the independent ejiiilma of miteiial SubtUnce and Caiue, illhough it ii teilly » lenution depeudet " " " "

" ■ incommode.' Cf. letl. 1 7,

" ■hypothciit nuihcmaiici.' '7. 35. j6— 41, fifi, 67 ; alio I

Thi.

~m-

C(. !

n,y nf MatI

lion luiiit up Bctkeley'i objectioi itig mar'rr witb real power. It may b* compted and hannoniEed with Hume, Di, Thomas BtDwn, and Mr. J. S. Mill so Cauialion. Bui Beikelej diffen rram lb« poiitite philosophm in recogr* efficiency In the spiritual world.

De Motu,

8:;

30. Datur res cogitans, activa, quam principium motus esse in nobis experimur. Hanc anlmam^ mentitn^ sfiritum dicimus. Datur etiam res extensa, iners, impenetrabilis, mobilis, quas a priori toto coelo diflFert, novumque genus -^^ oonstituit. Quantum intersit inter res cogitantes et extensas, primus omnium deprehendens Anaxagoras,vir longe sapientissimus, asserebat mentem nihil habere cum corporibus commune, id quod constat ex primo librq Aristo- telis I>€ Anhna^^, Ex neotericis idem optime animadvertit Cartesius'^^ Ab eo alii =^7 rem satis claram vocibus obscuris impeditam ac difScilem reddiderunt.

31. Ex dictis manifestum est eos qui vim activam, actionem, motus principium, in corporitus revera inesse affirmant, sententiam nulla experientia fundatam amplecti, eamque terminis obscuris et generalibus adstruere, nee quid sibi velint satis intelligere. E contrario, qui mentem esse principium motus volunt, sententiam propria experientia munitam proferunt, hominumque omni aevo doctissimorum suffragiis comprobatam.

31. Primus Anaxagoras^s rhv vovv introduxit, qui motum inerti materia? imprimeret. Quam quidem sententiam probat etiam Aristotelcs''^, pluribusque confirmat, aperte pronuncians primum movens esse immobile, indivisibile, ct nullam habens magnitu- dinem. Dicere autem, omne motivum esse mobile, recte anim-' advertit idem esse ac siquis diccrct, omne aedificativum esse xdificabile, Fhystc. lib. viii. Plato insuper in Timaeo^o tradit machi- nam hanc corpoream, seu mundum visibilem, agitari et animari a mente, qux sensum omnem fugiat. Quinetiam hodie philosophi Qutesiani ^ ' principium motuum naturalium Deum agnoscunt. Et

•• • nommque genus.' Cf. sect. 3 1 . We bare here Berkeley's antithesis of mind and matter— active personality in contrast tu the passive ideas which stream through the senses. This distinction he supports by eiq>erience and authority, sect. 31, 32.

* Z)r Anima, I. ii. 13, 2a, 24.

" 'Cartesius.* The antithesis of extended and thinking Substance pervades the philo* sophy of Des Cartes. See e.g. Principia, P. I. §5 63. 64.

* • Ab eo alii.' Does he refer to Locke, who soggeats the possibility of matter thinking ?

* See Aristotle, De Anima, 1. ii. 5, 13 ; Di'>genes Laertius. lib. VI. i. 6.

*• Nat. Ausc.t VIII. 15 ; also De Anima^ III. X. 7.

*® Hardly any passage in the Tinutut exactly corresponds to this. The following is, perhaps, the most pertinent : — Kinfo-ty yhp kwfvnifitv airr^ r^¥ rov ffiufiaros olxtlav, r&p iirrdi t^k ir«pl ¥0V¥ irol ^p6tn/i(ri¥ li^Kirra oZtrav (p. 34 a). Aristotle quotes the Timans in the same connection, De Anima, I. iii. li.

*^ * philosophi Cartesiani.* Secundum Cartesium causa generalis omnium motuum et quietum est Dcus. — Derodon, Pbysica, I. ix. 30.

De Motu.

Ncwtonus^'' passim nee obscure inouit, non solummodo motumi initio a numine profectum esse, vemm adbuc systema oiundaiua ab ecxlem actu moveri. Hoc sacris Uteris consonum est: scholasticorum calculo comprobatur. Nam etsi Peripatetic! natui tradant esse prliicipiuin molus ct quietis, intcrprctantur tamcn naturam naturantem esse Deum '■>. Intelligunt nimirum corpora omnia systematis hujusce mtindani a mente ptaepotenti juxta certain et constantem rationem" moveri.

33. Cteterum qui principium vitalc corporibus tribuunt, c^scunim aliquid et rebus parum conveniens fingunt. Quid enim aliud est viiali principicj pficditam esse quam vivere? aut vivere quam se movere, sistcre, ct statum suum mutare ? Philosophi auteai hujus S9Bculi doctissimi pro principio indubitatu ponunt, omne corpus perseverarc in statu suo, vel quietus vel motus unifbrmis in directum, nisi quatenus aliunde cogitur statum ilium mutare: c contrario, in aiiima sentimus esse facultatem tarn statum suum quam aliarum rerum mutandi ; id quod propric dicitur vitale, animamque a corporibus longe discriminat.

34. Motum et quietem in corporibus recentiorcs considerant velut duos status existendi, in quorum utrovis corpus omne sua natura iners permaueret'^, nulla vi externa urgente. Undc colligerc licet, eandcm esse causam motus et quietis, quae est existentix cor- porum, Neque enim quasrenda videtur alia causa existentia; corporis successive in diversis partibus spatii, quam ilia unde derivatur existentia ejusdem corporis successiva in diversis partibus temporis. De Deo autcm Optimo Maximo rerum omnium Conditorc ct Conservatorc tractare, et qua rationc res cunctx a summo et vero Ente pendeant demoostrare, quam vis pars sit scicntix liumanae precellcntissima, spectat tamen potius ad pbilosopbiam primam scu metaphysicam ct theologiam, quam ad philosophiam naturalem "■, quK hodie fere omnis continetur in experimentis et

	phyiial imivene ii the eipreoJOQ of Ab-
er»U.	(olule Reikiu.
" ■ nMurim naturantem e«Dcum'—i.e.
God coniidaed ii the iiuminent ciuie of
the uiiJTcne. Sh St. ThDmii Aqulmi.	irn melaphTiicsm el theologixm. quam id
Optra, irol. XXll. Qiie.t.6. p. ay.
** 'ioiiti ceitiuo et conttintem rationem'	etwy D* Mom u to diitingoiih the merdir
—i.e. while Ifae chinga in initlei are	phyucil Kiences of phenomena from the
	fiiib in an originating Ciule with which
but 1 niioiul will. The very Brbitratinew	the ipiritDiI itudenti of Metaphriict lud
of th* Lingiuge of Niture ii telitiie to oi.
and from oui point of view. In iiitlf, the	M

De Motu.

87

medisuiica. Itaque cognitionem de Deo vel supponit philosophia aatoralis, vel mutuatur ab aliqua scientia superior!. Quanquam Tcrissimum sit, naturse investigationem scientiis altioribus argu- meata ^^a ad sapientiam, bonitatem, et potentiam Dei illus- tiaodam et probandam undequaque subministrare.

35* Quod hxc minus intelligantur, in causa est, cur nonnulli immerito repudient physicas principia mathematica, eo scilicet Qomine quod ilia causas rerum e£E[cientes non assignant: quum tamen revera ad physicam aut mechanicam spectet regulas'*7 solum- modo, non causas efficientes, impulsionum attractionumve, et ut vcrbo dicam, motuum leges tradere; ex iis vero positis phxno- menon particularium solutionem^^^, non autem causam e£Bcientem assignare.

36. Multum intererit^s considerasse quid proprie sit principium, et quo sensu intelligenda sit vox ilia apud philosophos. Causa qaidem vera efficiens et conservatrix rerum omnium jure optimo af^Uatur fons et principium earundem. Principia vero philoso- phic experimentalis proprie dicenda sunt fundamenta quibus ilia ionititur, seu fbntes unde derivatur, (non dico existentia, sed) cognitio rerum corporearum, sensus utique ex experientia. Simi- liter, in philosophia mechanica, principia dicenda sunt, in quibus fiindatur et continetur universa disciplina, leges illae motuum primariae, quas experimentis comprobatac, ratiocinio etiam excultx sunt et redditas universales ^9. Hx motuum leges commode dicun- tur principia, quoniam ab iis tam theoremata mechanica generalia quam particulares -ttAv (paivofiivrnv explicationes derivantur.

37. Tum niminim dici potest quidpiam explicari mechanice, cum reducitur ad ista principia simplicissima et universalissima, et per accuratum ratiocinium, cum iis consentaneum et connexum esse ostenditur. Nam inventis semel naturx legibus, deinceps

• 'regoUs' — 'folutionem.* Of. iSirw, $«ct.

** *iDiiltii]n intererit/ Having, in the prcoediag sections, contrasted sensible mo- tMMM and their efficient cause — matter and mind — physics and metaphysics, Berkeley proceeds in this and the seven following lectioos to explain what he means by origin {prmaptatm), and the two meanings (meta- ^jFsical and mechanical) of tolutio. By

the real source and origin of things he understands their efficient and preserving cause. In natiu-al philosophy the term if applied to law or order manifested in effects, and not to causes proper at all.

^ 'experimentis comprobatse, ratiocinio etiam excoltse sunt et redditse universales.* The reUtions in knowledge of the facts given in the senses to the universalizing reason are here recognized.

88 De Motu.

monstrandum est philosopho, ex constant! harum legiim obser- vatione, hoc est, ex lis principiis phasnomenon quodvis necessujt consequi : id quod est phaenomena explicare et solvere^ causamiae^ id est rationem cur fiant, assignare.

38. Mens humana gaudet scientiam suam extend ere et dilatamJ Ad hoc autem notiones et propositiones generates efibrznaodc sunt, in quibus quodam modo continentur propositiones et cog- nitiones particulares, quae turn demum intelligi creduntur cum cs primis illis continuo nexu deducuntur. Hoc geometris notissinuun est. In mechanica ctiam prxmittuntur notiones, hoc est det nitiones, et cnunciationes de motu primx et generates, ex quibus postmodum methodo mathematica conclusiones magis remotxet minus generales colliguntur. Et sicut per applicationem theore- matum geomctricorum, corporum particu'arium magnitudines mcnsurantur; ita etiam per applicationem theorematum mechaniccs universalium, syst:*matis mundani partium quarumvis motus, ct phaenomena inde pendentia^ innotescunt et determinantur: ad qucm scopum unice collineandum physico.

39. £t qucmadmodum gcometrae, discipline causa, multa com- miniscuntur, quae nee ipsi describere possunt, nee in rerum natura invcnire; simili prorsus ratione mechanicus voces quasdam abstractas ct generales adhibct, fingitque in corporibus ww, Jf- ttonem^ attractionem^ solicit atton em ^ &c. qux ad theorias et enun- ciationcs, ut ct computationes de motu apprime utiles sunt, ctiamsi in ipsa rerum veritate ct corporibus actu existentibus frustra quaercrentur, non minus quam quae a geometris per abstrac- tioncm mathematicam finguntur.

40. Revera ope scnsuum nil nisi efFectus seu qualitates sensi- bilcs, et res corporeas omnino passivas, sive in motu sint sive in quietc, percipimus : ratioque et expericntia activum nihil praeter mentcm aut animam esse suadct. Quicquid ultra fingitur, id cjusdcm generis esse cum aliis hypothesibus et abstracrionibus mathematicis existimandum: quod penitus animo infigere oportct. Hoc ni fiat, facile in obscurum scholasticorum subtilitatem, quae per tot saecula, tanquam dira quaedam pcstis, philosophiam corrupit, relabi possumus.

41. Principia mechanica legcsque motuum aut naturae univer- silcs, saeculo ultimo feliciter inventx, et subsidio geometriae tractatae et applicatx, miram lucem in philosophiam intulerunt.

De Motu.

89

Principia vero metaphysica causaeque reales efiicientes motus et cxistentiae corponun attributorumve corporeorum nuUo modo ad mccfaanicam aut experimenta pertinent; neque eis lucem dare nisi quatenus, velut praecqgnita inserviant ad limites prxfiniendo% eaque ratione ad tollendas difficultates quses- tionesque peregrinas.

42. Qui a spiritibus motus principium petunt, ii vel rem cor- porcam vcl incorpoream vcx:e sphitus intelligunt. Si rem corpoream, quantumvis tenuem, tamen redit difEcultas : si incorpoream, quan- tumvis id verum sit, attamen ad physicam non proprie pertinet. QiKxl si quis philosophiam naturalem ultra limites experimentorum et mechanicae extenderit, ita ut rerum etiam incorporearum, et inextensarum cognitionem complectatur, latior quidem ilia vocis acceptio tractationem de anima, mente, seu principio vitali ad- mittit. Cacterum commodius erit, juxta usum jam fere receptum, ita distinguere inter scientias, ut singular propriis circumscribantur cancellis, et philosophus naturalis totus sit in experimentis, legi- busque motuum, et principiis mechanicis, indeque depromptis ratiociniis ; quidquid autem de aliis rebus protulerit, id superiori alicui scientias acceptum referat. Etenim ex cognitis naturae l^ibus pulcherrimae theoriac, praxes etiam mechanicae ad vitam titiles consequuntur. Ex cognitione autem ipsius naturae Auctoris considerationes longe praestantissimoe quidem illae, sed meta- physicae, theologicae, morales oriuntur.

43. De frimifih hactenus: nunc dicendum de natura motus ^^. Atque is quidem, cum sensibus clare percipiatur, non tarn natura sua, quam doctis philosophorum commentis obscuratus est. Motus nunquam in sensus nostros incurrit sine mole corporea, spatio, et tempore. Sunt tamen qui motum, tanquam ideam quandam simplicem et abstractam, atque ab omnibus aliis rebus sejunctam, contemplari student. Verum idea ilia tenuissima et subtilissima^i

■• • natnra motus.' Sect. 43 — 66 treat of the nature of the tffeet — sensible motion, as distinguished from its true cause or origin {principivm) — mind or spirit. As the origin of motion belongs to Metaphysics, the nature or definition of motion, constituted by the bws of ill sensible phenomena, belongs to .Mechanics or Physics. A true definition of motion is what Berkeley is in search of in this

second part of the De Molv, Cf. PrincipUs 0/ Human Knowledge, sect, iii — 116.

'^* * idea ilia tenuissima et subtilissima.' Here again the difficulty is attributed to abstractions, and to the inclination of the scholastic mind to prefer these to the con- crete motions of which we have sensible experience, and which may be universalized by reason.

90

De Motu.

incmH^

intellcctus aciem eludit: id quod quilibet secum meditando ex- J periri potest. Hiac iiascuntur magnae diificultates de natura motus, et definitiones, ipsa re quam illustrare debeot longc I obscuriores. Hujusmodi sunt definitiones illie Aristotelis rt Scholasticorum^f, qui motum dicunt esse actum mollis qHMniu tst moliile, vel actum enth 'm potent'ia quatenus in potent'ia. Hujusmodi ctiam est illud viri'* inter reccntiores Celebris, qui asserit vikil'a motu eist reaie prieter momentaneum illitd ^uod in vi ad mutatintM nlttKte eonst'itui debet. Porro constat, horum et similium defini- tionum auctores in animo habuisse abstractam motus Daturam, seclusa omni temporis et spatii consideratione, explicare: seii qua ratione abstracta ilia motus quintesscntia (ut ita dicam) io- telligi possit, non video.

44. Nequc hoc content!. Ulterius pergunt, partesque ip motus a se invicem dividunt et secernunt, quaram ideas distinct tanquam entium revera distinctorum, eSbrmarc conantur. Etenira sunt qui motionem a motu distinguant, illam velut instantancum motus elemcntum spcctantes. Velocitatem insuper, conatum, vim, impetum totidem res essentia diversas esse volunt, quarum quacque per propriam atque ab aliis omnibus segregatam et abstractam ideam intellectul objiciatur. S^d in hisce rebus discutiendis, stantibus iis quse supra disseruimus", non est cur diutius imjno- remur.

45. Multi ctiam per transitum ''* motum definiunt, obliti, scilicet, transitum ipsum sine motu intelUgi non posse, et per motum definiri oportere. Verissimum adeo est delinitiones, sicut noD- nullis rebus lucem, ita vicissim aliis tencbras afFerre. Et profecto, quascumque res sensu percipimus, eas clariores aut notiores defi- niendo eiScere vix quisquam potuerit. Cujus rei vana spe allecti res faciles difficillimas*^ reddiderunt philosophi, mentesque suas dilScultatibus, quas ut plurimum ipsi peperissent, implicavere. Ex hocce definiendi, simul ac abstrahendi studio, multx tam de motu quam de aliis rebus natx subtil issima: quxstiones, exdemque nullius utilitatis, hominum ingenia frustra tofserunt^ adeo ut Aristoteles ultro et sxpius fateatur motum esse Mtum ^uendam

' Motion u ihui dctincJ by AriiiotU ; — •* Cf. tect 3 — 41.

•unrroJ j iirniTii'. Nal. ■Iwi.inilni. Sccal»Dc«

* Dm Cirto. Printipai, P. II. | 1 j ; I

; Mc Bocellui, Dt Fi Ptrauiim

t ficilet lUScUlimu.' Cf> J

De Moiu. 91

tiputm Mffidlem '7, et nonnulli ex veteribus usque eo nugis exercitati deveniebant, ut motum omnino esse negarent^^.

46. Sed hujusmodi minutiis distineri piget. Satis sit fontes soiutiomim indicasse: ad quos etiam illud adjungere libet: quod ea quae de infinita divisione temporis et spatii in mathesi tra- Amtur, ob cx)ngeuitam rerum naturam paradoxa et theorias spinosas (quales sunt iUse omnes in quibus agitur de infinite ^9) in qiecnlationes de motu intulerunt. Quidquid autem hujus generis at, id omne motus commune habet cum spatio et tempore, vel potius ad ea refert acceptum.

47. £t quemadmodum ex una parte nimia abstractio seu divisio renim vere inseparabilium, ita ab altera parte compositio seu potius confusio rerum diversissimarum motus naturam perplexam feddidit. Usitatum enim est motum cum causa motus efficiente CDofuiidere^. Unde accidit ut motus sit quasi bifbrmis, unam &ci€m sensibus, obviam, alteram caliginosa nocte obvolutam habeas. Inde obscuritas et confusio, et varia de motu paradoxa originem trahunt, dum efFectui perperam tribuitur id quod revera causae solummodo competit.

48. Hinc oritur opinio ilia, eandem semper motus quantitatem conscrvari^*; quod, nisi intelligatur de vi et potentia causae, sive causa ilia dicatur natura, sive vovsj vel quodcunque tandem agens sit, falsum esse cuivus facile constabit. Aristoteles^^ quidem 1. 8. Physicorum, ubi quaerit utrum motus f actus sit et corruptuSy an veto ab atermo tamjuam v$ta smmartalis insit rehus omnthuSy vitale principium potius, quam effectum externum, sive mutationem loci ^ intellexisse vidctur.

49* Hinc etiam est, quod multi suspicantur motum non esse meram passionem in corporibus* Quod si intelligamus id quod in motu corporis sensibus objicitur, quin omnino passivum sit nemo dubitare potest. Ecquid enim in se habet successiva cor- poris existentia in diversis locis, quod actionem referat, aut aliud sit quam nudus et iners effectus ?

* fUdk Ztk rovro 9^ x^*^^*' uifT^iv •* The modem doctrine of the 'conier- XMfi*a' rl icrim. Nai, Ause. III. i. vation of force.'

" e.g. Zrno. •' Aristotle states the queition NeU. A use.

• *6e iofinito, Sec* Cf. Principles, sect. VIII. cap. i., and solves it in cap. iv. 130-^131, and the Analyst passim, for *>* mutatio loci' is tie effect, i.e. motion Bcrkclcj'f theory of infinitesimals. proper ; * vitale principium ' the efficient

** ' cobfondere.' Cf. sect. 3 — 43 in iU cause, i.e. vital and personal agency. hittratioo of this confusion.

92 De Moiu.

50. Peripatetici, qui dicunt motum esse actum unum utriusqaCy moventis et moti^', non satis discriminant causam ab efitcto. Similiter, qui nisum aut conatum in motu fingunt, aut idem corpus simul in contrarias partes ferri putant, eadem idearum confusione^ eadem vocum ambiguitate ludificari videntur.

51. Juvat multum, sicut in aliis omnibus, ita in sdentia de motu accuratam diligentiam adhibere, tam ad aliorum concepts intelligendos quam ad suos enunciandos : in qua re nisi peccatum esset, vix credo in disputationem trahi potuisse, utrum cofpus indifferens sit ad motum et ad quietem, necne. Quoniam enixn experientia constat, esse legem naturae primarium, ut corpus perinde p^rseverat in statu motus ac quietisy fuamJiu mlkmde mkU accidat ad statum istum mutandum i et propterea vim inertix sub di verso respectu esse vcl resistentiam, vel impetum, colligitur: hoc sensu profecto corpus dici potest sua natura iudiflFerens ad motum vel quietem. Niminmi tam difficile est quietem in corpus motum, quam motum in quiescens inducere: cum vero corpus pariter conservet statum utrumvis, quidni dicatur ad utrumvis se habere indiflferenter ?

52. Peripatetici pro varietate mutationum, quas res aliqua subire potest, varia motus genera distinguebant. — Hodie de motu agentcs intelligunt solummodo motum localem^^. Motus autem localis intelligi nequit nisi simul intelligatur quid sit locus: is vero a ncotericis^*^' definitur/4rr/ jr/tf/// quam corpus occupat : unde dividitur in relativum et absolutum pro ratione spatii. Distinguunt cnim inter spatium absolutum sive verum, ac relativum sive apparens. Volunt scilicet dari spatium undequaque immensum, immobile, insensibile, corpora universa permeans et continens, quod vocant spatium absolutum. Spatium autism a corporibus comprehensum vel definitum, sensibusque adco subjectum, dicitur spatium rela- tivum, apparens vulgare.

^^. Fingamus itaque corpora cuncta dcstrui, et in nihilum

'*' 'moventis et moti,' i.e. as concauscs. so-called absolute space is mere negation.

•* 'motum localem.' Sect. 52 —65 discuss Cf. /Vf««/>/«, sect. 1 1 6, 1 1 7. See Locke's

the alleged distinctions of real or absolute Esuty^ Bk. II. ch. 13; Corre$pondenc9 be-

and apjnrent or relative motion, and of real tween Clarke and Leibnitz ; and Newton's

or absolute and apparent or relative space. Principia.

The abstract impossibility of absolute space ** Newton's Principia, Def. Sch, III.

und motion is argued. With Berkeley space Sec also Derodon, Pby^ica^ P. I. cap. Ti.

is identified with sensible exteuMOn ; and § i .

De Motu. 93

Quod reliquum est vocant spatium absolutum, omni fdatione quae a situ et distantiis corponim oriebatur, simul cum ipsis corporibus, sublata. Porro spatium illud est infinitum, im^ mobile, indivisibile, insensibile, sine relatione et sine distinctione. Hoc est, omnia ejus attributa sunt privativa vel negativa: videtur igitur esse m^ram nihil '^^ Parit solummodo difficultatem aliquam fiod extensum sit. Extensio autem est qualitas positiva. Verum quails tandem extensio est ilia quse nee dividi potest, nee men- saiari, cujus nullam partem, nee sensu percipere, nee imaginatione depii^ere possumus? Etenim nihil in imaginationem cadit, quod, ex natura rei, non possibile est ut sensu percipiatur; si- quidem imagmath^** nihil aliud est quam facultas representatrix rerum sensibilium, vel actu existentium, vel saltem possibilium. Fugit insuper intellectum furum^^^ quum facultas ilia versetur tantum drca res spirituales et inextensas, cujusmodi sunt mentes nostras, canimque habitus, passiones, virtutes, et similia. Ex spatio igitur absolute auferamus modo vocabula, et nihil remanebit in sensu, imaginatione, aut intellectu : n^hil aliud ergo iis designatur, quam pura privatio aut negatio, hoc est, merum nihil.

54. Confitendum omnino est nos circa banc rem gravissimis praejudiciis teneri, a quibus ut liberemur, omnis animi vis exercenda. Etenim multi, tantum abest quod spatium absolutum pro nihilo ducant, ut rem esse ex omnibus (Deo excepto) unicam existiment, quas annihilari non possit : statuantque illud suapte natura neces- sario existere, aetemumque esse et increatum, atque adeo attribu- torum divinonim particeps'*^ Vcrum enimvero quum certissimum sit, res omnes, quas nominibus designamus, per qualitates aut relationes, vel aliqua saltem ex parte cognosci (ineptum enim foret vocabulis uti quibus cogniti nihil, nihil notionis, ideae vel conceptus subjiceretur), inquiramus diligenter, utrum formare liceat iJeam uUam spatii illius purl, realis, absoluti, quod post omnium corporum annihilationem perseveret existere. Ideam porro talem paulo acrius intuens, reperio ideam esse nihili purissimam, si modo idea appellanda sit« Hoc ipse summa adhibita diligentia expcrtus sum: hoc alios pari adhibita diligentia experturos reor.

" Cf. Locke'f account of absolute space, the df<r(h}o'i^, ^aifrcurla, and yovs of Aristo-

Sfsojr, Bk. II. ch. 13, 15, 17. tdian psychology.

** Note the definitions here given of "* * attributorum divinonim particeps.'

hmagmaiiom and inulltet, as distinguished Dr. Samuel Clarke, for instance, in his De*

from umUt which may be compared with monstnUiim,

94

De Mohi.

55. Decipcre nos nonnuaquam solet, quod altis omnibus coc- poribus imaginatione sublatis, naitmrn'^ tamcn manere suppontimis. C^o supposito, motum membrorum ab omnt parte liberrimum imaginamur. Motus autem sine spatio concipi non potest. Nihilominus si rem attento animo recolamus, constabit primo concipi spatium reJalivum partibus nostri corporis defiaitum: 2°. movendi membra potestatem liberrimam nulio obstaculo retusam: et pneter hsc duo nihil. Falso tamcn crcdimus tertium aliquod, spatium videlicet immensum realiter existere, quiid liberam potestatem nobis faciat movendi corpus nostrum : ad hoc enim rcquiritur absentia solummodo aliorum corporum. Quani absentiam, sive prlvationem corporum, nihil esse positivum fatca- mur necesse est".

56. Castcrum hasce res nisi quis libcro et acri examine per- spexerit, verba et voces parum valent. Meditanti vero, et ratione secum reputanti, ni fallor, manifestum crit, qujccunque de sp;itio puro et absolute prjedicantur, ea omnia de nihilo praedicari posse. Qua ratione mens humana faciUipie liberatur a magnis difficid- tatibus simulque ab ea absurditate tribuendi existentiam neccs- sariam"* ulli rei prstcrquam soli Deo optimo maximo.

57. In proclivi esset scntentiam nostram argumentis a posteriori (ut IcKjuuntur) ductis coniirmare, quKstiones de spatio absolute proponcndo; exempli gratia, utrum sit substantia vel accidcns? utrum creatum vel increatum? et absurditatcs ex utravis p^ite. consequcntes demnnstrando. Sed brevitati consulendum, tamcn omitti non debet, quod s;ntcntiam hancce Democril olim calcub suo comprohavit, uti auctor est Aristoteles I. i. PhyaC ubi Hec habct : Uemoaitus seHJum et inane fon'n prinfifia^ tfwanm aliud qiiiJem ut ^uoJ est, aliud ul quad non est esse dial, Scrupiilum si forte injiciat, quod distinctio ilia inter spatium absolutum et relativum a magni nominis philosophis usurpetur, eique quasi fijndamento inadiiicentur multa prazclara theoremata, scrupulum istum vanum esse, ex iis quae secutura sunt, apparebit.

58. £k pncmissis patet, non convenire, ut definiamus locum verum corporis esse partem spatii absoluti quam occupat corpus^

IIMJ ocritiifl

libio Dt PrintipiU Cogni

nute ingliciiio decim ibhinc

riA nr the Pr1>i«>I(i.

" Pure (pice being « in«re Motion >ci»iblc tyitet ■ niinir«ntion «i ' if Supreme Riiionil WiK.

De Motu. 95

motumque venun seu absolutum esse mutationem loci veri et absoluti. Siquidem omnis locus est relativus, ut et omnis motus. Venintamen ut hoc clarius appareat, animadvertendum est, motum nullum intelligi posse sine determinatione aliqua seu directione, fuse quidem intelligi nequit, nisi praeter corpus motum, nostrum ctiam corpus, aut aliud aliquod, simul intelligatur existere. Nam smsum, deorsum, sinistrorsum, dextrorsum, omnesque plagae et r^ones in relatione aliqua flindantur, et necessario corpus a moto diversum connotant et supponunt* Adeo ut^ si reliquis oorporibus in nihilum .redactis, globus, exempli gratia, unicus existere supponatur; in illo motus nullus concipi possit: usque adeo necesse est, ut detur aliud corpus, cujus situ motus deter- minari intelligatur. Hujus sententise Veritas clarissime elucebit^ modo corporum omnium tam nostri quam aliorum, praster globum istum unicum, annihilationem recte supposuerimus.

59* Concipiantur porro duo globi, et prseterea nil corporeum, existere. Concipiantur deinde vires quomodocunque applicari : quioquid tandem per applicatiooem virium intelligamus, motus drcularis duorum globorum circa commune centrum nequit per imaginationem conqipi. Supponamus deinde coelum fixarum creari: subito ex concepto appulsu globorum ad diversas coeli istius partes motus concipietur. Scilicet cum motus natura sua sit relativus, concipi non potuit priusquam darentur corpora correlata. Quemadmodum nee ulla relatio alia sine correlatis concipi potest.

60. Ad motum circularem quod attinet, putant multi, crescente motu vero circularly corpus necessario magis semper magisque ab axe niti. Hoc autem ex eo provenit, quod, cum motus cir- cularis spectari possit tanquam in omni momeiito a duabus direc- tionibus ortum trahens, una secundum radium^ altera secundum tangentem; si in hac ultima tantum directione impetus augeatur, turn a centro recedet corpus motum, orbita vero desinet esse cir- cularis. Quod si sequaliter augeantur vires in utraque directione, manebit motus circularis, sed acceleratus conatu, qui non magis arguet vires recedendi ab axe, quam accedendi ad eundem, auctas esse. Dicendum igitur, aquam in situla circumactam ascendere ad latera vasis, propterea quod, applicatis novis viribus in di- rectione tangentis ad quamvis parti culam aquse, eodem instanti non applicentur novae vires asquales centripetas. Ex quo expert-

96

De Motu.

mento nullo modo sequitur, motum absolutum circuarem per vir5~ recedendi ab axe motus necessario dignijsci. Porru qua rationc intelligendx sunt voces istK, virts eorporum et conatus^ ex prxmJssb satis sijf)erque inmitescU,

6i. Quo modo curva considerari potest tanquam constans ^ rectis iniinitis, ctiamsi revera ex lllis non constet, sed quod \ hypothesis ad gcometriam utilis sit eodem modo motus circulai spectari potest tanquam a direct ioni bus rcctilineis infinitis ortuq ducens, quae suppositio utilis est in philosophia mechanica. NtH tamen ideo affirmandum, impossibile ess?, ut centrum gravitatft corporis cujusvis successive existat in singulis punctis peripherili circularis, nulla ratione habita directionis uUius rectilinex, in tangents sive in radio.

62. Haud omittcndum est, motum lapidis in fuada, aut aqa|| in situla circumacta, dici non posse motum vere circularem, juxtt. I mefltem eonim qui p:r partes spatii absoluti definiunt loca vera corporum; cum sit mire compositus ex motibus non solum situlx vel fund^, sed etiam tdluris diurno circa proprium axem, mcn- struo circa commune centrum gravitatis terr^ ct lunx, ct annuo circa soiem: et propterea particula qusevis lapidis vel aqua describat lineam a circular! longe abhorrcntem, Neque rcvcia est, qui creditur, conatus axifiJgus, quoniam non respicit unum aliqucm axem ratione spatii absoluti, supposito quod detur tale spatium: proinde non video quomodo appollari possit conal unicus, cui motus vere circularis tanquam proprio et adequl eftectui respondit.

6j. Motus nullus dignosci potest, aut mcnsurari, nisi per r^~ sensibiles. Cum ergo spatium absolutum nullo modo in sensus incurrat, necesse est ut inutile prorsus sit ad distinctionem motuum. Praeterca dctcrminatio sive dircctio motui ess^atialis J est, i!!a vero in relatione consistlt. Ergo impossibile est Ut moUM absolutus concipiatur. "^

64. Porro quoniam pro dlversitate loci rclativi varius sit motus ejusdem corporis, quinimo uno respectu moveri, altero quiescere diciquidpiam possit '•'; ad determinandum motum verum ct quictcm veram, quo scilicet tollatur ambiguitas, et consulatur mcchania philosophorum, qui systcma rcrum latius contemplantur, satil

>na£H

,. Bk. ll.ch. 13. f J-

De Motu, 97

fberit spatium relativum ftxarum coelo, tanquam quiescente spec- tator condusum adhibere, loco spatii absoluti. Motus autem et quies tali spatio relativo definiti, commode adhiberi possunt loco absolutorum^ qui ab illis nuUo symptomate discemi possunt. ftenim imprimantur utcunque vires, sint quicunque conatus, con- cedamus motum distingui per actiones in corpora exercitas ; nun- quam tamen inde sequetur, dari spatium illud et locum absolutum, ejusque mutationem esse locum verum.

65. Leges motuum, effectusque, et theoremata eorundem pro- portiones et calculos continentia, pro diversis viarum figuris, accelerationibus itidem et directionibus diversis, mediisque plus minusve resistentibus, hsec omnia constant sine calculatione motus absoluti. Uti vel ex eo patet quod, quum secundum illorum principia qui motum absolutum inducunt, nullo symptOr mate scire liceat, utrum integra rerum compages quiescat, an moveatur unifbrmiter in directum, perspicuum sit motum abso- lutum nuilius corporis cognosci posse.

66. Ex dictis patet ad veram motus naturam perspiciendam sununopere juvaturum, lo. Distinguere inter hypotheses mathe- maticas et naturas rerum: 2". Cavere ab abstractionibus : y. Considerare motum tanquam aliquid sensibile, vel saltem imagi- nabile; mensurisque relativis esse contentos. Quas si fecerimus, simul clarissima quacque philosophise mechanicx theoremata, quibus reserantur naturas recessus, mundique systema calculis humanis subjicitur, manebunt intemerata , et motus contemplatio a mille minutiis, subtilitatibus, ideisque abstractis libera evadet. Atque haec de natura motus dicta sufficiant.

67. Restat, ut disseramus de causa communicationis motuum^^. Esse autem vim impressam in corpus mobile causam motus in CO, plenque existimant. Veruntamen illos non assignare causam motus cognitam, et a corpore motuque distinctam, ex praemissis

^ Sect. 67 — 7 a treat of the cause of the explanations of phenomena may be de-

tnnanisMon of motion among bodies. While duced from mechanical principles. Cf.Siris,

thn is held to be, strictly speaking, the active sect. 253, 153, in which an interpreta-

Intelligence which moves and embraces the tion of physical signs, sufficient for prc-

wbole material universe, nevertheless, for vision, is allowed to be an explanation

the purposes of merely natural philosophy, of them.

VOL. III. H

De Mottt.

istani^

y

constat. Patet insupcr vim non esse rem certam et detcrminSta ex CO quod viri summi de ilia multuju diversa, immo contraria, profcrant, salva tamen in consequentiis vcritate. Siquidcm Ncwtonus*-* ait vim impressam consistere in actione sola, esscquc \ actionem exercitam in corpus ad stalum ejus mutandum, ne^ post actionem manere. Torricellius"'' cumulum quendam ^VJ aggregatum virium Impressarum per percussionem in coqnis mobile recipi, ibidemque manere atque Impctum constituere contcndi^ Idem fere Borellus"' aliique prxdicanL At vero, tametsi inter ■ pugnare videantur Newtonus et Torriccllius, nihi!ominus, dun singuli sibi consentanca profcrunt, res satis commode ab utrisqiil e:(plicatur. Quippe vires omnes corporibus altributic tarn suotl liypotheses mathematics quam vires attractivse in planetis ct sole. Carterum entia mathematica in rerum natura stabilem ess^ntiam non habent: pendent autem a notione definientis; undc cadcni res diversimixie explicari potest.

68. Statuamus motum novum in corporc peraissn conservar sive per vim insitam, qua corfRis quodlibet perseverat in stabTI suo vcl mutus vel quietis uniformis in directum; sive per vim impressam, durante percussione in corpus percussum receptam ibidemque pcrmanentcm; idem erit quoad rem, differentia exis- tente in nominibus tantum. Similiter, ubi mobile percuticns perdit, et percussum acquirit motum, parum refert disputarc, utrum motus acquisitus sit idem numero cum motu perdito, dudt enim in minutias metaphysicas et prorsus nominales dc identitate. Itaque sive dicamus motum transire a percuticnte in percussum, sive in percusso motum de novo generari, descroM autem in percutiente, res eodem recidit. Utrobique intelligif^| unum corpus motum pcrdere, alterum acquirere, et prscterea nihil^l

69. Mentem, qux agitat et continet universam hancce molafl corporcam, estque causa vera efficiens motus, eandem esse, pru^vin et stride loquendo, causam communicationis cjusdem haud^ oegaverim. In philosophia tamen physica, causas et solutiones pharnomcnon a principiis mechanicis petcrc oportct. Physicc igitur res explicatur non assignando ejus causam vere agentem et incorpoream, sed demonstrando ejus amnexionem cum pri cipiis mechanicis: cujusm<xli est illiid, actionem et rtaetiontm «

" Prmiipia. Def. IV. " Laiani Actajtmubt.

De Motu, 99

semper camtrarias et equates f^^ a. quo, tanquam ibnte et principio primario, eniuntur regular de motuum communicatione, quas a neotericis, magno scientiarum bono, jam ante repertse sunt et demonstrataB.

70. Nobis satis fuerit, si innuamus principium illud alio modo dedarari potuisse. Nam si vera rerum natura potius quam abstracta mathesis spectetur, videbitur rectius dici, in attractione vel pcrcussione passionem corporum, quam actionem, esse utro- bique acqualem. Exempli gratia, lapis fune equo aliigatus tantum trahitur versus equum, quantum equus versus lapidem: corpus etiam motum in aliud quiescens impactum, patitur eandem mutationem cum corpore quiescente. Et quoad effectum realem, percutiens est item percussum, percussumque percutiens. Mutatio autem ilia est utrobique, tam in corpore equi quam in lapide, tarn in moto quam in quiescente, passio mera. Esse autem vim, virtutem, aut actionem corpoream talium eflFectuum vere et proprie causatricem non constat. Corpus motum in quiescens impingitur; loquimur tamen active, dicentes illud hoc impellere: nee absurde in mechanicis, ubi ideas mathematical potius quam verac rerum naturae spectantur.

71. In physica, sensus et experientia, quae ad eflFectus apparentes solummodo pertingunt, locum habent; in mechanica, notiones abstractae mathematicorum admittuntur. In philosophia prima, scu metaphysica, agitur de rebus incorporeis, de causis, veritate, et existcntia rerum. Physicus series sive successiones rerum sensibilium contemplatur, quibus legibus connectuntur, et quo ordine, quid praecedit tanquam causa, quid sequitur tanquam cfiectus, animadvertens. Atque hac ratione dicimus corpus motum esse causam motus in altero, vel ei motum imprimere, trahere etiam, aut impellere. Quo sensu causae secundae corporeae intelligi debent, nulla ratione habita verae sedis virium, vel potentiarum actricimi, aut causae realis cui insunt. Porro dici possunt causae vel principia mechanica, ultra corpus^ figuram, motum, etiam axiomata scientiae mechanicae primaria, tanquam causae consequentium spectata.

72. Causae vere activae meditatione tantum ct ratiocinio e tenebris erui quibus involvuntur possunt, et aliquatenus cognosci.

'• This is Ncwlon's third law of motion.

II Z

lOO

De Motu.

Spectat autem ad philosophiam primam, seu metaphysicam, de iis agere* Quod si cuique scientise provincia sua 79 tribuatur, limites assignentur, principia et objecta accurate distinguantur, quse ad singulas pertinent, tractare licuerit majore, cum facilitate, turn perspicuitate.

^ 'provincia lua.' The D$ Motu is a treatise on the theory of Physics, if, with Aristotle, we regard Physics as conversant at>oat the manifestations of the principle of motion, in contrast to Mathematics, which deals with things immutable but not tran- scendent, and to Theology or Metaphysics, concerned with the eternal and transcen-

dental. Berkeley treats of motion as the phenomenal expression of its Divine Cause — as the first link in the chain which con- nects the sensible and intelligible worlds, a conception unfolded more comprehensively in his SiriSj more than twenty years after- wards. Of. Sirit, sect 296, 397, 347 — 349.

MISCELLANEOUS WORKS

ENGLISH.

PASSIVE OBEDIENCE:

OR,

THE CHRISTIAN DOCTRINE OF NOT RESISTING THE SUPREME POWER, PROVED AND VINDICATED,

UPON

THE PRINCIPLES OF THE LAW OF NATURE,

LV A DISCOURSE DELIVERED AT THE COLLEGE-CHAPEL,

1712.

/ v .

TO THE READER.

iT an absolute passive obedience ought not to be paid any ower, but that submission to government should be measured lited by the public good of the society ; and that therefore sub- lay lawfully resist the supreme authority, in those cases where dHc good shall plainly seem to require it ; nay, that it is their duty »o, inasmuch as they are all under an indispensable obligation to te the common interest: — these and the like notions, which I help thinking pernicious to mankind, and repugnant to right , having of late years been industriously cultivated, and set in the dvantageous lights by men of parts and learning, it seemed neces- ) arm the youth of our University against them, and take care :o into the world well principled; — I do not mean obstinately iced in favour of a party, but, from an early acquaintance with iuty, and the clear rational grounds of it, determined to such es as may speak them good Christians and loyal subjects, his view, I made three Discourses not many months since in the e-chapel S which some who heard them thought it might be of make more public : and, indeed, the false accounts* that are gone

inity College, Dublin.] — Author.

; publication of the Dueouru did

«] these rumours, as we learn from

cdote recorded by Bishop Stock.

a, the principles inculcated in Mr.

Tvfo Trtatists of Oovemnunt seem

turned his attention to the doctrine

ve Obedience; in support of which

ed the substance of three Common-

Iclivcred by him that year in the

chapel, a work which afterwards

irly done him some injury in his

For, being presented by Mr.

1] Molyneux to their late M.ijcstit>,

then Prince and Princess of Wales (whose Secretary Mr. Molyneux had been at Han- over), he was then recommended to Lord Galway for some preferment in the Church of Ireland. But Lord Galway having heard of these Sermons, represented him as a Jacobite; an impression which Mr. Moly- neux, as soon as he was apprised of it, took care to remove from the minds of their Highnesses, by producing the work in question, and shewing that it contained nothing but principles of loyalty to the present happy Kitablishment.' (Stock's Life o/ Berkthy.)

io6 PREFACE.

abroad concerning them have made it necessary. Accordingly, I r send them into the world under the form of one entire Discourse.

To conclude : as in writing these thoughts it was my endeavour preserve that cool and impartial temper which becomes every sine inquirer after truth, so I heartily wish they may be read with the sa disposition.

PASSIVE OBEDIENCE*.

Romans, chap. xiii. ver. 2.

* Whosoever rcsisteth the Power, resisteth the ordinance of God.'

I. It is not my design to enquire into the particular nature of the government and constitution of these kingdoms ; much less to pretend to determine concerning the merits of the different parties now reigning in the state. Those topics I profess to lie out of my sphere, and they will probably be thought by most men improper to be treated of in an audience almost wholly made up of young persons, set apart from the business and noise of the

' The first two editions of this Discourse appeared in London in 1 71 2. A third, •corrected and enlarged,' followed in 1713, introducing, besides a few trifling verbal amendments, sect. 53, and the note to sect.

48. The Discourse is interesting for its in-

geniouslj-argued defence of Non-resistance

ai a duty of>posed to the sin of lawlessness,

bat especially for the most distinct and rea-

sooed account in Berkeley's writings of his

jFcoeral theory of moral obligation.

* Self-love ' he represents (sect. 5) as the

deepest and most universal motive of human

actioiL We call actions good or ivil as they

are fitted to promote or hinder our own

happiness. For distinguishing eternal good

from present enjoyment, we must refer

them, by means of reason, to universal law.

Now, it is * a truth evident by the light of

nature, that there is a sovereign omniscient

Spirit, who alone can make us for ever

happy or for -ever miserable.' (Sect. 6.)

The universal laws o( nature must, accord-

mgly, be referred to the nature of God, and

the end which He designs to accomplish by

human actions. This end must be * the

food of men' (sect, f), who are thus com-

manded to promote, by the * concurring actions of each individual,' the * general wellbeing of all men, of all nations, of all ages, of the world. The rational deduction of the goodness of actions is thus founded on their essential fitness to promote the wellbeing of mankind.'

Submission to the supreme authority is afterwards dedu'^ed as one of the most im- portant consequences from these principles.

The chief divisions and subdivisioiu of the Discourse are unfolded in sect. 2, 3.

The same theory of the duty of absolute unlimited submission to supreme civil au- thority as a fundamental article of Ethics, is enforced in Berkeley's Discourse to Ma- gistrcUes, published nearly a quarter of a century later, and which should be com- pared with this Discourse.

What follows may be compared with Gvardiartt No. 55, and Alciphron, Dial. 11., III. See also Locke's Treatises on Govern' ment, published more than twenty years previously, as well as his suggestions of a theory of Ethics, and of a demonstrated method of forming Ethical Science, in his Essay and Correspondence with Molyncux.

io8 Passive Obedience: upon the

world, for their more convenient instruction in learning ai piety* But surely it is in no respect unsuitable to the circun stances of this place to inculcate and explain every branch < the Law of Nature ^ or those virtues and duties which are equal binding in every kingdom or society of men under heaven; an of this kind I take to be that Christian Duty of not rcsistic the supreme Power implied in my text — ^'Whosoever resistel the Power, resisteth the ordinance of God.*

In treating on which words I shall observe the followin method : —

2. First, I shall endeavour to prove that there is an absolut unlimited non-resistance or passive obedience due to the suprem civil power, wherever placed in any nation*-.

Secondly, I shall inquire into the grounds and reasons of tb contrary opinions

Thirdly, I shall consider the objections drawn from the pre tended consequences of non-resistance to the supreme power ^.

In handling these points I intend not to build on the authorit of Holy Scripture, but altogether on the principles of Reasoi common to all mankind ; and that, because there are some ver rational and learned men, who, being verily persuaded an absolut passive subjection to any earthly power is repugnant to rigli Reason, can never bring themselves to admit such an interpretatio of Holy Scripture (however natural and obvious from the words as shall make that a part of Christian religion which seems t them in itself manifestly absurd, and destructive of the origin* inherent rights of human nature.

3. I do not mean to treat of that submission which men ar( either in duty or prudence, obliged to pay inferior or executiv powers; neither shall I consider where or in what persons th supreme or legislative power is Icxlged in this or that governmeni Only thus much I shall take for granted — that there is in ever civil community, somewhere or other, placed a supreme powe

^ Sect. 3—32- ' S<xt. 33—40- * Sect. 41—56.

Principles of the Law of Nature. 109

of making laws, and enforcing the observation of them. The fulfilling of those laws, either by a punctual performance of what IS enjoined in them^ or, if that be inconsistent with reason or conscience, by a patient submission to whatever penalties the sapreme power hath annexed to the neglect or transgression of tfaem^ is termed loyalty i as, on the other hand, the making use of force and open violence, either to withstand the execution of the laws^ or ward off the penalties appointed by the supreme power, is properly named rehellsm.

Now, to make it evident that every degree of rebellion is

criminal in the subject, I shall, in the first place, endeavour to

prove that loyalty is a natural or moral duty; and disloyalty,

or rebellion, in the most strict and proper sense, a vice or breach

of the law of nature. And, secondly, I propose to show that the

prohibitions of vice, or negative precepts of the law of nature,

as, *Thou shalt not commit adultery. Thou shalt not forswear

thyself. Thou shalt not resist the supreme power,' and the like,

ought to be taken in a most absolute^ necessary, and immutable

sense: insomuch that the attainment of the greatest good, or

deliverance from the greatest evil, that can befal any man or

number of men in this life, may not justify the least violation

of them.

First then, I am to show that loyalty is a Moral Duty, and disloyalty or rebellion, in the most strict and proper sense, a Vice, or breach of the Law of Nature '».

4. Though it be a point agreed amongst all wise men, that there are certain moral rules or laws of nature, which carry with them an eternal and indispensable obligation; yet, concerning the proper methods for discovering those laws, and distinguishing them from others dependent on the humour and discretion of men, there are various opinions. Some direct us to look for them in the Divine Ideas; others in the natural inscriptions on the mind : some derive them from the authority of learned men, and the universal agreement and consent of nations. Lastly,

' Sect. 4—25.

no Passive Obedience: upon the

others hold that they are only to be discovered by the deductions of reason. The three first methods must be acknowledged to labour under great difficulties ; and the last has not, that I know, been anywhere distinctly explained, or treated of so fully as the importance of the subject doth deserve.

I hope therefore it will be pardoned, if, in a discourse of passive obedience, in order to lay the foundation of that duty the deepefy we make some inquiry into the origin, nature, and obligation of Moral Duties in general, and the criterions whereby they are to be known fi.

5. Self-love being a principle of all others the most univeisal, and the most deeply engraven in our hearts, it is natural for us to regard things as they arc fitted to augment or impair our own happiness; and accordingly we denominate them good or enl Our judgment is ever employed in distinguishing between these two, and it is the whole business of our lives to endeavour, by a proper application of our faculties, to procure the one and avoid the other. At our first coming into the world, we arc entirely guided by the impressions of sense ; sensible pleasure being the infallible characteristic of present good, as pain is of evil. But, by degrees, as we grow up in our acquaintance with the nature of things, experience informs us that present good is afterwards often attended with a greater evil ; and, on the other side, that present evil is not less frequently the occasion of procuring to us a greater future good. Besides, as the nobler faculties of the human soul begin to display themselves, they discover to us goods far more excellent than those which afFect the senses^. Hence an alteration is wrought in our judgments; we no loi^r comply with the first solicitations of sense, but stay to consider the remote consequences of an action — what good may be hoped, or what evil feared from it, according to the wonted course of

* In what follows (sect. 5 — 1 4), the nidi- The later Idealism of Berkeley, ts in

ments of Berkeley's ethical doctrine of iS'im, presents more distinctly than in this

Theological or Ontological Utilitarianism Discfiuru the Eternal Law of Reason (by

are unfolded. Berkeley's eternal moral rules whatever method discovered) as the essence

accord so far with Ix)cke*s hints of a de- of morality.

monstrative Ethics, while his motive of ^ Cf. Aleipbron, Dial. I. sect. 14— 16:

moral action coincides with that of Paley. II. sect. 13 — 16.

Prificiplcs of the Law of Nature, 1 1 1

things. This obliges us frequently to overlook present momentary enjoyments, when they come in competition with greater and more lasting goods — though too far oS^ or of too refined a nature to afiect our senses.

6. But, as the whole earth, and the entire duration of those perishing things contained in it is altogether inconsiderable, or, in the prophefs expressive style, ' less than nothing ' in respect of Eternity, who sees not that every reasonable man ought so to frame his actions as that they may most effectually contribute to promote his eternal interest ? And, since it is a truth evident by the light of nature, that there is a sovereign omniscient Spirit, who alone can make us for ever happy, or for ever miserable j it {riainly follows that a conformity to His will, and not any prospect of temporal advantage, is the sole rule whereby every man who acts up to the principles of Reason must govern and square his actions. — The same conclusion doth likewise evidently result from the relation which God bears to his creatures. God alone is maker and preserver of all things. He is, therefore, with the most undoubted right, the great legislator of the world ; and man- kind are, by all the ties of duty, no less than interest, bound to obey His laws.

7. Hence we should above all things endeavour to trace out the Divine will, or the general design of Providence with regard to mankind, and the methods most directly tending to the accom- plishment of that design ; — and this seems the genuine and proper way for discovering the laws of nature. For, laws being rules directive of our actions to the end intended by the legislator, in order to attain the knowledge of God's laws, we ought first to inquire what that end is which He designs should be carried on by human actions. Now, as God is a being of infinite goodness, it is plain the end He proposes is good. But, God enjoying in Himself all possible perfection, it follows that it is not His own

good, but that of His creatures Again, the moral actions of men

are entirely terminated within themselves, so as to have no in- fluence on the other orders of intelligences or reasonable crea- tures ; the end therefore to be procured by them can be no other than the good of men. But, as nothing in a natural state can entitle one man more than another to the favour of God, except only moral goodness; which, consisting in a conformity to the laws

112 Passive Obedience: upon the

of God, doth presuppose the being of such laws, and law ever sup- piising an end, to which it guides our actions — it follows thai, antecedent to the end proposed by God, no distinction can be conceived between men ; that end therefore itseif, or general de- sign of Providence, is not determined or limited by any respect of persons. It is not therefore the private good of this or that man, nation, or age, but the general well-being of all men, of all nations, of all ages of the world, which God designs should he procured by the concurring actions of each individual.

Having thus discovered the great end to which all moral obli- gations are subordinate, it remains that we inquire what methods are necessary tor the obtaining that end.

8. The well-being of mankind must necessarily be carried oo in one of these two ways : — cither, first, without the injunction of any certain universal rules of morality, only by obliging every one, upon each particular occasion, to consult the public gtxxi, and always to do that which to him shall seem, in the present time and circumstances, most to conduce to it. Or, secondly, by enjoining the observation of some determinate, established laws, which, if universally practised, have, from the nature of things, an essential fitness to procure the well-being of mankind; though, in their particular application, they are sometimes, through un- toward accidents, and the perverse irregularity of human wills, the occasions of great sufferings and misfortunes, it may be, to very many good men.

Against the former of these methods there lie several strong objections. For brevity I shall mention only two: —

9. First, it will thence follow that the best men, for want of judgment, and the wisest, for want of knowing all the hidden cir- cumstances and consequences of an action, may very often be ai a loss how to behave themselves; — which they would not be, in case they judged of each action by comparing it with some par- ticular precept, rather than by eiiamining the gaod or evil which in that single instance it tends to procure: it being far more easy to judge with certainty, whether such or such an action be a Uans- gression of this or that precept, than whetlicr it will be attended with more g(x>d or ill consequences. In short, to calculate the events of each particular action is impossible; and, thwigh it '

Principles of the Law of Nature, 113

not, would yet take up too much time to be of use in the affairs of life.

Secondly^ ii that method be observed, it will follow that we can have no sure standard to which, comparing the actions of another, we may pronounce them good or bad, virtues or vices. For, since the measure and rule of every good man's actions is suj^xxsed to be nothing else but his own private disinterested opinion of what makes most for the public good at that juncture ; and, since this opinion must unavoidably in different men, from their particular views and circumstances, be very diflPerent : it is impossible to know, whether any one instance of parricide or per- jury, for example, be criminal. The man may have had his reasons for it, and that which in me would have been a heinous sin may be in him a duty. Every man's particular rule is buried in his own breast, invisible to all but himself, who therefore can only tell whether he observes it or no. And, since that rule is fitted to particular occasions, it must ever change as they do : hence it is not only various in different men, but in one and the same man at different times.

10. From all which it follows, there can be no harmony or agreement between the actions of good men : no apparent steadiness or consistency of one man with himself, no adhering to principles : the best actions may be condemned, and the most villainous meet with applause. In a word, there ensues the most horrible confusion of vice and virtue, sin and duty, that can pos- sibly be imagined. It follows, therefore, that the great end to iriiich God requires the concurrence of human actions must of necessity be carried on by the second method proposed, namely, the observation of certain, universal, determinate rules or moral precepts, which, in their own nature, have a necessary tendency to promote the well-being of the sum of mankind, taking in all nations and ages, from the beginning to the end of the world.

11. Hence, upon an equal comprehensive survey of the general nature, the passions, interests, and mutual respects of mankind ; — whatsoever practical proposition doth to right reason evidently appear to have a necessary connexion with the universal well- being included in it is to be looked upon as enjoined by the will of God. For, he that willeth the end doth will the necessary means conducive to that end; but it hath been shewn that G(xi willeth

VOL. III. I

1 1 4 Passive Obedteiue : upon ihe

the universal well-being of mankind should be promoted by the concurrence of each particular person ; therefore, every such prac- tical proposition necessarily tending thereto is to be esteemed a decree of God, and is consequently a law to man.

12. These propositions are called lavis of naturf, because they are universal, and do not derive their obligation from any civil sanction, but immediately from the Author of nature himself. They are said to be stamped on the mind, to be tngravtn on the tables of the hearty because they are well known to mankind, and suggested and inculcated by conscience. Lastly, they are termed eternal rules of reasoti^ because they necessarily result from the nature of things, and may be demonstrated by the infallible de- ductions of reason".

13. And, notwithstanding that these rules are too often, citli by the unhappy concurrence of events, or more especially by t wickedness of perverse men who will not conform to them, made accidental causes of misery to those good men who do, yet this doth not vacate their obligation : they are ever to be esteemed the fixed unalterable standards of moral good and evil ; no privatttJ interest, no love of friends, no regard to the public good, should^ make us depart from them. Hence, when any doubt arises con- cerning the morality of an action, it is plain this cannot be deter- mined by computing the public good which in that particular case it is attended with, but only by comparing it with the Eternal Law of Reason. He who squares his actions by this rule can never do amiss, though thereby he should bring himself to poverty, death, or disgrace: no, not though he should involve his ^unily, tus-i friends, his country, in all those evils which are accounted 1 greatest and most insupportable to human nature. Tender and benevolence of temper are often motives to the best and ' greatest actions; but wc must not make them the sole rule of our actions; they are passions rooted in our nature, and, like all other

• The Theoloi^cat Ulililiriiniun of Berkeley berc, «« eliewhete, encouiaget Ihii lereiencc lo ' ettnul rul«J of leucm,' uid to the immutubilily, uiiivenalilj, and nccn- fily of motxl diiUncliont — bngiitge foreign to the purely tecular ind »perinitii] uiilitariui theory. Hit reverence for Uau, in ciiiilriit iu eiiijHriol cnlciil.iliniii by indi-

fidnab of the pcrionRi utility of f> acliotii, ii apparent in ihete [niiagei. mil if the criterion of the« "eternal lam' it iheit lendency to promote general happj- DCii. a ctour ii ilill open lo Ihe queitioni of catuiilrv, in the endeavour to detenuoe and

J

Prituipks of t!u- Lazv of Nature.

115

tsions, must be restrained and kept under, otherwise they may

Rsibiy betray us into as great enormities as any otiier unbridled

Nay, they are more daug^-rous than other passions, inso-

> as they are more plausible, and apt to da7.ile and corrupt

i with the appearance of goodness and generosity ''. . For the illustration of what has been said, it will not be ss, if from the mora! we turn our eyes on the natural world. ) ffrtus est (says Balbus in Cicero "J^ ad munJum contemplandumj ium. And, surely, it is not possible for free intellectual aits to propose a nobler pattern for their imitation than Nature, 1 is nothing else but a series of free actions produced by the t and wisest Agent'i. But, it is evident that those actions are . adapted to particular views, but al! conformed to certain taeral rules, which, being collected from observation, are by lilosophers termed laws of nature. And these indeed are excel- lently suited to promote the general well-being of the creation: ' but, what from casual combinations of events '-, and what from the voluntary motions '= of animals, it often fells out, that the natural good not only of private men but of entire cities and nations would be better promoted by a particular suspension, or contra- diction, than an exact observation of those laws. Yet, for all that, nature still takes its course; nay, it is p'ain that plagues, famines, inundations, earthquakes, with an infinite variety of pains and sorrows — in a word, all kinds of calamities public and private, do arise from a uniform steady observation of those General Laws, which are once established by the Author of nature, and which He will not change or deviate from upon any of those accounts, how wise or benevolent soever it may be thought by foolish men to do so. As for the miracles'-' recorded in Scripture, they were always wrought for confirmation of some doctrine or

* So Bullef, «ho cegirdi the beneToIenl aircc:iu[u It 1 tubordiuite pirt only of ihit ideal htuniti lulure lo which our icliom ibouM conform. B«ie»oleiit moiivci may be ipriiiEi of ndout laioai.

* [pt Naiura Dtanm. lib. ii, ( .17-1 — AUTKOl

d dat>«

oft!

we tuTC in gtmi ihc c pbiloiopby. iTcinding lo which h tb« material univntc, it umply a phenuotciia (ftrtifi bdng their ui

in Iheii miifnrn] order of cocxiitence and (uccetsion. eiprew Suprerne Reason 1111! Will. Cf. Prineiples 0/ HumaH Knouilidgt, leet. 26— il: Dialo/rMio/Hylas and I'bilanaut ; Dt Morn; TbiOTf 0/ Viaan Vindicaud pauim; AUipbrOH. Dial. IV. ; Sh-h pauim.

" ■ciiualcombhiitiontofercali.' What are they?

" ■ rolunlary motions.' i e. the inlemip- timii produced liy finite agency or will.

" Cl.AlcifbroH.DM.W.: Sn-man htfori Ibt S. P G.

ii6

Passive Obedience : upon the

mission from God, and not for the sake of the particular natural goods, as health or life, which some men might have reaped from them. From all which it seems sufficiently plain that we can- not be at a loss which way to determine, in case we think God's own methods the properest to obtain His ends, and that it is our duty to copy after them, so far as the frailty of our nature will permit.

lonin

ij. Thus far in general, of the nature and necessity of Mor Rules, and the criterion or mark whereby they may be known.

As for the particulars, from the foregoing discourse, the principal of them may without much difficulty be deduced. It hath been shewn that the Law of Nature is a system of such rules or precepts as that, if they be all of them, at all times, in all places, and by all men observed, they will necessarily promote the well-being of mankind, so far as it is attainable by hum.TTi actions. Noiu let any one who hath the use of reason take but an impartufl survey of the general frame and circumstances of human natureil and it will appear plainly to him that tlie constant observation of truth, for instance, of justice, and chastity hath a necessary connexion with their universal well-being; that, therefore, they are to be esteemed virtues or duties; and that 'Thou shalt not forswear thyself,' 'Thou shalt not commit adultery,' 'Thou shalt not steal,' are so many unalterable moral rules, which to vio!atc in the least degree is vice or sin. I say, the agreement of these particular practical propositions with the definition or criterion premised doth so clearly result from the nature of things that, it were a needless digression, in this place, to enlarge upon it. ■

And, from the same principle, by the very same reasoning^ d follows that Loyalty is a moral virtue, and 'Thou shalt not resisP the supreme powei ' a rule or law of nature, the least breach whereof hath the inherent stain of moral turpitude.

16. The miseries inseparable from a state of anarchy are easily imagined. So insufficient is the wit or strength of any single man, either to avert the evils, or procure the blessings of life, and so apt are the wills of different persons to contradict and thwart each trther, that it is absolutely necessary several in- dependent powers be combined together, under the direction (if ! may so speak) of one and the same will— I mean the law of t

ftl^

Principles of the Law of Nature, 1 17

society. Without this there 13 no politeness, no order, no peace, among men, but the world is one great heap of misery and confusion ^ the strong as well as the weak, the wise as well as the foolish, standing on all sides exposed to all those calamities which man can be liable to in a state where he has no other security than the not being possessed of any thing which may raise envy or desire in another. A state by so much more in- eligible than that of brutes as a reasonable creature hath a greater reflection and foresight of miseries than they. From all which it plainly follows, that loyalty, or submission to the supreme authority, hath, if universally practised in conjunction with all other virtues, a necessary connexion with the well-being of the whcde sum of mankind ; and, by consequence, if the criterion we have laid down be true, it is, strictly speaking, a moral duty, or Inanch of natural religion. And, therefore, the least degree of rebellion is, with the utmost strictness and propriety, a sin: not only in Christians, but also in those who have the light of reason alone for their guide. Nay, upon a thorough and impartial view, this submission will, I think, appear one of the very first and fundamental laws of nature ; inasmuch as it is civil govern- ment which ordains and marks out the various relations between men, and regulates property, thereby giving scope and laying a foundation for the exercise of all other duties. And, in truth, whoever considers the condition of man will scarce conceive it possible that the practice of any one moral virtue should obtain, in the naked, forlorn state of nature.

1 7. But, since it must be confessed that in all cases our actions come not within the direction of certain fixed moral rules, it may possibly be still questioned, whether obedience to the supreme power be not one of those exempted cases, and consequently to be regulated by the prudence and discretion of every single person rather than adjusted to the rule of absolute non-resist- ance. I shall therefore endeavour to make it yet more plain, that 'Thou shalt not resist the supreme power' is an undoubted precept of morality j as will appear from the following consider- ations : —

First, then, submission to government is a point important enough to be established by a moral rule. Things of insignificant and trifling concern arc, for that very reason, exempted from the rules

Tl8

Passive Obedience: -upon the

of morality. But government, on which so much depend the \ peace, order, and well-being, of mankind, cannot surely be thougjil 1 of too small importance to be secured and guarded by a moral I rule. Government, I say, which is itself the principal source under heaven of those particular advantages for the procurement and conservation whereof several unquestionable moral rules were prescribed to men.

iK. Secondly, obedience to government is a case universal enough to fall under the direction of a law of nature. Numberless rules there may be for regulating affairs of great concernment, at certain junctures, and to some particular persons or societies, which, notwitlistanding, are not to be esteemed moral or natural laws, but may be either totally abrogated or dispensed with;— because the private ends they were intended to promote respect only some particular persons, as engaged in relations not founded in the general nature of man, who, on various occasions, and in different postures of things, may prosecute their own designs by different measures, as in human prudence shall seem convenicnL But what relation is there more extensive and universal than that J of subject and law? This is confined to no particular age ou climate, but universally obtains, at all times, and in all places; | wherever men live in a state exalted above that of brutes. It is, therefore, evident that the rule forbidding resistance to the law or supreme power is not, upon pretence of any defect in point of universality, to be excluded from the number of the laws of nature.

19. Thirdly, there is another consideration which confirms the necessity of admitting this rule ft>r a moral or natural law: namely, because the case it regards is of too nice and difficult a nature to be left to the judgment and determination of each private person. Some cases there are so plain and obvious to judge of that they may safely be trusted to the prudence of every reasonable man. But in all instances to determine, whether a civil law is fitted to promote the public interest; or whether submission or resistance will prove most advantageous in the consequence ; or when it is that the genera! good of a nation may require an alteration of government, cither in its form, or in the hands which administer it ; — these arc points too arduous and intricate, and which require too great a degree of pai

part^

Principles of the Law of Nature. 119

leisure, and liberal education, as well as disinterestedness and thorough knowledge in the particular state of a kingdom, for CTcry subject to take upon him the determination of them. From wfaidi it follows that, upon this account also, non-resistance, which, in the main, nobody can deny to be a most profitable and whole- some duty, ought not to be limited by the judgment of private persons to particular occasions, but esteemed a most sacred law (rf* nature.

ao. The foregoing aiguments do, I think, make it manifest, tiiat tiie precept against rebellion is on a level with other moral rules. Which will yet further appear from this fourth and last consideration. It cannot be denied that right reason doth require some common stated rule or measure, whereby subjects ought to shape their submission to the supreme power; since any clashing or disagreement in this point must unavoidably tend to weaken and dissolve the society. And it is unavoidable that there should be great clashing, where it is left to the breast of each individual to suit his fancy with a different measure of obedience. But this common stated measure must be either the general precept for- bidding resistance, or else the public good of the whole nation ; which last, though it is allowed to be in itself something certain and determinate, yet, forasmuch as men can regulate their conduct only by what appears to them, whether in truth it be what it appears or no ; and, since the prospects men form to themselves of a country*s public good are commonly as various as its land- scapes, which meet the eye in several situations: it clearly follows, that to make the public good the rule of obedience is, in effect, not to establish any determinate, agreed, conmion measure of loyalty, but to leave every subject to the guidance of his own pguticular mutable fancy.

21. From all which arguments and considerations it is a most evident conclusion, that the law prohibiting rebellion is in strict truth a law of nature, universal reason, and morality. But to this it will perhaps be objected by some that, whatever may be concluded with regard to resistance from the tedious deductions of reason, yet there is I know not what turpitude and deformity in some actions, which at first blush shews them to be vicious; but tbey, not finding themselves struck with such a sensible and immediate horror at the thought of rebellion, cannot think it on

120 Passiw Oiedimce: npem the

a level with other crimes against nature. To which 1 answer: — that it is true, there are certain natural antipathies implanted in the soul, which are ever the most lasting and insurmountable; but, as custom is a second nature, whatever aversions are from our early childhood continually infijsed into the mind give tt so deep a stain as is scarce to be distinguished from natural com- plexion. And, as it doth hence follow, that to make all the inward horrors of soul pass for infallible marks of sin were the way to establish error and superstition in the world; so, on the other hand, to suppose all actions lawful which are unattended with those starts of nature would prove of the last dangerous con- sequence to virtue and morality. For, these pertaining to us as men, we must not be directed in respect of them by any emotions in our blood and spirits, but by the dictates of sober and impartial reason. And, if there be any who find they have a less abhorrence of rebellion than of other villanies, all that can be inferred from it is, that this part of their duty was not so much reflected on, or so early and frequently inculcated into their hearts, as it ought to have been. Since without question there are other men who have as thorough an aversion for that as for any other crime'*.

22. Again, it will probably be objected that submission to government differs from moral duties in that it is founded in a contract 1% which, upon the violation of its conditions, doth of course become void, and in such case rebellion is lawful: it hath not therefore the nature of a sin or crime, which is in itself absolutely unlawful, and must be committed on no pretext what- soever.— Now, passing over all inquiry and dispute concerning the first obscure rise of government, I observe its being founded on a contract may be understood in a twofold sense : — either, first, that several independent persons, finding the insufferable in- convenience of a state of anarchy, where every one was governed by his own will, consented and agreed together to pay an absolute submission to the decrees of some certain legislative j which.

'[•iidi

nondt, ou pour tnlti lu let gnndi chcDuiu. cl quVnItn >l n'y

n.lurti; He (Mi. Pucul) u«J lo u> h*

had at great in abborTcucc dF rebellion at of minder, or robbing on ihc high-waj, aiid thai there «at nothing moie shocking to hl> natiue. — Vidt M. Pauol, p. 44,]— AuTBoa. " Cr. Locke'i TrKtiit M Oovmimml, Bk. II. di, S.

J

Principles of the Law of Nature. 121

thou^ sometimes they may bear hard on the subject, yet must surely prove easier to be governed by than the violent humours and unsteady opposite wills of a multitude of savages. And, in case we admit such a compact to have been the original foundation of civil government, it must even on that supposition be held sacred and inviolable.

23. Or, secondly, it is meant that subjects have contracted with their respective sovereigns or legislators to pay, not an absolute, but conditional and limited, submission to their laws, that is, upon condition, and so far forth, as the observation of them shall contribute to the public good : reserving still to themselves a right of superintending the laws, and judging whether they are fitted to promote the public good or noj and (in case they or any of them think it needful) of resisting the higher powers, and changing the whole frame of government by force : which is a right that all mankind, whether single persons or societies, have over those that are deputed by them. But, in this sense, a contract cannot be admitted for the ground and measure of civil obedience, except one of these two things be clearly shewn: — cither, first, that such a contract is an express known part of the fundamental constitution of a nation, equally