The works of George Berkeley, D.D. late Bishop of Cloyne in Ireland. To which is added, an account of his life [by J. Stock] and several of his letters to Thomas Prior, Esq., Dean Gervais, and Mr. Pope, etc

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rated ' by the motion of potnis, planes by the motion os lines, and soli is by the motion os planes. And Whereas quantities generaled in

equat times are greater or lesier according to the greater or lesier veloci ty wherewith they increase and are genera ted, a method hath been und to determine quantities stoin the velocities of their generat ing motions. And lacti velocities are called fluxions: and the quantities generaled are called Bowing quantities. These fluxions are Oid to benearly as the increments of the flo ing quantities, generaled in theleast equat particles of time; and to be accurately in the first proportion of the nascent, or in the last of the evanescent increments. Some- times, instead os velocities, the momentaneous increments Or decrements of undetermined flowing quantities are considered, under the appellation os momentS.IU. By moments we are not to understand finite particies. Theseare seid not to be momenis, but quantities generaled from momenis,

which last a re only the nascent principies of finite quantities. It is seid,

that the minutest errors are not to be neglected in mathematics: that thefluxions are celerities, not proportional to the finite increments thoughever so smali; but only to the moments or nascent incrementS, Where-

of the proportion alone, and not the magnitude, is considered. Andof the asoresaid fluxions there be other fluxions, whicli fluxions os fluxions are called second fluxions. And the fluxions of these secondfluxions are called third fluxions: and se on, urth, fifth, fixili, &c. ad in itum. NON as our sense is strained and puggled with the perception of Objects extremely minute, even so the imagination, whicli faculty derives froin sense, is very much strained and pu ZZled to, frame clear ideas of the least particles of time, or the least increments generaled therein: and much more se to comprehend the momenis, or those increments of the flowing quantities in satu notenti,

Introd. ad quadraturam curVarum.

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in their very first origin or beginning to exiit, besere they hecomefinite particles. Atid it stems stili more difficult to conceive theabstracted velocities of suci, nascent impersect entities. But the velocities of the velocities, the second, third, urth and fifth velocities,&c. exceed, is I mistahe not, ali human understanding. Τhe fur-ther the mind analyseth and pursveth these fugitive ideas, the more it is tost and hewildered, the objects, at first fleeting and minute seon vanishing out of sight. Certainly, in any sense, a second or third fluxion seems an obscure mystery. The incipient celeri ty of an incipient celeri ty, the nascent augment of a nascent augment, i. e. of a thing whichliath no magnitude ; take it in What light you please, the clear conception os it will, is I mistahe not, be sound impossibie: whether it he so orno I appeal to the trial of every thinhing reader. And is a second flux ion be inconceivabie, What are We to thin k of third, urth, fifth fluxions, and so onward without end tV. The foretgn mathematicians are supposed by some, even of ourown, to proceed in a manner lese accurate perhaps and geometrical, yet more intelligibie. Instead of flowing quantities and their fluxions, theyconsider the variabie finite quantities, as increa sing or dimini ming by the continuat addition or subduction of infinitely smali quantities. Instead

of the velocities Where illi increments are generaled, they consider the increments or decrements them selves, whicli they cali disserenoes, and

whicli are supposed to be infinitely smali. The disseretice of a line is an infinitely litile line; of a plane an infinitely litile plane. They suppose finite quantities to consist of paris infinitely litile, and curves to bepolygons, whereos the sides a re infinitely litile, which by the angies theymahe one With another determine the Curvi ty of the line. Now to conceive a quantity infinitely smali, that is, infinitely lese than any sensi bleor imaginable quantity, or any the least finite magnitude, is, I consess, a bove my capaci ty. But to conceive a part of such infinitely sin aliquantity,

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The Anal s. a 6 Iquantity, that stiali be stili infinitely lesi than it, and consequently thoughmultiplied infinitely mali never equat the minutest finite quantity, is, I suspect, an infinite dissiculty to any man what ever; and will heallowed lach by thoso who candidly say what they thinli , providedihey reatly thirili and reflect, and do not take things upon trust. VI. And yet in the calculus disserentialis, whicli method serves to allthe fame intenis and ends With that os fluxions, our modem analysis arenot content to consider only the differences of sinite quantities : they also consider the disserences of those disserenoes, and the differences of the disserences of the first disserenoes. And se on ad Disnitum. That is,

they consider quantities infinitely lese than the least discernible quantity iand others infinitely lest than those infinitely smali ones , and stili others

And which is most strange) although you mould take a million os million sos these infinitesiimal, each Whereos is supposed infinitely greater than me other real magnitude, and add them to the least given quantity, it mali be never the bigger. For this is orae of the modest posulata of our

modern mathematicians, and is a corner-stone or ground-work of their

speculations. .

VII. Ait these potnis, I say, are supposed and belle ved by certain rugorous exactors of evidence in religion, men who pretend to belleve nosurther than they can see. That men, Who have been conversant on lyabout clear potnis, sinould with difficulty admit obscure ones might notsecm altogether unaccountable. But he who cari digest a second or

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third fluxion, a secondor third disserence, need not, methinlis, be squeam- isti about any potnt in divinity. There is a natural presumption that mens faculties are made alike. It is on this supposition that they attempt to amgue and convince one another. What, there re, mali appear evidently impossibie and repugnant to one, may be presumed the fame to another.

But with what appea rance of reason mali any man presume to say, that mysteries may not be objects of faith, at the fame time that he himselfadmiis such obscure mysteries to be the object os science tVlII. It must indeed be aclino ledged, the modern mathematicians do not consider these potnis as mysteries, but as clearly conceived and mastered by their comprehensive minds. They scrupte not to say, that by the help of these new analytics they can penetrate into infinity itself: that they can even extend their views beyond infinity : that their artcomprehends not only infinite, but infinite of infinite as they express it or an infinity of infinites. But, notwithstanding ali these assertions and pretensions, it may be justly questioned whether, as other men in

other inquiries are osten deceived by words or terms, se they lihewiseare not wonderiasty deceived and deluded by 'their own peculiar signs, symbias, or species. Nothing is easier than to devise expressions or no lations sor fluxions and infinitesimais of the firit, second, third, soarth, and subsequent orders, proceed ing in the fame regular sorm Without

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ne Analbi. I 63 IX. Having considered the object, I proceed to consider the principies of this new analysis by momentums, fluxions, or infinitesimals ; where- in is it mali appear that your capital potnis, upon whicli the rest aresupposed to depend, include error and false rea ning ; it mill then fol-low that you, who are at a tost to conduct yourseives, cannot with any decency set up sor guides to other men. The main pol ni in the methodos fluxions is to obtain the fluxion or momentum of the rectangle orproduct of two indeterminate quantities. lnasmuch as froin thence are derived rules sor obtaining the fluxions os ali other products and powers ;be the coefficients or the indexes What they Will, integers or fractions, rational or lard. Now this fundamental potnt one would thinti mould bevery clearly ma de ovi, considering how much is bulli upon it, and that iis influence extends throughout the whole analysis. But let the readerjudge. This is gi ven for demonstration. ' Suppose the product or rectan- E A B increased by continuat motione and that the momentaneous increments of the sides A and B are a and b. When the sides A and Bwere deficient, or lesier by one half of their momenis, the rectanglewas a κ B- b i. e. A B-' a B - b AFI a b. And as seon as thesides A and B a re increased by the other two halves of their momenis,

the rectangle becomes AH . a X BF b or AB aB b A ab. Fromthe lalter rectangle subduct the sormer, and the remaining disserenoe willhe a Bib A. There re the increment of the rectangle generaled by the intire increments a and b is a Blb A. Q. E. D. But it is plain that thedirect and true method to obtain the moment or increment of the rectangle A B, is to take the sides as increa sed by their whiae incremenis, and se multiply them together, Aψ a by B b, the product whereos AB aB i b A i a b is the augmented rectangle : whence is we subduct A B, there mainder a Blb A s a b will be the true increment of the rectan gle, eX eding that which was obtained by the former illegitimate and indirect

' Naturalis philosophiae principia mathematica, l. 2. 4 m. a.

method

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ties es and b be what they wili, big or litile, finite or infinitesimal, incrementS, momenis, or velocities. Nor Will it avail to say that a b is aquantity exceed ing smali: since we are told that in rebus mathematicis frrores quam minimi non sunt contemnendi. X. Such rea ning as this for demonstration, nothing but the obscuri ty of the subject could have encouraged or induced the great author of the fluxionary method to put upon his follo ers, and nothingbut an implicit deserenoe to author ity could move them to admit. Thecase indoed is difficult. There can be nothing done tili you have got ridos the quantity a b. In order to this the notion os fluxions is misted :it is placed in various lights : potnts whicli mould be clear as first priniaci pies are pugHed ; and term s which should be Readily used are ambiguous. But notwith standing ait this address and stili the potnt of gettingrid os a b cannot be obtained by legitimate reaining. Is a man by methods, not geometrical or demonstrative, mali have satisfied himself of theusefulnest os certain rules ; which he aster ards mali propose to his dis cipies for undoubted truths; which he underlahes to demonstrate in asubiit manner, and by the help of nice and intricate notions ; it is nothard to conceive that suci, his discipies may, to save them selves the trou-ble of thini ing, he inclined to con und the u fulta est os a rule withthe certain ty of a truth, and accepi the one sor the other ; especialty isthey are men accustomed rather to compute than to thin k ; earnest ra-ther io go on fast and far, than solicitous to set out warily and see their

XI. The potnis or meer limits of nascent lines are undoubtedly equat,as having no more magnitude one than another. a limit as such being no 'quantity. Is by a momentum you mean more than the initiat limit, it

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quantities are expressy excluded Domithe notion os a momentum. There re the momentum must be an infinitesimah And indeed though much artifice hath been employed to escape or avoid the admisson os quantities infinitely small, yet it seems inessectual. Forought I see, you can admit no quantity as a medium belween a finite quantity and nothing, without admitting infinitesii mais. An increment generaled in a finite particle of time, is it self a finite par

ticle ; and cannot there re be a momentum. You must there re talio an infinitesimal part of timc wherein to generate yOur momentum.

It is seid, the magnitude of moments is not considered. And yetthese same moments are supposed to be divided into paris. Thisis not east to conceive, nor more than it is why we mould tahe quantities test than A and B in order to obta in the increment of A B, of whicli proceeding it must be o ned the final cause or motive is obvious , hut it is.not se obvious or easy to explain a justand legitimate reason sor it, or me it to be geometricat. XII. From the foregoing principie se demonstrated, the generat rulesor finding the suxion of any po er os a fio ing quantity is derived '. But, as there stems to have been seme in ard scrupte orconsci ou esse of defect in the soregoing demonstration, and as this finding the fiugion os a given poWer is a potnt of primary importance, it halli there re been judged proper to demonstrate the semein a disserent manner independent of the foregoing demonstration. But whether this other method be more legitimate and conclusi vethan the former, I proceed now to examine; and in order thereto stati premi se the following lemma. Is with a vieis to demonstrate' any proposition, a certain pol ni is supposed, by virtve of which certain other potnis are attained , and lacti supposed potnt be itself

Philosophiae naturalis principia mathematica, lib. 2. Iem. 2.

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last proportion will he 1 to n xη- . Eut it mould seem that this rea- ning is not fair Or conclusive. For when it is seid, let the incremenis Vanish, i. e. let the increments be nothing, or let there he noincremenis, the former supposition that the increments were sonaething, or that there were incremenis, is destroyed, and yet a consequence of that supposition, i. e. an expression got by virtve thereos,

is retained. Whicli, by the soregoing lemma, is a false way of rea- ning. Certa inly when we suppose the increments to vanisti, wemust suppose their proportions, their expressions4 and e very thing et sedcrived from the supposition of their existence to vanish with them. XIV. To malle this poliat plainer, I mali uti sold the rea soning, and propose it in a fuller light to your vi eis. It amounts there re tot his, or may in other words be thus expressed I suppose that thequantity x flows, and by fio ing is increased, and iis increment ICall

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tae. And is hom the two augmented quantities we subduct theroot and the power respectively, we mali have rema ining the two in-

menis, being both divided by the common divisor o, yield the quoti-

of the ratio of the increments. Vitherio I have supposed that x flows, that x hath a reat increment that o is semething. And I have proceededali along on that supposition, without whicli I smould not have been ableto have made so much as one single st . From that supposition it is that I get at the increment of that I am able to compare it with the increment of x, and that I find the proportion belween the two increments. I now beg leave to malie a new supposition contrary to the first, i. e. I wili suppost that there is no increment of x, or that o is no thing ; whicli second supposition destroys my first, and is inconsistent with it, and there re with every thing that supposeth it. I do neveriheless heg leave to retain n xn - I, whicli is an expression obtained in virtve of my firsi supposition, whicli necessarily pre-supposed such supposition, and which could not be obtained without it. All which seems amost inconsistent Way of arguing, and such as would not be allowed of in XV. Nothing is pia iner than that no just conclusion can be directlydra n stom two inconsistent suppositions. You may indeed suppose any

thing

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stroys what you first supposed. Or is you do, you must hegin de novo. Is therelare you suppost that tho augments vanish, i. e. that there are noaugments, you are to begin again, and see what follows Dom fuch supposition. But nothing wili sello to your purpose. You cannot by that means CVer arrive at your Conclusion, or succeed in, What is called by the celebrated author, the investigation of the first or last proportions os nascent and evanescent quantities, by instituting the analysis in siniteones. I repeat it again : you are at liberty to mahe any possibie supposition : and you may destroy one supposition by another: but then youmay not retain the consequences, or any part of the consequences of your first supposition se destroyed. Ι admit that signs may be made to denote ei ther any thing or nothing: and consequently that in the original notation x - - o, o might have signified either an increment or nothin: But then whicli of these. ever you malie it signisy, you mustargue conssistently with lach iis signification, and not proceed upon adorabie meaning : which to do were a manifest sophism. Whether youargue in symbolf or in words, the rules of right rea n are stili the fame. Nor can it be supposed, you will plead a privilege in mathematiclis to be

XVI. Is you assume at first a quantity increased by nothing, and in

the expression xl o, o stands for nothing, upon this supposition as thereis no increment of the root, so there Will he no increment of the power; and consequently there will be none except the first, of ali those mem-hers of the series constituting the power of the binomial ; and willthere re never come to your expression of a fluxion legitimately bysuch method. Hence you are dri ven into the fallacious way of pro- ceeding to a certain potnt on the supposition os an increment, and then at once staming your supposition to that of no increment. There

may stem great skill in doing this at a certain pol ni or period. Since isthis