George Berkeley

“ As you, who are a skilful Computist or Analyst, may not therefore be deemed skilful in Anatomy: or vice versa, as a Man who can dissect with Art, may, nevertheless, be ignorant in your Art of computing Even so you may both, notwithstanding your peculiar Skill in your respective Arts, be alike unqualified to decide upon Logic, or Metaphysics, or Ethics, or Religion. ”

“ The Velocities or Fluxions are said to primò and ultimò, as the Augments nascent and evanescent. Take therefore the Ratio of the evanescent Quantities, it is the same with that of the Fluxions. It will therefore answer all Intents as well. Why then are Fluxions introduced? Is it not to shun or rather to palliate the Use of Quantities infinitely small? But we have no Notion whereby to conceive and measure various Degrees of Velocity, beside Space and Time, or when the Times are given, beside Space alone. We have even no Notion of Velocity prescinded from Time and Space. ”

“ Whether there can be Science of the Conclusion, where there is not Science of the Principles? And whether a Man can have Science of the Principles, without understanding them? And therefore whether the Mathematicians of the present Age act like Men of Science, in taking so much more pains to apply their Principles, than to understand them? ”

“ I observe in the last place, that in case the Differences are supposed finite Quantities ever so great, the Conclusion will nevertheless come out the same: inasmuch as the rejected Quantities are legitimately thrown out, not for their smallness, but for another reason, to wit, because of contrary errors, which destroying each other do upon the whole cause that nothing is really, though something is apparently thrown out. ”

“ Now if we are content to come at the Conclusion in a summary way, by supposing that the Ratio of the Fluxions of x and xn are found [1] to be 1 and , and that the Ordinate of the Area is considered as its Fluxion; we shall not so clearly see our way, or perceive how the truth comes out, that Method as we have shewed before being obscure and illogical. But if we fairly delineate the Area and its Increment, and divide the latter into two Parts BCFD and CFH [2] and proceed regularly by Equations between the algebraical and geometrical Quantities, the reason of the thing will plainly appear. ”

“ Whether those Philomathematical Physicians, Anatomists, and Dealers in the Animal Oeconomy, who admit the Doctrine of Fluxions with an implicit Faith, can with a good grace insult other Men for believing what they do not comprehend? ”

“ IX. Having considered the Object, I proceed to consider the Principles of this new Analysis by Momentums, Fluxions, or Infinitesimals; wherein if it shall appear that your capital Points, upon which the rest are supposed to depend, include Error and false Reasoning; it will then follow that you, who are at a loss to conduct your selves, cannot with any decency set up for guides to other Men. The main Point in the method of Fluxions is to obtain the Fluxion or Momentum of the Rectangle or Product of two indeterminate Quantities. ”

“ It is therefore plain, that the supposing the rejectaneous algebraical Quantity to be an infinitely small or evanescent Quantity, and therefore to be neglected, must have produced an Error, had it not been for the curvilinear Spaces being equal thereto, and at the same time subducted from the other Part or Side of the Equation agreeably to the Axiom, If from Equals you subduct Equals, the Remainders will be equal. For those Quantities which by the Analysts are said to be neglected, or made to vanish, are in reality subducted. ”

“ An Increment generated in a finite Particle of Time, is it self a finite Particle; and cannot therefore be a Momentum. You must therefore take an Infinitesimal Part of Time wherein to generate your Momentum. It is said, the Magnitude of Moments is not considered: And yet these same Moments are supposed to be divided into Parts. This is not easy to conceive, no more than it is why we should take Quantities less than A and B in order to obtain the Increment of AB, of which proceeding it must be owned the final Cause or Motive is very obvious ”

“ Whether it be necessary to consider Velocities of nascent or evanescent Quantities, or Moments, or Infinitesimals? And whether the introducing of Things so inconceivable be not a reproach to Mathematics?
Qu. 23. Whether Inconsistencies can be Truths? Whether Points repugnant and absurd are to be admitted upon any Subject, or in any Science? And whether the use of Infinites ought to be allowed, as a sufficient Pretext and Apology, for the admitting of such Points in Geometry? ”

“ Now as our Sense is strained and puzzled with the perception of Objects extremely minute, even so the Imagination, which Faculty derives from Sense, is very much strained and puzzled to frame clear Ideas of the least Particles of time, or the least Increments generated therein: and much more so to comprehend the Moments, or those Increments of the flowing Quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite Particles. And it seems still more difficult, to conceive the abstracted Velocities of such nascent imperfect Entities. ”

“ There can be nothing done till you have got rid of the Quantity ab. In order to this the Notion of Fluxions is shifted: It is placed in various Lights: Points which should be clear as first Principles are puzzled' and Terms which should be steadily used are ambiguous. But notwithstanding all this address and skill the point of getting rid of ab cannot be obtained by legitimate reasoning. If a Man by Methods, not geometrical or demonstrative, shall have satisfied himself of the usefulness of certain Rules ”

“ But several who make the loudest Claim to those Qualities, do, nevertheless, the very thing they would seem to despise, clothing themselves in the Livery of other Mens opinions, and putting on a general deference for the Judgment of you, Gentlement, who are presumed to be of all Men the greatest Masters of Reson, to be most conversant about distinct Ideas, and never to take things upon trust, but always clearly to see your way, as Men whose constant Employment is the deducing Truth by the justest inference from the most evident Principles. ”

“ It should seem therefore upon the whole, that your Conclusions are not attained by just Reasoning from clear Principles; consequently that the Employment of modern Analysts, however useful in mathematical Calculations, and Constructions, doth not habituate and qualify the Mind to apprehend clearly and infer justly; and consequently, that you have no right in Virtue of such Habits, to dictate out of your proper Sphere, beyond which your Judgment is to pass for no more than that of other Men. ”

“ Nor should any Man, in virtue of being conversant in such obscure Analytics, imagine his rational Faculties to be more improved than those of other Men, which have been exercised in a different manner, and on different Subjects; much less erect himself into a Judge and an Oracle, concerning Matters that have no sort of connexion with, or dependence on those Species, Symbols or Signs, in the Management whereof he is so converfant and expert. ”

“ IV. By Moments we are not to understand finite Particles. These are said not to be Moments, but Quantities generated from Moments, which last are only the nascent Principles of finite Quantities. It is said, that the minutest Errors are not to be neglected in Mathematics: that the Fluxions are Celerities, not proportional to the finite Increments though ever so small; but only to the Moments or nascent Increments, whereof the Proportion alone, and not the Magnitude, is considered. And of the aforesaid Fluxions there be other Fluxions, which Fluxions of Fluxions are called second Fluxions. ”

“ So on the other hand it would be equally wrong to imagine, that in a geometrical Demonstration we may be allowed to admit any Error, though ever so small, or that it is possible, in the nature of Things, an accurate Conclusion should be derived from inaccurate Principles. Therefore o cannot be thrown out as an Infinitesimal, or upon the Principle that Infinitesimals may be safely neglected. But only because it is destroyed by an equal Quantity with a negative Sign, whence o - qo is equal to nothing. ”

“ XLV. One would think that Men could not speak too exactly on so nice a Subject. And yet, as was before hinted, we may often observe that the Exponents of Fluxions or Notes representing Fluxions are confounded with the Fluxions themselves. Is not this the Case, when just after the Fluxions of flowing Quantities were said to be the Celerities of their increasing, and the second Fluxions to be the mutations of the first Fluxions or Celerities, we are told that [1] represents a Series of Quantities, whereof each subsequent Quantity is the Fluxion of the preceding ”

“ I observe in the second place, that whatever is rejected, be it ever so small, if it be real and consequently makes a real error in the Premises, it will produce a proportional real error in the Conclusion. Your Theorems therefore cannot be accurately true, nor your Problems accurately solved, in virtue of Premises, which themselves are not accurate, it being a rule in Logic that Conclusio sequitur partem debiliorem. ”

“ And forasmuch as it may perhaps seem an unaccountable Paradox, that Mathematicians should deduce true Propositions from false Principles, be right in the Conclusion, and yet err in the Premises; I shall endeavour particularly to explain why this may come to pass, and shew how Error may bring forth Truth, though it cannot bring forth Science.
XXI. In order therefore to clear up this Point, we will suppose for instance that a Tangent is to be drawn to a Parabola, and examine the progress of this Affair, as it is performed by infinitesimal Differences. ”

“ And it must be owned, that when the Definitions are clear; when the Postulata cannot be refused, nor the Axioms denied; when from the distinct Contemplation and Comparison of Figures, their Properties are derived, by a perpetual well-connected chain of Consequences, the Objects being still kept in view, and the attention ever fixed upon them; there is acquired an habit of reasoning, close and exact and methodical: which habit strengthens and sharpens the Mind, and being transferred to other Subjects, is of general use in the inquiry after Truth. ”

“ V. The foreign Mathematicians are supposed by some, even of our own, to proceed in a manner, less accurate perhaps and geometrical, yet more intelligible. Instead of flowing Quantities and their Fluxions, they consider the variable finite Quantities, as increasing or diminishing by the continual Addition or Subduction of infinitely small Quantities. Instead of the Velocities wherewith Increments are generated, they consider the Increments or Decrements themselves, which they call Differences, and which are supposed to be infinitely small. The Difference of a Line is an infinitely little Line ”

“ III. The Method of Fluxions is the general Key, by help whereof the modern Mathematicians unlock the secrets of Geometry, and consequently of Nature. And as it is that which hath enabled them so remarkably to outgo the Ancients in discovering Theorems and solving Problems, the exercise and application thereof is become the main, if not sole, employment of all those who in this Age pass for profound Geometers. ”

“ But, notwithstanding all these Assertions and Pretensions, it may be justly questioned whether, as other Men in other Inquiries are often deceived by Words or Terms, so they likewise are not wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species. Nothing is easier than to devise Expressions or Notations for Fluxions and Infinitesimals of the first, second, third, fourth and subsequent Orders, proceeding in the same regular form without end or limit &c. ”

“ Whether from this, and other concurring Causes, the Minds of speculative Men have not been born downward, to the debasing and stupifying of the higher Faculties? And whether we may not hence account for that prevailing Narrowness and Bigotry among many who pass for Men of Science, their Incapacity for things Moral, Intellectual, or Theological, their Proneness to measure all Truths by Sense and Experience of animal Life?
Qu. 58. Whether it be really an Effect of Thinking, that the same Men admire the great Author for his Fluxions, and deride him for his Religion? ”

“ But if this Point be throughly considered, we shall find there is even here a double mistake, and that one compensates or rectifies the other. For in the first place, it was supposed, that when NO is infinitely diminished or becomes an Infinitesimal, then the Subsecant NM becomes equal to the Subtangent NL. But this is a plain mistake, for it is evident, that as a Secant cannot be a Tangent, so a Subsecant cannot be a Subtangent. Be the Difference ever so small, yet still there is a Difference. ”

“ Whether we can form an Idea or Notion of Velocity distinct from and exclusive of its Measures, as we can of Heat distinct from and exclusive of the Degrees on the Thermometer, by which it is measured? ”

“ Leibnitz and his Followers in their calculus differentialis making no manner of scruple, first to suppose, and secondly to reject Quantities infinitely small: with what clearness in the Apprehension and justness in the reasoning, any thinking Man, who is not prejudiced in favour of those things, may easily discern. ”

“ Or whether any one will say, that if the Equation 2 × o = 5 × o, be divided by o, the Quotients on both Sides are equal? Whether therefore a Case may not be general with respect to all Quantities, and yet not extend to Nothings, or include the Case of Nothing? And whether the bringing Nothing under the Notion of Quantity may not have betrayed Men into false Reasoning?
Qu. 41. Whether in the most general Reasonings about Equalities and Proportions, Men may not demonstrate as well as in Geometry? Whether in such Demonstrations, they are not obliged to the same strict Reasoning as in Geometry? ”

“ They are the Foundations on which the Moderns build, the Principles on which they proceed, in solving Problems and discovering Theorems. It is with the Method of Fluxions as with all other Methods, which presuppose their respective Principles and are grounded thereon. Although the Rules may be practised by Men who neither attend to, nor perhaps know the Principles. ”