The Analyst: a Discourse addressed to an Infidel
Mathematician
By George Berkeley
Contents
1. Mathematicians presumed to be the great masters of reason. Hence an undue deference to their decisions where they have
no right to decide. This one cause of infidelity.
2. Their principles and methods to be examined with the same freedom which they assume with regard to the principles and
mysteries of religion. In what sense and how far geometry is to be allowed an improvement of the mind.
3. Fluxions the great object and employment of the profound geometricians in the present age. What these fluxions are.
4. Moments or nascent increments of flowing quantities difficult to conceive. Fluxions of different orders. Second and third
fluxions obscure mysteries.
5. Differences, i.e. increments or decrements infinitely small, used by foreign mathematicians instead of fluxions or velocities of
nascent and evanescent increments.
6. Differences of various orders, i.e. quantities infinitely less than quantities infinitely little; and infinitesimal parts of infinitesimals
of infinitesimals, &c. without end or limit.
7. Mysteries of faith unjustly objected against by those who admit them in science.
8. Modern Analysts supposed by themselves to extend their views even beyond infinity: deluded by their own species or
symbols.
9. Method for finding the fluxion of a rectangle of two indeterminate quantities, shewed to be illegitimate and false.
10. Implicit deference of mathematical men for the great author of fluxions. Their earnestness rather to go on fast and far, than
to set out warily and see their way distinctly.
11. Momentums difficult to comprehend. No middle quantity to be admitted between a finite quantity and nothing, without
admitting infinitesimals.
12. The fluxion of any power of a flowing quantity. Lemma premised in order to examine the method for finding such fluxion.
13. The rule for fluxions of powers attained by unfair reasoning.
14. The aforesaid reasoning farther unfolded, and shewed to be illogical.
15. No true conclusions to be justly drawn by direct consequence from inconsistent suppositions. The same rules of right
reason to be observed, whether men argue in symbols or in words.
16. An hypothesis being destroyed, no consequence of such hypothesis to be retained.
17. Hard to distinguish between evanescent increments and infinitesimal differences. Fluxions placed in various lights. The great
author, it seems, not satisfied with his own notions.
18. Quantities infinitely small supposed and rejected by Leibnitz and his followers. No quantity, according to them, greater or
smaller for the addition or subduction of its infinitesimal.
19. Conclusions to be proved by the principles, and not principles by the conclusions.
20. The geometrical Analyst considered as a logician; and his discoveries, not in themselves, but as derived from such
principles and by such inferences.
21. A tangent drawn to the parabola according to the calculus differentialis. Truth shewn to be the result of error, and how.
22. By virtue of a twofold mistake Analysts arrive at truth, but not at science: ignorant how they come at their own conclusions.
23. The conclusion never evident or accurate, in virtue of obscure or inaccurate premises. Finite quantities might be rejected as
well as infinitesimals.
24. The foregoing doctrine farther illustrated.
25. Sundry observations thereupon.
26. Ordinate found from the area by means of evanescent increments.
27. In the foregoing case, the supposed evanescent increment is really a finite quantity, destroyed by an equal quantity with an
opposite sign.
28. The foregoing case put generally. Algebraical expressions compared with the geometrical quantities.
29. Correspondent quantities algebraical and geometrical equated. The analysis shewed not to obtain in infinitesimals, but it
must also obtain in finite quantities.
30. The getting rid of quantities by the received principles, whether of fluxions or of differences, neither good geometry nor
good logic. Fluxions or velocities, why introduced.
31. Velocities not to be abstracted from time and space: nor their proportions to be investigated or considered exclusively of
time and space.
32. Difficult and obscure points constitute the principles of the modern Analysis, and are the foundation on which it is built.
33. The rational faculties whether improved by such obscure analytics.
34. By what inconceivable steps finite lines are found proportional to fluxions. Mathematical infidels strain at a gnat and
swallow a camel.
35. Fluxions of infinitesimals not to be avoided on the received principles. Nice abstractions and geometrical metaphysics.
36. Velocities of nascent or evanescent quantities, whether in reality understood and signified by finite lines and species.
37. Signs or exponents obvious; but fluxions themselves not so.
38. Fluxions, whether the velocities with which infinitesimal differences are generated.
39. Fluxions of fluxions or second fluxions, whether to be conceived as velocities of velocities, or rather as velocities of the
second nascent increments.
40. Fluxions considered, sometimes in one sense, sometimes in another; one while in themselves, another in their exponents:
hence confusion and obscurity.
41. Isochronal increments, whether finite or nascent, proportional to their respective velocities.
42. Time supposed to be divided into moments: increments generated in those moments: and velocities proportional to those
increments.
43. Fluxions, second, third, fourth, &c., what they are, how obtained, and how represented. What idea of velocity in a
moment of time and point of space.
44. Fluxions of all orders inconceivable.
45. Signs or exponents confounded with the fluxions.
46. Series of expressions or of notes easily contrived. Whether a series of mere velocities, or of mere nascent increments
corresponding thereunto, be as easily conceived.
47. Celerities dismissed, and instead thereof ordinates and areas introduced. Analogies and expressions useful in the modern
quadratures, may yet be useless for enabling us to conceive fluxions. No right to apply the rules without knowledge of the
principles.
48. Metaphysics of modern Analysts most incomprehensible.
49. Analysts employed about notional shadowy entities. Their logics as exceptionable as their metaphysics.
50. Occasion of this address. Conclusion. Queries.
THE ANALYST:
A Discourse addressed to an Infidel Mathematician
1. Though I am a stranger to your person, yet I am not, Sir, a stranger to the reputation you have acquired in that branch of
learning which hath been your peculiar study; nor to the authority that you therefore assume in things foreign to your profession;
nor to the abuse that you, and too many more of the like character, are known to make use of such undue authority, to the
misleading of unwary persons in matters of the highest concernment, and whereof your mathematical knowledge can by no
means qualify you to be a competent judge. Equity indeed and good sense would incline one to disregard the judgement of
men, in points which they have not considered or examined. 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 men's opinions, and putting on
a general deference for the judgement of you, Gentlemen, who are presumed to be of all men the greatest masters of reason, to
be most conversant about distinct ideas, and never to take things on 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. With this bias on their
minds, they submit to your decisions where you have no right to decide. And that this is one short way of making Infidels, I am
credibly informed.
2. Whereas then it is supposed that you apprehend more distinctly, consider more closely, infer more justly, and conclude
more accurately than other men, and that you are therefore less religious because more judicious, I shall claim the privilege of a
Free-thinker; and take the liberty to inquire into the object, principles, and method of demonstration admitted by the
mathematicians of the present age, with the same freedom that you presume to treat the principles and mysteries of Religion; to
the end that all men may see what right you have to lead, or what encouragement others have to follow you. It hath been an old
remark, that Geometry is an excellent Logic. 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 a 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. But how far this is the case of our
geometrical analysts, it may be worth while to consider.
3. 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 ...
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