https://www.superphysics.org/research/berkeley/analyst/ Recent content in The Analyst: a Discourse addressed to an Infidel Mathematician on Superphysics Hugo en https://www.superphysics.org/research/berkeley/analyst/section-001/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-001/ <!-- It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith. By the Author of The Minute Philosopher. --> <!-- First cast out the beam out of thine own Eye; and then shalt thou see clearly to cast out the mote out of thy brother's eye.S. Matt. c. vii. v. 5. LONDON: Printed for J. Tonson in the Strand. 1734. Ornamentation for p7 of Berkeley's 'The analyst' --> <!-- SECT. 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 Quntities infinitely little; and infinitesimal Parts, of infinitesimals of infinitesimals, &. without end or limit. 7. Mysteries in 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 Mathematicalmen for the great Author of Fluxions. Their earnestiness 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 the Fluxions of Powers attained by unfair reasoning. 14. The aforesaid reasoning farther unfolded and shew'd to be illogical. 15. No true Conclusion 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 porved 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 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 Geometrical Quantities. 29. Correspondent Quanities Algebraical and Geomtrical equated. The Analysis shewed not to obtain in Infintesimals, 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 Analysi, 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 or 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, cerresponding 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. Analysis employ'd about notional shadowy Entities. Their Logics as exceptionable as their Metaphysics. 50. Occasion of this Address. Conclusion. Queries. --> <!-- 1. 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 Charader, are known to make 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 Judgment 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 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. 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, 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 older remark that Geometry is an excellent Logic. 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. But how far this is the case of our Geometrical Analysts, it may be worth while to consider. --> <ol start="3"> <li>The Method of Fluxions is the general Key that modern Mathematicians use to unlock the secrets of Geometry, and consequently of Nature.</li> </ol> <!-- 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 whether this Method be clear or obscure, consistent or repugnant, demonstrative or precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own Judgment, and that of every candid Reader. --> <ul> <li>Lines are supposed to be generated[1] by the motion of Points</li> <li>Planes by the motion of Lines</li> <li>Solids by the motion of Planes</li> </ul> <p>Quantities generated in equal times are greater or lesser, according to the greater or lesser Velocity, wherewith they increase and are generated, a Method hath been found to determine Quantities from the Velocities of their generating Motions.</p> https://www.superphysics.org/research/berkeley/analyst/section-007/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-007/ <p>⁷ All these Points are believed by Men who believe only what they see.</p> <!-- certain rigorous Exactors of Evidence in Religion, who pretend to believe no further than they can see. --> <p>They hesitate to admit that there might be obscure Points.</p> https://www.superphysics.org/research/berkeley/analyst/section-015/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-015/ <p>¹⁵ No just Conclusion can be directly drawn from 2 inconsistent Suppositions.</p> <p>You cannot suppose any thing that destroys what you first supposed. If you do, you must begin de novo.</p> https://www.superphysics.org/research/berkeley/analyst/section-024/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-024/ <ol start="24"> <li>For the fuller illustration of this Point, I shall consider it in another light, and proceeding in finite Quantities to the Conclusion, I shall only then make use</li> </ol> <!-- P37 diagram for Berkeley's 'The Analyst'.png --> <p>of one Infinitesimal.</p> https://www.superphysics.org/research/berkeley/analyst/section-030/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-030/ <p>³⁰ On the whole, the Conclusion cannot be right:</p> <ol> <li>If in order thereto any Quantity be made to vanish, or be neglected, except that either one Error is redressed by another; or</li> <li>On the same Side of an Equation equal Quantities are destroyed by contrary Signs, so that the Quantity we mean to reject is first annihilated; or lastly, that from the opposite Sides equal Quantities are subducted.</li> </ol> <p>Therefore to get rid of Quantities by the received Principles of Fluxions or of Differences is neither good Geometry nor good Logic. When the Augments vanish, the Velocities also vanish.</p> https://www.superphysics.org/research/berkeley/analyst/section-037/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-037/ <p>³⁷ It is easiest to assign Names, Signs, or Expressions to these Fluxions.</p> <p>It is not difficult to compute and operate by means of such Signs.</p> <p>But it is much more difficult to omit the Signs and yet retain in our Minds the things which we suppose to be signified by them.</p> https://www.superphysics.org/research/berkeley/analyst/section-050/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-050/ <ol start="50"> <li>These modern Analytics were not scientific. I expressed this to the Public about 25 Years ago.</li> </ol> <!-- Since which time, I have been diverted by other Occupations, and imagined I might employ my self better than in deducing and laying together my Thoughts on so nice a Subject. --> <p>Recently, I have been called on to make good my Suggestions.</p> https://www.superphysics.org/research/berkeley/analyst/section-050b/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/berkeley/analyst/section-050b/ <h4 id="query--40-whether-it-be-not-a-general-case-or-rule-that-one-and-the-same-coefficient-dividing-equal-products-gives-equal-quotients">Query 40. Whether it be not a general Case or Rule, that one and the same Coefficient dividing equal Products gives equal Quotients?</h4> <p>And yet whether such Coefficient can be interpreted by o or nothing? 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?</p>