The Analyst: a Discourse addressed to an Infidel Mathematician

50. These modern Analytics were not scientific. I expressed this to the Public about 25 Years ago.

Recently, I have been called on to make good my Suggestions; yet as the Person, who made this Call, doth not appear to think maturely enough to understand, either those Metaphysics which he would refute, or Mathematics which he would patronize, I should have spared my self the trouble of writing for his Conviction.

Nor should I now have troubled you or my self with this Address, after so long an Intermission of these Studies; were it not to prevent, so far as I am able, your imposing on your self and others in Matters of much higher Moment and Concern.

And to the end that you may more clearly comprehend the Force and Design of the foregoing Remarks, and pursue them still further in your own Meditations, I shall subjoin the following Queries.

Query 1. Whether the Object of Geometry be not the Proportions of assignable Extensions? And whether, there be any need of considering Quantities either infinitely great or infinitely small?

Qu. 2. Whether the end of Geometry be not to measure assignable finite Extension? And whether this practical View did not first put Men on the study of Geometry?

Qu. 3. Whether the mistaking the Object and End of Geometry hath not created needless Difficulties, and wrong Pursuits in that Science?

Qu. 4. Whether Men may properly be said to proceed in a scientific Method, without clearly conceiving the Object they are conversant about, the End proposed, and the Method by which it is pursued?

Qu. 5. Whether it doth not suffice, that every assignable number of Parts may be contained in some assignable Magnitude? And whether it be not unnecessary, as well as absurd, to suppose that finite Extension is infinitely divisible?

Qu. 6. Whether the Diagrams in a Geometrical Demonstration are not to be considered, as Signs of all possible finite Figures, of all sensible and imaginable Extensions or Magnitudes of the same kind?

Qu. 7. Whether it be possible to free Geometry from insuperable Difficulties and Absurdities, so long as either the abstract general Idea of Extension, or absolute external Extension be supposed its true Object?

Qu. 8. Whether the Notions of absolute Time, absolute Place, and absolute Motion be not most abstractedly Metaphysical? Whether it be possible for us to measure, compute, or know them?

Qu. 9. Whether Mathematicians do not engage themselves in Disputes and Paradoxes, concerning what they neither do nor can conceive? And whether the Doctrine of Forces be not a sufficient Proof of this?[18]

Qu. 10. Whether in Geometry it may not suffice to consider assignable finite Magnitude, without concerning our selves with Infinity? And whether it would not be righter to measure large Polygons having finite Sides, instead of Curves, than to suppose Curves are Polygons of infinitesimal Sides, a Supposition neither true nor conceivable?

Qu. 11. Whether many Points, which are not readily assented to, are not nevertheless true? And whether those in the two following Queries may not be of that Number?

Qu. 12. Whether it be possible, that we should have had an Idea or Notion of Extension prior to Motion? Or whether if a Man had never perceived Motion, he would ever have known or conceived one thing to be distant from another?

Qu. 13. Whether Geometrical Quantity hath coexistent Parts? And whether all Quantity be not in a flux as well as Time and Motion?

Qu. 14. Whether Extension can be supposed an Attribute of a Being immutable and eternal?

Qu. 15. Whether to decline examining the Principles, and unravelling the Methods used in Mathematics, would not shew a bigotry in Mathematicians?

Qu. 16. Whether certain Maxims do not pass current among Analysts, which are shocking to good Sense? And whether the common Assumption that a finite Quantity divided by nothing is infinite be not of this Number?

Qu. 17. Whether the considering Geometrical Diagrams absolutely or in themselves, rather than as Representatives of all assignable Magnitudes or Figures of the same kind, be not a principal Cause of the supposing finite Extension infinitely divisible; and of all the Difficulties and Absurdities consequent thereupon?

Qu. 18. Whether from Geometrical Propositions being general, and the Lines in Diagrams being therefore general Substitutes or Representatives, it doth not follow that we may not limit or consider the number of Parts, into which such partiticular Lines are divisible?

Qu. 19. When it is said or implied, that such a certain Line delineated on Paper contains more than any assignable number of Parts, whether any more in truth ought to be understood, than that it is a Sign indifferently representing all finite Lines, be they ever so great. In which relative Capacity it contains, i. e. stands for more than any assignable number of Parts? And whether it be not altogether absurd to suppose a finite Line, considered in it self or in its own positive Nature, should contain an infinite number of Parts?

Qu. 20. Whether all Arguments for the infinite Divisibility of finite Extension do not suppose and imply, either general abstract Ideas or absolute external Extension to be the Object of Geometry? And, therefore, whether, along with those Suppositions, such Arguments also do not cease and vanish?

Qu. 21. Whether the supposed infinite Divisibility of finite Extension hath not been a Snare to Mathematicians, and a Thorn in their Sides? And whether a Quantity infinitely diminished and a Quantity infinitely small are not the same thing?

Qu. 22. 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?

Qu. 24. Whether a Quantity be not properly said to be known, when we know its Proportion to given Quantities? And whether this Proportion can be known, but by Expressions or Exponents, either Geometrical, Algebraical, or Arithmetical? And whether Expressions in Lines or Species can be useful but so far forth as they are reducible to Numbers?

25. Whether the finding out proper Expressions or Notations of Quantity be not the most general Character and Tendency of the Mathematics? And Arithmetical Operation that which limits and defines their Use?

Qu. 26. Whether Mathematicians have sufficiently considered the Analogy and Use of Signs? And how far the specific limited Nature of things corresponds thereto?

Qu. 27. Whether because, in stating a general Case of pure Algebra, we are at full liberty to make a Character denote, either a positive or a negative Quantity, or nothing at all, we may therefore in a geometrical Case, limited by Hypotheses and Reasonings from particular Properties and Relations of Figures, claim the same Licence?

Qu. 28. Whether the Shifting of the Hypothesis, or (as we may call it) the fallacia Suppositionis be not a Sophism, that far and wide infects the modern Reasonings, both in the mechanical Philosophy and in the abstruse and fine Geometry?

Qu. 29. 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? And whether this be not supposed in the Reasonings of modern Analysts?

Qu. 30. Whether Motion can be conceived in a Point of Space? And if Motion cannot, whether Velocity can? And if not, whether a first or last Velocity can be conceived in a mere Limit, either initial or final, of the described Space?

Qu. 31. Where there are no Increments, whether there can be any Ratio of Increments? Whether Nothings can be considered as proportional to real Quantities? Or whether to talk of their Proportions be not to talk Nonsense? Also in what Sense we are to understand the Proportion of a Surface to a Line, of an Area to an Ordinate? And whether Species or Numbers, though properly expressing Quantities which are not homogeneous, may yet be said to express their Proportion to each other?

Qu. 32. Whether if all assignable Circles may be squared, the Circle is not, to all intents and purposes, squared as well as the Parabola? Or whether a parabolical Area can in fact be measured more accurately than a Circular?

Qu. 33. Whether it would not be righter to approximate fairly, than to endeavour at Accuracy by Sophisms?

Qu. 34. Whether it would not be more decent to proceed by Trials and Inductions, than to pretend to demonstrate by false Principles?

Qu. 35. Whether there be not a way of arriving at Truth, although the Principles are not scientific, nor the Reasoning just? And whether such a way ought to be called a Knack or a Science?

Qu. 36. 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?

Qu. 37. Whether the greatest Genius wrestling with false Principles may not be foiled? And whether accurate Quadratures can be obtained without new Postulata or Assumptions? And if not, whether those which are intelligible and consistent ought not to be preferred to the contrary? See Sect. XXVIII and XXIX.

Qu. 38. Whether tedious Calculations in Algebra and Fluxions be the likliest Method to improve the Mind? And whether Mens being accustomed to reason altogether about Mathematical Signs and Figures, doth not make them at a loss how to reason without them?

Qu. 39. Whether, whatever readiness Analysts acquire in stating a Problem, or finding apt Expressions for Mathematical Quantities, the same doth necessarily infer a proportionable ability in conceiving and expressing other Matters?

Qu. 40. Whether it be not a general Case or Rule, that one and the same Coefficient dividing equal Products gives equal Quotients? 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?

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? And whether such their Reasonings are not deduced from the same Axioms with those in Geometry? Whether therefore Algebra be not as truly a Science as Geometry?

Qu. 42. Whether Men may not reason in Species as well as in Words? Whether the same Rules of Logic do not obtain in both Cases? And whether we have not a right to expect and demand the same Evidence in both?

Qu. 43. Whether an Algebraist, Fluxionist, Geometrician or Demonstrator of any kind can expect indulgence for obscure Principles or incorrect Reasonings? And whether an Algebraical Note or Species can at the end of a Process be interpreted in a Sense, which could not have been substituted for it at the beginning? Or whether any particular Supposition can come under a general Case which doth not consist with the reasoning thereof?

Qu. 44. Whether the Difference between a mere Computer and a Man of Science be not, that the one computes on Principles clearly conceived, and by Rules evidently demonstrated, whereas the other doth not?

Qu. 45. Whether, although Geometry be a Science, and Algebra allowed to be a Science, and the Analytical a most excellent Method, in the Application nevertheless of the Analysis to Geometry, Men may not have admitted false Principles and wrong Methods of Reasoning?

Qu. 46. Whether although Algebraical Reasonings are admitted to be ever so just, when confined to Signs or Species as general Representatives of Quantity, you may not nevertheless fall into Error, if, when you limit them to stand for particular things, you do not limit your self to reason consistently with the Nature of such particular things? And whether such Error ought to be imputed to pure Algebra?