40.If you defend Sir Isaac's notions,as delivered in his'Principia,'it must be on the rigorous foot of rejecting nothing,neither admitting nor casting away infinitely small quantities.If you defend the Marquis,whom you also style your Master,it must be on the foot of admitting that there are infinitesimals,that they may be rejected,that they are nevertheless real quantities,and themselves infinitely subdivisible.
But you seem to have grown giddy with passion,and in the heat of controversy to have mistaken and forgot your part.I beseech you,Sir,to consider that the Marquis (whom alone,and not Sir Isaac,this double error in finding the subtangent doth concern)rejects indeed infinitesimals,but not on the foot that you do,to wit,their being inconsiderable in practical geometry or mixed mathematics.But he rejects them in the accuracy of speculative knowledge:in which respect there may be great logical errors,although there should be no sensible mistake in practice;which,it seems,is what you cannot comprehend.He rejects them likewise in virtue of a postulatum,which I venture to call rejecting them without ceremony.And,though he inferreth a conclusion accurately true,yet he doth it,contrary to the rules of logic,from inaccurate and false premises.And how this comes about,I have at large explained in the'Analyst,'and shewed in that particular case of tangents,that the rejectaneous quantity might have been a finite quantity of any given magnitude,and yet the conclusion have come out exactly the same way;and,consequently,that the truth of this method doth not depend on the reason assigned by the Marquis,to wit,the postulatum for throwing away infinitesimals;and,therefore,that he and his followers acted blindfold,as not knowing the true reason for the conclusions coming out accurately right,which I shew to have been the effect of a double error.
41.This is the truth of the matter,which you shamefully misrepresent and declaim upon,to no sort of purpose but to amuse and mislead your reader.For which conduct of yours throughout your remarks,you will pardon me if I cannot otherwise account,than from a secret hope that the reader of your'Defence'would never read the'Analyst.'
If he doth,he cannot but see what an admirable method you take to defend your cause:how,instead of justifying the reasoning,the logic,or the theory of the case specified,which is the real point,you discourse of sensible and practical errors:and how all this is a manifest imposition upon the reader.He must needs see that I have expressly said,"I have no controversy except only about your logic and method:that I consider how you demonstrate;what objects you are conversant about;and whether you conceive them clearly.''That I have often expressed myself to the same effect,desiring the reader to remember,"that I am only concerned about the way of coming at your theorems,whether it be legitimate or illegitimate,clear or obscure,scientific or tentative:that I have,on this very occasion,to prevent all possibility of mistake,repeated and insisted that I consider the geometrical analyst as a logician,i.e.so far forth as he reasons and argues;and his mathematical conclusions,not in themselves but in their premises;not as true or false,useful or insignificant,but as derived from such principles,and by such inferences.''[`Analyst,'sect.20.]
You affirm (and indeed what can you not affirm?)that the difference between the true subtangent and that found without any compensation is absolutely nothing at all.I profess myself of a contrary opinion.My reason is,because nothing cannot be divided into parts.But this difference is capable of being divided into any,or into more than any given number of parts;for the truth of which consult the Marquis de l'Hospital.And,be the error in fact or in practice ever so small,it will not thence follow that the error in reasoning,which is what I am alone concerned about,is one whit the less,it being evident that a man may reason most absurdly about the minutest things.
42.Pray answer me fairly,once for all,whether it be your opinion that whatsoever is little and inconsiderable enough to be rejected without inconvenience in practice,the same may in like manner be safely rejected and overlooked in theory and demonstration.if you say No ,it will then follow that all you have been saying here and elsewhere,about yards,and inches,and decimal fractions,setting forth and insisting on the extreme smallness of the rejectaneous quantity,is quite foreign to the argument,and only a piece of skill to impose upon your reader.If you say Yes ,it follows that you then give up at once all the orders of fluxions and infinitesimal differences;and so most imprudently turn all your sallies and attacks and veterans to your own overthrow.If the reader is of my mind,he will despair of ever seeing you get clear of this dilemma.The points in controversy have been so often and so distinctly noted in the'Analyst'that I very much wonder how you could mistake,if you had no mind to mistake.It is very plain,if you are in earnest,that you neither understand me not your masters.And what shall we think of other ordinary analysts,when it shall be found that even you,who like a champion step forth to defend their principles,have not considered them?
