21.In my opinion the greatest men have their prejudices.
Men learn the elements of science from others:and every learner hath a deference more or less to authority,especially the young learners,few of that kind caring to dwell long upon principles,but inclining rather to take them upon trust:and things early admitted by repetition become familiar:and this familiarity at length passeth for evidence.Now to me it seems there are certain points tacitly admitted by mathematicians which are neither evident nor true.And such points or principles ever mixing with their reasonings do lead them into paradoxes and perplexities.If the great author of the fluxionary method were early imbued with such notions it would only shew he was a man.And if,by virtue of some latent error in his principles,a man be drawn into fallacious reasonings,it is nothing strange that he should take them for true:and nevertheless,if,when urged by perplexities and uncouth consequences,and driven to arts and shifts,he should entertain some doubt thereof,it is no more than one may naturally suppose might befall a great genius grappling with an insuperable difficulty:
which is the light in which I have placed Sir Isaac Newton.[`Analyst,'sect.18.]Hereupon you are pleased to remark that I represent the great author not only as a weak but as an ill man,as a deceiver and an impostor.
The reader will judge how justly.
22.As to the rest of your colourings and glosses,your reproaches and insults and outcries,I shall pass them over,only desiring the reader not to take your word,but read what I have written,and he will want no other answer.It hath been often observed that the worst cause produceth the greatest clamour;and indeed you are so clamorous throughout your defence that the reader,although he should be no mathematician,provided he understands common sense,and hath observed the ways of men,will be apt to suspect that you are in the wrong.It should seem,therefore,that your brethren the analysts are but little obliged to you for this new method of declaiming in mathematics.Whether they are more obliged by your reasoning I shall now examine.
23.You ask me (p.32)where I find Sir Isaac Newton using such expressions as the velocities of velocities,the second,third,and fourth velocities,&c.This you set forth as a pious fraud and unfair representation.I answer,that if according to Sir Isaac Newton a fluxion be the velocity of an increment,then according to him I may call the fluxion of a fluxion the velocity of a velocity.But for the truth of the antecedent see his'Introduction to the Quadrature of Curves,'where his own words are,Motuum vel incrementorum velocitates nominando fluxiones .
See also the second lemma of the second book of his Mathematical Principles of Natural Philosophy,where he expresseth himself in the following manner:Velocitates incrementorum ac decrementorum quas etiam,motus,mutationes,et fluxiones quantitatum nominare licet.And that he admits fluxions of fluxions,or second,third,fourth fluxions,&c.,see his Treatise of the Quadrature of Curves.I ask now,Is it not plain that if a fluxion be a velocity,then the fluxion of a fluxion may,agreeably thereunto,be called the velocity of a velocity?In like manner,if by a fluxion is meant a nascent augment,will it not then follow that the fluxion of a fluxion or second fluxion is the nascent augment of a nascent augment?
Can anything be plainer?Let the reader now judge who is unfair.
24.I had observed that the great author had proceeded illegitimately,in obtaining the fluxion or moment of the rectangle of two flowing quantities;and that he did not fairly get rid of the rectangle of the moments.In answer to this,you allege that the error arising from the omission of such rectangle (allowing it to be an error)is so small that it is insignificant.This you dwell upon and exemplify to no other purpose but to amuse your reader and mislead him from the question;which in truth is not concerning the accuracy of computing or measuring in practice,but concerning the accuracy of the reasoning in science.That this was really the case,and that the smallness of the practical error nowise concerns it,must be so plain to anyone who reads the'Analyst'that I wonder how you could be ignorant of it.
25.You would fain persuade your reader that Imake an absurd quarrel against errors of no significancy in practice,and represent mathematicians as proceeding blindfold in their approximations,in all which I cannot help thinking there is on your part either great ignorance or great disingenuity.If you mean to defend the reasonableness and use of approximations or of the method of indivisibles,I have nothing to say.But then you must remember this is not the doctrine of fluxions:
it is none of that analysis with which I am concerned.That I am far from quarrelling at approximations in geometry is manifest from the thirty-third and fifty-third queries in the'Analyst.'And that the method of fluxions pretends to somewhat more than the method of indivisibles is plain;because Sir Isaac disclaims this method as not geometrical.[See the Scholium at the end of the first section.Lib.i.,'Phil.Nat.Princip.Math.']And that the method of fluxions is supposed accurate in geometrical rigour is manifest to whoever considers what the great author writes about it;especially in his'Introduction to the Quadrature of Curves,'where he saith,In rebus mathematicis errores quam minimi non sunt contemnendi .
Which expression you have seen quoted in the'Analyst,'and yet you seem ignorant thereof,and indeed of the very end and design of the great author of this his invention of fluxions.
