43.The impartial reader is entreated to remark throughout your whole performance how confident you are in asserting,and withal how modest in proving or explaining:how frequent it is with you to employ figures and tropes instead of reasons:how many difficulties proposed in the'Analyst'are discreetly overlooked by you,and what strange work you make with the rest:how grossly you mistake and misrepresent,and how little you practise the advice which you so liberally bestow.Believe me,Sir,I had long and maturely considered the principles of the modern analysis,before I ventured to publish my thoughts thereupon in the'Analyst.'
And,since the publication thereof,I have myself freely conversed with mathematicians of all ranks,and some of the ablest professors,as well as made it my business to be informed of the opinions of others,being very desirous to hear what could be said towards clearing my difficulties or answering my objections.But,though you are not afraid or ashamed to represent the analysts as very clear and uniform in their conception of these matters,yet I do solemnly affirm (and several of themselves know it to be true)that I found no harmony or agreement among them,but the reverse thereof -the greatest dissonance,and even contrariety of opinions,employed to explain what after all seemed inexplicable.
44.Some fly to proportions between nothings.Some reject quantities because infinitesimal.Others allow only finite quantities,and reject them because inconsiderable.Others place the method of fluxions on a foot with that of exhaustions ,and admit nothing new therein.
Some maintain the clear conception of fluxions.Others hold they can demonstrate about things incomprehensible.Some would prove the algorism of fluxions by reductio ad absurdum ,others a priori .Some hold the evanescent increments to be real quantities,some to be nothings,some to be limits.
As many men,so many minds:each differing one from another,and all from Sir Isaac Newton.Some plead inaccurate expressions in the great author,whereby they would draw him to speak their sense;not considering that if he meant as they do,he could not want words to express his meaning.
Others are magisterial and positive,say they are satisfied,and that is all;not considering that we,who deny Sir Isaac Newton's authority,shall not submit to that of his disciples.Some insist that the conclusions are true,and therefore the principles;not considering what hath been largely said in the'Analyst'[Sect.19,20,&c.]on that head.Lastly,several (and those none of the meanest)frankly owned the objections to be unanswerable.
All which I mention by way of antidote to your false colours:and that the unprejudiced inquirer after truth may see it is not without foundation that I call on the celebrated mathematicians of the present age to clear up these obscure analytics,and concur in giving to the public some consistent and intelligible account of their great Master:for if they do not,I believe the world will take it for granted that they cannot.
45.Having gone through your defence of the British mathematicians,I find,in the next place,that you attack me on a point of metaphysics,with what success the reader will determine.I had upon another occasion many years ago wrote against abstract general ideas.
[Introduction to the'Treatise concerning the Principles of Human Knowledge.']
In opposition to which,you declare yourself to adhere to the vulgar opinion -that neither geometry nor any other general science can subsist without general ideas (p.74).This implies that I hold that there are no general ideas.But I hold the direct contrary -that there are indeed general ideas,but not formed by abstraction in the manner set forth by Mr.Locke.To me it is plain there is no consistent idea of the likeness whereof may not really exist:whatsoever therefore is said to be somewhat which cannot exist,the idea thereof must be inconsistent.Mr Locke acknowledgeth it doth require pains and skill to form his general idea of a triangle.He farther expressly saith it must be neither oblique nor rectangular,neither equilateral,equicrural nor scalenum;but all and none of these of these at once.He also saith it is an idea wherein some parts of several different and inconsistent ideas are put together.[`Essay on Human Understanding,'bk.iv,ch.vii,sect.9.]All of which looks very like a contradiction.
But,to put the matter past dispute,it must be noted that he affirms it to be somewhat imperfect that cannot exist;consequently the idea thereof is impossible or inconsistent.
46.I desire to know whether it is not impossible for anything to exist which doth not include a contradiction:and,if it is,whether we may not infer that what may not possibly exist,the same doth include a contradiction:I further desire to know,whether the reader can frame a distinct idea of anything that includes a contradiction?For my part,I cannot,nor consequently of the above-mentioned triangle;though you (you it seems know better than myself what I can do)are pleased to assure me of the contrary.Again I ask whether that which it is above the power of man to form a complete idea of may not be called incomprehensible?
And whether the reader can frame a complete idea of this imperfect impossible triangle?And,if not,whether it doth not follow that it is incomprehensible?
it should seem that a distinct aggregate of a few consistent parts was nothing so difficult to conceive or impossible to exist;and that,therefore,your comment must be wide of the author's meaning.You give me to understand (p 82)that this account of a general triangle was a trap which Mr.Locke set to catch fools.Who is caught therein let the reader judge.
