32.Afterwards to remove (as you say)all scruple and difficulty about this affair,you observe that the moment of the rectangle determined by Sir Isaac Newton,and the increment of the rectangle determined by me are perfectly and exactly equal,supposing a and b to be diminished ad infinitum :and,for proof of this,you refer to the first lemma of the first section of the first book of Sir Isaac's principles.I answer that if a and b are real quantities then ab is something,and consequently makes a real difference:
but if they are nothing,then the rectangles whereof they are coefficients become nothing likewise:and consequently the momentum or incrementum ,whether Sir Isaac's or mine,are in that case nothing at all.As for the above-mentioned lemma,which you refer to,and which you wish I had consulted sooner,both for my own sake and for yours;I tell you I had long since consulted and considered it.But I very much doubt whether you have sufficiently considered that lemma,its demonstration,and its consequences.For,however that way of reasoning may do in the method of exhaustions ,where quantities less than assignable are regarded as nothing:yet,for a fluxionist writing about momentums to argue that quantities must be equal because they have no assignable difference seems the most injudicious step that could be taken:it is directly demolishing the very doctrine you would defend.For,it will thence follow that all homogeneous momentums are equal,and consequently the velocities,mutations,or fluxions,proportional thereto,are all likewise equal.There is,therefore,only one proportion of equality throughout,which at once overthrows the whole system you undertake to defend.Your moments (I say)not being themselves assignable quantities,their differences cannot be assignable:and,if this be true,by that way of reasoning it will follow,they are all equal;upon which supposition you cannot make one step in the method of fluxions.It appears from hence,how unjustly you blame me (p.32)for omitting to give any account of that first section of the first book of the'Principia,'wherein (you say)the foundation of the method of fluxions is geometrically demonstrated and largely explained,and difficulties and objections against it are clearly solved.All which is so far from being true that the very first and fundamental lemma of that section is incompatible with and subversive of the doctrine of fluxions.And,indeed,who sees not that a demonstration ad absurdum more veterum ,proceeding on a supposition that every difference must be some given quantity,cannot be admitted in,or consist with,a method wherein quantities,less than any given,are supposed really to exist,and be capable of division?
33.The next point you undertake to defend is that method for obtaining a rule to find the fluxion of any power of a flowing quantity,which is delivered in his'Introduction to the Quadratures,'and considered in the'Analyst.'And here the question between us is,whether I have rightly represented the sense of those words evanescant jam augmenta illa ,in rendering them,"let the increments vanish,''i.e.let the increments be nothing,or let there be no increments?This you deny;but,as your manner is,instead of giving a reason you declaim.I,on the contrary,affirm,the increments must be understood to be quite gone,and absolutely nothing at all.My reason is,because without that supposition you can never bring the quantity or expression down to ,the very thing aimed at by supposing the evanescence.Say whether this be not the truth of the case?Whether the former expression is not to be reduced to the latter?
And whether this can possibly be done so long as o is a real quantity?
I cannot indeed say you are scrupulous about your affirmations,and yet I believe that even you will not affirm this;it being most evident,that the product of two real quantities is something real;and that nothing real can be rejected either according to the of geometry,or according to Sir Isaac's own Principles;for the truth of which I appeal to all who know anything of these matters.Further,by evanescent must either be meant,let them (the increments)vanish and become nothing,in the obvious sense,or let them become infinitely small.But that this latter is not Sir Isaac's sense is evident from his own words in the very same page,that is,in the last of his'Introduction to the Quadratures,'where he expressly saith,volui ostendere quod in methodo fluxionum non opus sit figuras infinite parvas in geometriam introducere .Upon the whole,you seem to have considered this affair so very superficially as greatly to confirm me in the opinion you are so angry with,to wit,that Sir Isaac's followers are much more eager in applying his method than accurate in examining his principles.You raise a dust about evanescent augments,which may perhaps amuse and amaze your reader,but I am much mistaken if it ever instructs or enlightens him.For,to come to the point,those evanescent augments either are real quantities,or they are not.If you say they are;I desire to know how you get rid of the rejectaneous quantity?If you say they are not;you indeed get rid of those quantities in the composition whereof they are coefficients;but then you are of the same opinion with me,which opinion you are pleased to call (p.58)"a most palpable,inexcusable,and unpardonable blunder,''although it be a truth most palpably evident.
