26.As oft as you talk of finite quantities inconsiderable in practice,Sir Isaac Newton disowns your apology.Cave ,saith he,intellexeris finitas .And,although quantities less than sensible may be of no account in practice,yet none of your masters,not will even you yourself,venture to say that they are of no account in theory and in reasoning.The application in gross practice is not the point questioned,but the rigour and justness of the reasoning.And it is evident that,be the subject ever so little,or ever so inconsiderable,this doth not hinder but that a person treating thereof may commit very great errors in logic;which logical errors are in nowise to be measured by the sensible or practical inconveniences thence arising,which,perchance,may be none at all.It must be owned that,after you have misled and amused your less qualified reader (as you call him),you return to the real point in controversy,and set yourself to justify Sir Isaac's method of getting rid of the above-mentioned rectangle.And here I must intreat the reader to observe how fairly you proceed.
27.First then you affirm (p.44),"that neither in the demonstration of the rule for finding the fluxion of the rectangle of two flowing quantities,nor in anything preceding or following it,is any mention,so much as once,made of the increment of the rectangle of such flowing quantities.''Now I affirm the direct contrary.For,in the very passage by you quoted in this same page,from the first case of the second lemma of the second book of Sir Isaac's Principles,beginning with Rectangulum quodvis motu perpetuo auctum ,and ending with igitur laterum incrementis totis a and b generatur rectanguli incrementum aB +bA.Q.E.D.in this very passage,I say,is express mention made of the increment of such rectangle.As this is matter of fact,I refer it to the reader's own eyes.Of what rectangle have we here the increment?
Is it not plainly of that whose sides have a and b for their incrementa tota ,that is,of AB .Let any reader judge whether it be not plain from the words,the sense,and the context,that the great author in the end of his demonstration understands his incrementum as belonging to the rectangulum quodvis at the beginning.Is not the same also evident from the very lemma itself prefixed to the demonstration?
The sense whereof is (as the author there explains it),that if the moments of the flowing quantities A and B are called a and b ,then the momentum vel mutatio geniti rectanguli AB will be aB +bA .Either therefore the conclusion of the demonstration is not the thing which was to be demonstrated,or the rectanguli incrementum aB +bA belongs to the rectangle AB .
28.All this is so plain that nothing can be more so;and yet you would fain perplex this plain case by distinguishing between an increment and a moment.But it is evident to every one who has any notion of demonstration that the incrementum in the conclusion must be the momentum in the lemma;and to suppose it otherwise is no credit to the author.It is in effect supposing him to be one who did not know what he would demonstrate.But let us hear Sir Isaac's own words:Earum (quantitatum scilicet fluentium)incrementa vel decrementa momentanea sub nomine momentorum intelligo .And you observe yourself that he useth the word moment to signify either an increment or decrement.Hence,with an intention to puzzle me,you propose the increment and decrement of AB ,and as which of these I would call the moment?The case you say is difficult.My answer is very plain and easy,to wit,Either of them.
You,indeed,make a different answer;and from the author's saying that by a moment he understands either the momentaneous increment or decrement of the flowing quantities,you would have us conclude,by a very wonderful inference,that his moment is neither the increment nor decrement thereof.
Would it not be as good an inference,because a number is either odd or even,to conclude it is neither?Can any one make sense of this?Or can even yourself hope that this will go down with the reader,how little soever qualified?It must be owned,you endeavour to intrude this inference on him,rather by mirth and humour than by reasoning.Your are merry,I say,and (p.46)represent the two mathematical quantities as pleading their rights,as tossing up cross and pile,as disputing amicably.You talk of their claiming preference,their agreeing,their boyishness,and their gravity.And after this ingenious digression you address me in the following words -Believe me,there is no remedy,you must acquiesce.But my answer is that I will neither believe you nor acquiesce;there is a plain remedy in common sense;and,to prevent surprise,I desire the reader always to keep the controverted point in view,to examine your reasons,and be cautious how he takes your word,but most of all when you are positive,or eloquent,or merry.
