The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations,had led me to imagine that all things,to the knowledge of which man is competent,are mutually connected in the same way,and that there is nothing so far removed from us as to be beyond our reach,or so hidden that we cannot discover it,provided only we abstain from accepting the false for the true,and always preserve in our thoughts the order necessary for the deduction of one truth from another.And I had little difficulty in determining the objects with which it was necessary to commence,for I was already persuaded that it must be with the simplest and easiest to know,and,considering that of all those who have hitherto sought truth in the sciences,the mathematicians alone have been able to find any demonstrations,that is,any certain and evident reasons,I did not doubt but that such must have been the rule of their investigations.I resolved to commence,therefore,with the examination of the simplest objects,not anticipating,however,from this any other advantage than that to be found in accustoming my mind to the love and nourishment of truth,and to a distaste for all such reasonings as were unsound.But I had no intention on that account of attempting to master all the particular sciences commonly denominated mathematics:but observing that,however different their objects,they all agree in considering only the various relations or proportions subsisting among those objects,I thought it best for my purpose to consider these proportions in the most general form possible,without referring them to any objects in particular,except such as would most facilitate the knowledge of them,and without by any means restricting them to these,that afterwards I might thus be the better able to apply them to every other class of objects to which they are legitimately applicable.Perceiving further,that in order to understand these relations I should sometimes have to consider them one by one and sometimes only to bear them in mind,or embrace them in the aggregate,I thought that,in order the better to consider them individually,I should view them as subsisting between straight lines,than which I could find no objects more simple,or capable of being more distinctly represented to my imagination and senses;and on the other hand,that in order to retain them in the memory or embrace an aggregate of many,I should express them by certain characters the briefest possible.In this way I believed that I could borrow all that was best both in geometrical analysis and in algebra,and correct all the defects of the one by help of the other.
And,in point of fact,the accurate observance of these few precepts gave me,I take the liberty of saying,such ease in unraveling all the questions embraced in these two sciences,that in the two or three months I devoted to their examination,not only did I reach solutions of questions I had formerly deemed exceedingly difficult but even as regards questions of the solution of which I continued ignorant,I was enabled,as it appeared to me,to determine the means whereby,and the extent to which a solution was possible;results attributable to the circumstance that Icommenced with the simplest and most general truths,and that thus each truth discovered was a rule available in the discovery of subsequent ones Nor in this perhaps shall I appear too vain,if it be considered that,as the truth on any particular point is one whoever apprehends the truth,knows all that on that point can be known.The child,for example,who has been instructed in the elements of arithmetic,and has made a particular addition,according to rule,may be assured that he has found,with respect to the sum of the numbers before him,and that in this instance is within the reach of human genius.Now,in conclusion,the method which teaches adherence to the true order,and an exact enumeration of all the conditions of the thing .sought includes all that gives certitude to the rules of arithmetic.
But the chief ground of my satisfaction with thus method,was the assurance I had of thereby exercising my reason in all matters,if not with absolute perfection,at least with the greatest attainable by me:
besides,I was conscious that by its use my mind was becoming gradually habituated to clearer and more distinct conceptions of its objects;and Ihoped also,from not having restricted this method to any particular matter,to apply it to the difficulties of the other sciences,with not less success than to those of algebra.I should not,however,on this account have ventured at once on the examination of all the difficulties of the sciences which presented themselves to me,for this would have been contrary to the order prescribed in the method,but observing that the knowledge of such is dependent on principles borrowed from philosophy,in which I found nothing certain,I thought it necessary first of all to endeavor to establish its principles..And because I observed,besides,that an inquiry of this kind was of all others of the greatest moment,and one in which precipitancy and anticipation in judgment were most to be dreaded,I thought that I ought not to approach it till I had reached a more mature age (being at that time but twenty-three),and had first of all employed much of my time in preparation for the work,as well by eradicating from my mind all the erroneous opinions I had up to that moment accepted,as by amassing variety of experience to afford materials for my reasonings,and by continually exercising myself in my chosen method with a view to increased skill in its application.
