DIFFERENTIAL AND INTEGRAL
The following Treatise will differ from most others, for better or worse, in several points.
In the first place, it has been endeavoured to make the theory of limits, or ultimate ratios, by whichever name it may be called, the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit, or not to be admitted in the usual consequences, when drawn independently of expansions, that is, of developments under assumed forms. The objection to the difficulty I have endeavoured to lessen in the introductory chapter ; that to the name by which a science founded on limits should be called, I cannot feel the force of, or see what is to be answered. I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical.
Secondly, I have introduced applications to mechanics as well as geometry, in cases where the preliminary notions are not of too difficult a character, and I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. The parts of the former science which can be understood by a learner at any stage of the latter, are, I suppose it will be allowed, necessary to a proper view even of so much of the latter as precedes the point supposed.
Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered?
If so, why are not multiplication and involution in arithmetic made to follow addition and precede subtraction ? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold—but I do not feel it necessary further to defend placing the question of finding the area of a parabola at an earlier period of the work than that of finding the lines of curvature of a surface. Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the line of easiest deduction. This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction ; but made one help out the other as he best could.
At the same time I am far from saying that this Treatise will be easy ; the subject is a difficult one, as all know who have tried it. The absolute requisites for the study of this work, as of most others on the same subject, are a knowledge of algebra to the binomial theorem at least (according to the usual arrangement), plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry. The Treatise entitled ' Elementary Illustrations of the Differential and Integral Calculus,' will be bound up with this Volume, and referred to in the proper places.
Augustus De Morgan
London, July 1, 1836.
Elementary illustrations of the differential and integral calculus
questa e' solo la parte in appendice del testo precedente, edita posteriormente (1909) in forma resa autonoma.
NdR ci sono arrivato poiche' visto nel libro Geometric Exercises in Paper Folding.