Foundations of Differential Calculus Euler Springer [...]... everything is kept within the bounds of pure analysis, so that in the explanation of the rules of this calculus there is no need for any geometric figures Euler Translator’s Introduction In 1748 Euler published Introductio in Analysin Infinitorum, which has been translated as Introduction to Analysis of the Infinite, in two books This can be thought of as Euler s “precalculus.” In 1755 he published Institutiones... part, after laying the foundations of differential calculus, I have presented the method for differentiating every kind of function, for finding not only differentials of the first order, but also those of higher order, and those for functions of a single variable as well as those involving two or more variables In the second part, I have developed very fully applications of this calculus both in finite... put it into the form of a discipline, collected its rules into a system, and gave a crystal-clear explanation From this there followed great aids in the further development of this calculus, and some of the open questions whose answers were sought were pursued through certain definite principles Soon, through the studies of both Leibniz and the Bernoullis, the bounds of differential calculus were extended... Newton must be given credit for that part of differential calculus concerned with irrational functions This was nicely deduced from his wonderful theorem concerning the general evolution of powers of a binomial By this outstanding discovery, the limits of differential calculus have been marvelously extended We are no less indebted to Leibniz insofar as this calculus at that time was viewed as individual... numerical coefficients of these formulas are the same as those of the binomial expansion Insofar as the first difference is determined by the first two terms of the series y, y I , y II , y III , , the second difference is determined by three terms, the third is determined by four terms, and so forth It follows that when we know the differences of all orders of y, likewise, differences of all orders of y I , y II... polynomial is the sum of several powers of x, we can find all of the differences of polynomials, provided that we know how to find the differences of these powers For this reason we will investigate the differences of powers of x in the following examples Since x0 = 1, we have ∆x0 = 0, because x0 does not change when x changes to x + ω Also, since as we have seen, ∆x = ω and ∆∆x = 0, all of the following differences... difference is for a polynomial function of x, then its sum (or the function of which it is the difference) can easily be found with these formulas Since the difference is made up of different powers of x, we find the sum of each term and then collect all of these terms Example 1 Find the function whose difference is ax2 + bx + c We find the sum of each term by means of the formulas found above: Σax2 = ax2... putting x + ω instead of x Furthermore, ∆y III is the difference, or increment, of y III , and so forth With this settled, from the series of values of y, namely, y, y I , y II , y III , , we obtain a series of differences ∆y, ∆y I , ∆y II , , which we find by subtracting each term of the previous series from its successor 1 On Finite Differences 3 6 Once we have found the series of differences, if we... the study of series In that part, I have also given a very clear explanation of the theorem concerning maxima and minima As to the application of this calculus to the geometry of plane curves, I have nothing new to offer, and this is all the less to be required, since in other works I have treated this subject so fully Even with the greatest care, the first principles of xii Preface differential calculus. .. which had in part already been discussed Then, too, the foundations of integral calculus were firmly established Those who followed in the elaboration of this field continued to make progress It was Newton who gave very complete papers in integral calculus, but as to its first discovery, which can hardly be separated from the beginnings of differential calculus, it cannot with absolute certainty be attributed . function of x Preface vii receives. Indeed, the investigation of this kind of ratio of increments is not only very important, but it is in fact the foundation of the whole of analysis of the infinite the work of integral calculus is to study changing motion in a finite space. It is my opinion that it is hardly necessary to show further the uses of differen- tial calculus and analysis of the infinite,. the founda- tions of differential calculus, I have presented the method for differentiating every kind of function, for finding not only differentials of the first order, but also those of higher