N oted historian of mathematics Carl Boyer called Euler's Introductio in Analysin Infinitorum "the foremost textbook of modern times" [1] guess what is the foremost textbook of all times. Published in two volumes in , the Introductio takes up polynomials and infinite series Euler regarded the two as virtually synonymous , exponential and logarithmic functions, trigonometry, the zeta function and formulas involving primes, partitions, and continued fractions. That's Book I, and the list could continue; Book II concerns analytic geometry in two and three dimensions. The Introductio was written in Latin [2] , like most of Euler's work.

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N oted historian of mathematics Carl Boyer called Euler's Introductio in Analysin Infinitorum "the foremost textbook of modern times" [1] guess what is the foremost textbook of all times. Published in two volumes in , the Introductio takes up polynomials and infinite series Euler regarded the two as virtually synonymous , exponential and logarithmic functions, trigonometry, the zeta function and formulas involving primes, partitions, and continued fractions.

That's Book I, and the list could continue; Book II concerns analytic geometry in two and three dimensions. The Introductio was written in Latin [2] , like most of Euler's work. This article considers part of Book I and a small part. The Introductio has been massively influential from the day it was published and established the term "analysis" in its modern usage in mathematics.

It is eminently readable today, in part because so many of the subjects touched on were fixed in stone from that day till this, Euler's notation, terminology, choice of subject, and way of thinking being adopted almost universally. Both volumes have been translated into English by John D. Blanton as Introduction to Analysis of the Infinite [3]. Blanton starts his short introduction like this:.

One of his remarks was to the effect that he was trying to convince the mathematical community that our students of mathematics would profit much more from a study of Euler's Introductio in Analysin Infinitorum , rather than of the available modern textbooks.

The Introductio is an unusual mix of somewhat elementary matters, even for , together with cutting-edge research. Euler went to great pains to lay out facts and to explain. Not always to prove either — he states at many points that a polynomial of degree n has exactly n real or complex roots with nary a proof in sight.

The point is not to quibble with the great one, but to highlight his unerring intuition in ferreting out and motivating important facts, putting them in proper context, connecting them with each other, and extending the breadth and depth of the foundation in an enduring way, ironclad proofs to follow.

Large sections of mathematics for the next hundred years developed almost as a series of footnotes to Euler and this book in particular, researchers expanding his work, proving or re-proving his theorems, and firming up the foundation. Consider the estimate of Gauss, born soon before Euler's death Euler - , Gauss - and the most exacting of mathematicians:. C hapter I, pictured here, is titled "De Functionibus in Genere" On Functions in General and the most cursory reading establishes that Euler's concept of a function is virtually identical to ours.

Here is his definition on page A function of a variable quantity functio quantitatis variabilis is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. Euler was not the first to use the term "function" — Leibnitz and Johann Bernoulli were using the word and groping towards the concept as early as , but Euler broadened the definition an analytic expression composed in any way whatsoever! He considers implicit as well as explicit functions and categorizes them as algebraic, transcendental, rational, and so on.

The concept of an inverse function was second nature to him, the foundation for an extended treatment of logarithms. He called polynomials "integral functions" — the term didn't stick, but the interest in this kind of function did.

He says that complex factors come in pairs and that the product of two pairs is a quadratic polynomial with real coefficients; that the number of complex roots is even; that a polynomial of odd degree has at least one real root; and that if a real decomposition is wanted, then linear and quadratic factors are sufficient.

Then he pivots to partial fractions, taking up the better part of Chapter II. Maybe he's setting up for integrating fractions of polynomials, that's where the subject came up in my education and the only place.

That's the thing about Euler, he took exposition, teaching, and example seriously. He was prodigiously productive; his Opera Omnia is seventy volumes or something, taking up a shelf top to bottom at my college library. Even the nature of the transcendental functions seems to be better understood when it is expressed in this form, even though it is an infinite expression. In the case of quotients of polynomials, his method is to assume an infinite series expansion, cross multiply, then equate coefficients for the respective powers there are an infinite number of them.

Let's go right to that example and apply Euler's method. We want to find A, B, C and so on such that:. That's a Fibonacci-like sequence known as the Lucas series , for which:. The proof is similar to that for the Fibonacci numbers. I learned the ratio test long ago, but not Euler's method, and the poorer for it. E uler's treatment of exponential and logarithmic functions is indistinguishable from what algebra students learn today, though a close reader can sense that logs were of more than theoretical interest in those days.

Briggs's and Vlacq's ten-place log tables revolutionized calculating and provided bedrock support for practical calculators for over three hundred years. How quickly we forget, beneficiaries of electronic calculators and computers for fifty years. In , taking a tenth root to any precision might take hours for a practiced calculator.

By , log tables at hand, seconds. From this we understand that the base of the logarithms, although it depends on our choice, still should be a number greater than 1. Furthermore, it is only of positive numbers that we can represent the logarithm with a real number. The second row gives the decimal equivalents for clarity, not that a would-be calculator knows them in advance. The calculation is based on observing that the next two lines imply the third:. Euler says that Briggs and Vlacq calculated their log table using this algorithm, but that methods in his day were improved keep in mind that Euler was writing years after Briggs and Vlacq.

This isn't as daunting as it might seem, considering that the Newton-Raphson method of calculating square roots was well known by the time of Briggs — it was stated explicitly by Hero of Alexandria around the time of Christ and was quite possibly known to the ancient Babylonians. He does an amortization calculation for a loan "at the usurious rate of five percent annual interest", calculating that a paydown of 25, florins per year on a , florin loan leads to a 33 year term, rather amazingly tracking American practice in the late twentieth century with our thirty year home mortgages.

When this base is chosen, the logarithms are called natural or hyperbolic. The latter name is used since the quadrature of a hyperbola can be expressed through these logarithms. Boyer says, "The concept behind this number had been well known ever since the invention of logarithms more than a century before; yet no standard notation for it had become common.

These two imply that:. He proceeds to calculate natural logs for the integers between 1 and But then:. The natural logs of other small integers are calculated similarly, the only sticky one between 1 and 10 being 7.

No problem! T rigonometry is an old subject Ptolemy's chord table! Notation varied throughout the 17 th and well into the 18 th century. Granted that spherical trig is a more complicated branch of the subject, it still illustrates the danger of entrusting notational decisions to one less brilliant than Euler. Here is a screen shot from the edition of the Introductio.

Of course notation is always important, but the complex trigonometric formulas Euler needed in the Introductio would quickly become unintelligible without sensible contracted notation. Euler uses arcs radians rather than angles as a matter of course. Chapter VIII on trigonometry is titled "On Transcendental Quantities which Arise from the Circle" and at its start he says let's assume the radius is 1 — second nature today, but not necessarily when he wrote and the gateway to the modern concept of sines and cosines as ratios rather than line segments.

My check shows that all digits are correct except the th , which should be 8 rather than the 7 he gives. This was the best value at the time and must have come from Thomas Fantet de Lagny's calculation in At this point, you can almost hear the "Eureka!

A tip of the hat to the old master, who does not cover his tracks, but takes you along the path he traveled. To my mind, that path is the one to understanding, truer and deeper than some latter day denatured and "elegant" generalized development with all motivation pressed right out of it.

There is another expression similar to 6 , but with minus instead of plus signs, leading to:. Applying the binomial theorem to each of those expressions in 7 results in the following, since all the odd power terms cancel:. Now he's in a position to prove the theorem that will be known as Euler's formula until the end of time.

Erit vero" — It follows how the sine and cosine of real arcs can be reduced to imaginary exponential quantities. In particular:. The last two are true only in the limit, of course, but let's think like Euler. Substituting into 7 and 7' :. We are talking about limits here and were when manipulating power series expansions as well , so those four expressions in the numerators can be replaced by exponentials, as developed earlier:.

This was a famous problem, first formulated by P. Mengoli in ; it had resisted the efforts of all earlier analysts, including Leibniz and the Bernoullis. With this procedure he was treading on thin ice, and of course he knew it p Euler was 28 when he first proved this result. Also that "for the next ten years, Euler never relaxed his efforts to put his conclusions on a sound basis" p By , when the Introductio went into manuscript, he was able to include "a full account of the matter, entirely satisfactory by his standards, and even, in substance, by our more demanding ones" Weil, p From the earlier exponential work:.

Continuing in this vein gives the result:. Truly amazing and if this isn't art, then I've never seen it. But not done yet. The master says, " The truth of these formulas is intuitively clear, but a rigorous proof will be given in the differential calculus".

So as asserted above:. I'd forgotten that:. His output, like his penetrating insight, is beyond understanding, over seventy volumes in the Opera Omnia and still coming. He established notations and laid down foundations enduring to this day and taught in high school and college virtually unchanged.

The foregoing is simply a sample from one of his works an important one, granted and would run four times as long were it to be a fair summary of Volume I, including enticing sections on prime formulas, partitions, and continued fractions.

Volume II of the Introductio was equally path-breaking in analytic geometry. No wonder his contemporaries and immediate successors were in awe of him. Struik, Dover 1 st ed.

A History of Mathematics , by Carl B. Skip to main content. Ex Libris. Main menu Home.

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## Introduction to the Analysis of Infinities

It seems that you're in Germany. We have a dedicated site for Germany. From the preface of the author: " I have divided this work into two books; in the first of these I have confined myself to those matters concerning pure analysis.

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## Introduction to Analysis of the Infinite

Introductio in analysin infinitorum Latin for Introduction to the Analysis of the Infinite is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in , the Introductio contains 18 chapters in the first part and 22 chapters in the second. Carl Boyer 's lectures at the International Congress of Mathematicians compared the influence of Euler's Introductio to that of Euclid 's Elements , calling the Elements the foremost textbook of ancient times, and the Introductio "the foremost textbook of modern times". The first translation into English was that by John D. Blanton, published in

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Introductio in analysin infinitorum, volume 1. Leonhard Euler. In the Introductio in analysin infinitorum this volume, together with E , Euler lays the foundations of modern mathematical analysis. Perhaps more importantly, the Introductio makes the function the central concept of analysis; Euler introduces the notation f x for a function and uses it for implicit as well as explicit functions, and for both continuous and discontinuous functions. In addition, he calls attention to the central role of e and e x in analysis. These formulations put e x and ln x on an equal basis for the first time. Additionally, Euler proves that every rational number can be written as a finite continued fraction and that the continued fraction of an irrational number is infinite.