In addition to textbooks on higher analysis, i.e. differential and integral calculus, the excellent and very extensive textbooks that we have in Polish, there are also smaller textbooks, also intended for self-educated students - a list of them is given in the last chapter of this book; nevertheless, we are publishing this book because it is different from the existing ones, because it differs in its method of teaching. The lack of a small Polish book introducing the reader to the beginnings of mathematical analysis in a didactic and, at the same time, rigorous way, has long been felt. It is known that students leaving secondary school are usually not well enough prepared to be able to read textbooks of higher analysis in Polish, much less in foreign languages. Moreover, there is a certain group of intelligent people who do not professionally deal with mathematics, but are still curious about differentials, derivatives and integrals.

With this in mind, we wrote a book in which we present several basic concepts of differential and integral calculus, namely, what a variable and constant quantity is, a function, its continuity and limit, what a derivative and differential are, the limit of a sequence and integral of the function. We did not want to accumulate more information, because we assumed that the reader did not have a great and extensive knowledge of elementary mathematics. No professional mathematician will be surprised that we have not covered a great deal of material on very many pages, as the saying goes: non multa, sed multum ["not many but much" meaning "not many things, but one thing deeply"]. Yes, if this volume is well received, we will publish a second volume that will complement the first one. We wanted, above all, the form of a lecture on the very difficult and fundamental concepts that we present. Our lecture is inductive; we give examples that are selected in such a way that they create an intuitive concept in the reader, then we formulate a precise, specific concept, and we give examples again. We provide concepts and as few "rules", "formulas" and "patterns" as possible - because we are of the opinion that nowhere can mechanical science prevail so easily and wreak such mental devastation as in mathematics that is not properly taught.

First of all, we would like to see this book in the hands of our young people who want to study exact sciences, i.e. mathematics, natural sciences and technology, who are already in the 6th grade of junior high school and also Real School students can use this booklet. We also believe that after thoroughly studying it, the reader can easily start studying the textbooks, a list of which is provided in the last chapter.

Antoni Hoborski and Antoni Wilk
October 1909

When writing the textbook "Higher Mathematics", I was aware of the greatness of the task I had undertaken, because popularising differential and integral calculus is a very difficult undertaking.

The author of each scientific textbook should, before starting his work, define as precisely as possible the group of people for whom he is writing it; this term is to be a guide for the author in the selection and content of the textbook and popularisation methods, and - which is also not an easy task - it will allow him to mark the resources of information that he accepts as known to the reader.

Well, when writing the textbook that I am putting at the service of our society today, I decided to devote it to students of academic technical schools and students of natural sciences. This group of readers, although so diverse, because it includes future engineers, botanists, chemists, physicists, etc., etc. , however, has this common feature that, by using mathematics, he turns it into a tool for scientific research: without considering it as the ultimate goal. This is where the definition of the scope of the textbook's content comes from, which should include primarily classical topics, although sometimes not general, but completely sufficient for those who today apply mathematics to various branches of knowledge. Therefore, many issues, although beautiful and tempting, although more general than classical issues and therefore simpler for a professional mathematician, were not taken into account. This is a utilitarian point of view, one might say, but it is dictated by rational considerations - not all parts of mathematics are applicable to issues of natural science or technology.

Of course, one could rightly object that it is difficult to predict the development of sciences using mathematics, so it may happen that knowledge of a branch of mathematics that is not yet useful for a scientific approach to practical life or natural issues will become necessary tomorrow; so will the reader of my textbook be prepared enough to be able to supplement his mathematical knowledge if necessary? This question is closely related to the second issue I raised above, i.e. the issue of the lecture method. Well, the lecture should be clear, accessible and precise. However, the quality and accessibility of the lecture are requirements so often elusive, so undefined, that I cannot go into their analysis. Moreover, what may be clear and accessible to one reader may be dark, complicated and difficult for another; in addition to the reader's mental dispositions, the degree of preparation with which he or she begins reading the textbook will also come into play, as discussed below.

However, the accuracy of the lecture is not subject to as much freedom of opinion as the clarity and accessibility; in this matter we already have good criteria, so that the judgment as to whether the textbook is precise or not can be the same for all professional mathematicians. But this is where the views of professional mathematicians and those who apply mathematics differ. The postulate of precision not only concerns the way of expressing definitions and theorems and the method of proof, but also causes an increase in the auxiliary material, and therefore requires auxiliary definitions, many lemmas without value for applications, the expression of axioms, the reduction of intuition in proofs to a minimum, etc. In one word, a postulate accuracy affects the volume of the textbook. The times that engineers constantly tell mathematicians that an engineer cannot devote much time to mathematics and therefore prefers to read Goschen's books rather than Kowalewski's or Rothe's textbooks.

Of course, there is some kind of misunderstanding here. After all, every mathematical theorem has the form of a sentence: from $P$ it follows $Q$. The most common inaccuracy of small textbooks consists in the fact that the assumption $P$ is either completely omitted or incompletely stated, as a result of which the textbook gives a different theorem: from $P$ it follows $Q$, which is usually untrue. Few realise that inaccurate textbooks contain false judgments (!). In addition, there may be imprecise terms, quasi-evidence, etc. Is it possible to build a correct theory based on imprecisely constructed concepts and false statements? After all, a building necessarily built on fragile foundations, will collapse under a gust of just criticism!

The second reason for misunderstanding is popular views on the use of mathematics. Some people think that the point is to put experimental data into mathematical formulas. Meanwhile, the pattern itself is indeterminate, because hundreds of patterns may be consistent with the experimental results. So the essence of the application of mathematics is not in the "formulas". A characteristic feature is the form of deduction, i.e. mathematics can be applied to a theory only when premises (hypotheses, axioms) that can be expressed mathematically have been found for the theory. Typical examples in this direction are provided by physics (i.e. mathematical physics), which is the highest among the sciences that use mathematics.

This actual task of applying mathematics must not be forgotten. And this cannot be taught by a mathematics textbook unless it shows $Q$ almost constantly, as follows from $P$. An imprecise textbook will never be able to teach this.

These are the main reasons why I preferred to be more precise in the lecture than many similar foreign textbooks.

Hence the need to limit ourselves to classic and not very extensive material. One more issue, which was and is often discussed in the reviews of previous years, concerns examples illustrating the theory. Those who use mathematics demand that examples be selected from physics (thermodynamics, mechanics), electrical engineering, the theory of the strength of materials, etc., etc.

Demands of this kind are only to a certain extent valid. After all, each of these sciences is based on certain principles, the knowledge of which is not general. A somewhat cautious author cannot introduce such issues without appropriate introductions and therefore often prefers to omit them. Moreover, the examples are intended to explain theoretical, not easy, things; so if we use an example that requires significant preparation, the actual goal will not be achieved by the author. Therefore, a certain measure in this direction is necessary.

The third, very important thing that the author of a textbook should always keep in mind is the level of preparation he assigns to the reader. There is quite a difficulty here. The recent post-war period does not allow for great optimism in this direction, and the matter is made worse by the lack of a well-known, good textbook of elementary mathematics that I could refer to without any reservations. That's where the "Introduction" in my textbook comes from.

I would also like to explain some details related to the creation of this manual.

The textbook was created from lectures entitled Higher Mathematics that I give every year at the Mining Academy and which have been expanded and supplemented in the textbook. Hence the name of the textbook, which may shock many mathematicians.

In addition to Higher Mathematics, the scientific programme of the Mining Academy includes algebra, trigonometry and analytical geometry as subjects of separate lectures, and many deficiencies in pre-academic preparation are eliminated during them. The "Introduction" in this textbook is an echo of these elementary lectures and sincere attempts to make the teaching of higher mathematics more accessible. The "Introduction" will become superfluous with the appearance of printed lectures on algebra, trigonometry and analytical geometry.

The lecture method used in the current textbook is similar to the one I used when writing with my colleague Prof Dr Antoni Wilk, now out of print: "Basic concepts of differential and integral calculus". This method, which is my idea, tries to create in the reader's mind an intuitive mathematical concept through appropriately selected and simple examples and to demonstrate the need and usefulness of this concept, and only then gives the vague definitions. The textbook has a very large number of examples.

We do not have a set of mathematical tasks intended for technical schools. Initially, I thought that such a collection could be created soon after the publication of this textbook. Today, I do not have this illusion - the financial conditions of the publishing house are so difficult for private individuals that it is impossible to set a date for such a publication today. It would be a collective effort; chemists, physicists and engineers, together with the mathematician, should be co-authors - the beginning has already been made, but there is no end in sight! Therefore, I provide the first part from § 1 to § 60 with a modest collection of tasks that were covered at the Academy at different training hours; from § 60, I provide the text with more examples and topics for exercises.

Whatever the set of tasks that will be presented in a few years, I will not be able to eliminate two gaps in my textbook. An attentive reader will notice that I have omitted the theory of determinants and the modern theory of irrational numbers. The first one will include an Algebra lecture; the second, my own, original one is ready in manuscript and I will publish it soon under the title: New theory of irrational numbers. [The theory of determinants can be found in the book of prof. Zaremba entitled Theory of determinants and linear equations (published by the Jagiellonian University Mathematics Society)].

Since this manual contains classical items, I consider myself exempt from citing sources. This does not mean that it is not possible to see an independent interpretation of many more or less important details (definitions of the indefinite integral on page 247, the inequality $c ≠ 0$ in the theorem on page 264, etc.).

When it comes to practical examples, apart from a few that I found myself, I took them from a number of textbooks:

Schlomilch, Übungsbuch zum Studien der höheren Analysis;
Nernst-Schonfłies, Einführung in die mahtematische Behandlung der Naturwissenschaften;
Scheffers, Lehrbuch der Mathematik für Studierende der Natu rwissenscbaften u. der Technik;
Mangoldt, Einführung in die höhere Mathematik;
Perry, Höhere Mathematik für Ingenieure (translated by Memrnck);
Egerer, Ingenieur-Mathematik;
Bouasse, Cours de mathématiques generales;
Townsend and Goodenough, Essentials of Calculus.

While Mangoldt's book should be recommended, Perry's book should be warned against; the last one, overloaded with practical examples, has 450 pages, and after reading it I very much doubt whether the reader will know the main principles of higher mathematics.

I cannot ignore the fact that it is certain and logical in nature and the concept of an aggregate of infinitesimals (§ 10), which greatly simplify the proofs of § 11-14, I owe to Professor Sleszyński, and the idea to solve the problem of uniform motion (p. 155) came to me from a lecture by Professor Zaremba, who solved this issue in a completely different way.

How to read this manual? a beginner reader will ask. It is difficult to give detailed instructions - I can only give general advice: you should carefully read intuitive remarks, definitions, theorems and shorter proofs (longer proofs should be postponed for repeated reading). I admit that reading the book is not easy, because for reasons of savings I could not afford to print it, which was the decoration of pre-war trials. I place the formulas in the text, which causes quite a few technical difficulties. The fact that the book was printed sparingly is a very important advantage today!

One more note for the Reader: if the Reader would like to expand his knowledge in any direction, he should obtain information in the Guide for the Self-Taught (Volume I: Mathematics); there you will find necessary tips and a list of books.

Concluding the preface, I would like to express my heartfelt thanks to all those who facilitated or supported me with the publication of the book with advice and cooperation. So, first of all, I would like to thank the Faculty of Science and its Head, Dr Stanisław Michalski, for providing a repayable subsidy that covered the entire cost of the publishing house.

The drawings for the textbook were made by my students Mr Stanisław and Zbigniew Gołąb; proofreading was carried out by Mr Stanisław Gołąb. I would like to express my sincere thanks to both of them for their effort and work.

I would like to thank the Management and Printing House of the Jagiellonian University for so willingly taking into account all my special wishes.

I am also pleased to express my appreciation to Mr Welanyk for making the plates for the drawings included in the text of the textbook.

Prof Dr A Hoborski.

The full title of this textbook is: Differential Geometry: Theory of Real and Regular Curves in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. Hence, it is understandable that the manual excludes complex curves, that it deals with curves represented by a real parameter, and that it does not assume that the curves are analytical. The assumption of analyticity of the curves is only necessary when the independent variable takes complex values; this time, I excluded this case in advance, because even the functions occurring in the following arguments are constantly real (with one exception in IV §6).

This textbook, which covers - as can be seen from the above - certain sections of classical differential geometry of curves, was created from lectures that I have been giving at the Jagiellonian University for several years. The first lectures in this field were published in 1924, as a lithographed script, published by the Mathematics and Physics Club of the Jagiellonian University, and edited by Dr Stanisław Gołab. This was - as it seems - the first attempt to strictly and correctly capture the basic considerations of the differential geometry of curves, considerations usually given without proper assumptions, which was first drawn to the attention of geometers by E Study in 1909.

This manual differs from that script in several respects. While the script used only the analytical method, in this textbook I prove theorems using the vectorial or kinematic method (rarely purely analytical); moreover, from the very beginning (i.e. from the second part of the chapter) at the so-called In border considerations I use the development of Peany. Also, in terms of content, this manual does not coincide with the quoted script.

For practical reasons, the content of the textbook has been divided into two parts: while the first part contains theoretical foundations and a few examples, the second part presents a large number of various geometric problems, as the application of general methods: analytical, vectorial and kinematic.

Since the detailed arrangement of the content is visible from the list of things, I limit myself to pointing out that in the first (introductory) chapter I summarise the principles of vector theory, leaving aside its full exposition, corresponding to today's requirements; similarly, I do not teach kinematics, even though many concepts and theorems from the theory of motion are necessary.

In chapters II and III I present the classical theory of curves of the Euclidean space $\mathbb{R}^{3}$; I therefore depart from the common custom of starting a lecture with the theory of plane curves, which I talk about randomly and several times, but always in such a way as not to repeat myself and not bore the reader. If it was allowed, I am writing some of my arguments in a "telegraphic" style!

* * *
Proofreading is a very unpleasant and quite tedious task; the number of overlooked errors becomes only minor when the author does not rely only on himself in his proofreading work. Well, I am pleased to thank Mr Dr Franciszek Leja, Professor of the Warsaw University of Technology for revising the finished sheets; I have used the results of this revision in the List of Errors, to which I draw the reader's attention. During the second correction of each sheet, Dr Stanisław Gołąb, Associate Professor of the Jagiellonian University, was very helpful to me; I would like to express my sincere gratitude to him at this point. I would like to thank P M Kłosowicz, student of metallurgy at the Mining Academy, for the beautifully and carefully made drawings.

Of the two volumes of the textbook, the first is published by the Mathematics and Physics Club of the Jagiellonian University, to whom I am grateful as the publisher; the second volume is published by me.

I would like to express my thanks to the National Printing House (for printing Part 1) and the Jagiellonian University Printing House (for printing Part II) for careful printing and taking into account my typographic wishes. The drawing plates were made very carefully by the Krakow company: Fotochemia.