Preface.

Ever since Leibniz's time, there has been speculation about introducing the calculus of situation or position into the scope of analysis, but Leibniz himself did not bring this about, and from time to time after his death the dream was considered vain.

Several decades ago, several papers appeared showing an almost unanimous tendency to give the so-called imaginary quantities the meaning of reliability in terms of their position in space. For eight years I too had been constantly occupied with this subject; it was not until 1856 that I came into possession of a work of greater size and great importance (Situationskalkül by Hermann Scheffler, Braunschweig, 1851), that I became convinced that as regards the first principles of mathematics no one had yet gone down the path I had outlined for myself, but that in the further development of the science, both in terms of newly introduced expressions and in terms of calculation in analytic geometry, I had been considerably surpassed. This circumstance not only did not discourage me from the work I had undertaken, but it even became an incentive to continue along this path, and instilled in me the conviction that from such an important subject for mathematics no benefits would flow to the general public until this method of teaching, which is taken from its very nature, had found application in books on elementary mathematics, and until it was shown thereby how much such a presentation of the matter not only enriches the content of elementary mathematics, but also gives to its entire structure that feature of completion and rigour that has been attributed to it for centuries.

Having been seized by this idea, I did not hesitate to base my lectures on elementary mathematics in a technical academy on the principles which I have outlined in this work, and five years of experience have convinced me that this method, from a didactic point of view, must be conceded superior to that which is commonly used.

In this work I have developed the beginnings of mathematics, so far as they seemed necessary, in order to lead the student to a point where he could venture on his own into the further domain of this knowledge, and to enable him, if not to pursue independent research in the natural sciences, at least to avail himself of the treasury of truths already established and duly illuminated.

I think that I should give the reader of the preface a brief view of the line of thought which I have developed in the course of this work; then indicate the position which elementary mathematics is to occupy in relation to the natural sciences, and what influence it should exert on the justification of their first beginnings; finally, to list those subjects which, either in form or in essence, I have developed according to my own ideas.

I. Starting from the statement of mathematics as the clearly established science of the relations of quantities, I have tried first of all to make the essence of spatial quantities accessible to the imagination, and this in § 1; then I have presented the measurement of the simplest spatial formations, the so-called rectilinear bands, by means of a rectilinear unit of measurement, in order thereby to represent the concept of extension and direction expressed in the obtained result of measurement, and I have recognised that it is precisely this result of measurement, as a natural representative of the ratio of arbitrary quantities, that leads to a number which is the rule according to which the size and direction of the measured quantity are guessed from the size and direction of the unit of measurement. And the act of creating this from the unit of measurement according to the rules of one or more numbers may be considered as a generalisation of what we usually call counting, and by means of which we usually obtain only such numbers which lie in two directions directly opposite to each other.

The first traces of the method of teaching mathematics based on the representation of quantities in space was already known in remote antiquity. But this so natural foundation, handed down from the founders of this knowledge, was abandoned by posterity and devised a new one, creating from the so-called absolute unity and from a part of it, absolute numbers, to which it adapted the rules of principal operations. Although the natural development of the said rules, undoubtedly drawn from the consideration of space, probably served as a model for deriving them from the formation of absolute numbers, - it is strange that they so carefully concealed the real source from which it came, seeing in it a sort of pride that on the basis of general numbers they were able to erect a structure of operations by means of pure reasoning. From that time on, the teaching of mathematics began with those abstractions, which were only later applied to real quantities, among which spatial quantities were also taken into account, either as such - or at least as representatives of others. Our method, however, according to which the present work was developed, proceeds in quite the opposite way: it deals primarily with the essence of spatial quantities, or their mutual relations, and consistently considers the cases obtained as reliable even when we substitute numbers for the quantities in the relationship.

While the results obtained according to our method seem obvious, it is not surprising that the student of the first method, tired by the abstractions themselves, finally considers this important science as a sterile and laborious immersion in speculation.

Ordinary absolute numbers indicate only two directly opposite directions; from spatial numbers one can understand the quantity of every possible direction.

The science of principal operations, built on spatial numbers, is characterised by its perfection and the uncommon advantage that it can be easily applied to all quantities - for it draws its fundamental rules from the consideration in space, in this dwelling, so to speak, of all quantities, and thus enables them to be clearly conceived.

The very consideration of all possible directions in a given plane diversifies and enlivens the science of arithmetical operations, and proves to be especially important in planimetry.

A careful elaboration of the theory of exponentiation allows us to derive from the binomial theorem many of series necessary for calculating logarithms, angular and arc-measure numbers, for creating tables useful for this purpose, and for indicating how to use them.

Goniometry, polygonometry and all mathematical formulas related to them are only direct results of the general theorems of addition, multiplication and exponentiation. The tools used for geometrical drawings, such as the ruler and compass, are here replaced by the operation of addition and multiplication.

The way in which such formulas are derived is so simple, and is repeated so often, that a pupil who listens attentively to the lecture almost considers himself the inventor of the rules. This circumstance, which will not escape the attention of an attentive and conscientious teacher, must be highly desirable for this reason alone, because it facilitates the retention in memory of formulas, namely, acquired methods.

The principles of analytical geometry, the inculcation of which is usually laboriously carried out, are familiarised by the youth in the theory of operations, and at the same time sufficiently supplied with the necessary means for a deeper introduction to this analysis.

The extension of the theory of operations, extended to quantities of different directions lying in one plane, is not, in comparison with the method hitherto used, any novelty, either in the way of deducing the events or in the events themselves. It is only a supplement to this inexact theory, or rather a proof of the fundamental theorems in the theory of operations long used, as will appear presently.

Before, in the science of square roots, and even now, in some places, they say:

The symbolic expression in the form of a binary: $(p + q\sqrt{-1})$ is called an imaginary form, that is, meaningless; nevertheless, it is necessary to introduce them into the calculation according to the rules concerning reliable numbers, because it often happens that such imaginary numbers lead to reliable cases, once some of the operations in question have been performed on them. This opinion persisted for a long time, until at last, after many, many years, the mathematician de Moivre hit upon a happy idea: he put the numbers of the addition and subtraction of the same angle into a binary, applied the rule of multiplication to it, and what did he get? - The operational rules borrowed for binary terms proved reliable for them.

The remarks quoted here can be summarised as follows:

The solution of every numerical problem must ultimately be represented by a number, or relatively by numbers with certain directions; these numbers are therefore either primary or secondary, appearing, among other things, in the form of a binary term $(p + q\sqrt{-1})$, and sometimes a ternary term $(p + q\sqrt{-1} + r\sqrt{-1}\sqrt{\dot{-}1})$, depending on whether they serve to designate a point lying on a plane or a point occupying any position in space.

These cases therefore always have a reliable meaning if we relate the conditions of a given problem to spatial quantities.

If, however, we apply these conditions to such quantities as only primary numbers presuppose, we can only regard primary numbers as reliable, while numbers in other directions, which remain irrelevant in this case, indicate that no number suitable for fulfilling the given conditions exists.

Let us imagine, for example, some numerical equation of the 2nd or 3rd or 4th degree, the coefficients of which are themselves primary numbers; let this equation be solved by a mathematician who supposes only primary numbers, and by one who has drawn numbers of any direction into the sphere of mathematics. The latter will attribute to the roots obtained in the form of binary terms reliability at least as far as space is concerned, while the former will consider them either as meaningless forms that have crept into the calculation and he cannot explain his own results, or, investigating their meaning more deeply, will refer them to space and will be forced to go beyond the limits that embrace only primary numbers and will also draw numbers of any direction into the sphere of mathematics.

In view of what has been said above, it will not be difficult to choose between the old method, together with the geometrical theorems derived from it, which, no matter how systematically arranged, always lack foundation, and the method in language, according to which we have justified the theory of operations in this work.

II. In the theory of operations I placed the notion of functions and developed from them the last two operational rules, that is: the rule of differentiation and integration, the last insofar as it suffices for understanding the definite integral. Without venturing into the development of more complicated methods of integration, I limited myself to indicating the primitive functions of differential formulas, which I gave as an example and derived for the purpose of practice in differentiation.

The beginnings of differentiation and integration, which are otherwise easily understood, I have explained in order to facilitate the student's thorough understanding of those mathematical theorems and their application in natural sciences, the essence of which is summarised by a differential formula that can be easily constructed; and finally by the integral appropriate to it, definite or indefinite. For even in such cases, when integration is difficult to perform, the beginner can, by means of differential calculus, convince himself of the reliability of the integral, either dictated by the teacher or in a book given without a proof: and, recognising the truth of the theorem, he will have no doubts, unless he is awakened by the desire to delve deeper into the calculus of integration, so that in similar cases he can do without outside help.

By standardizing the method of solving problems of this kind, which are not infrequently encountered in solid surveying and in the natural sciences, these sciences become easier; for by frequently repeating arguments of the same form, the matter is understood more clearly and remembered better.

Before the invention of differential and integral calculus, attempts were made to solve at least some of the detailed problems of this kind in an elementary way. We gratefully acknowledge the merit in this; - but despite being in possession of this simple and so useful tool of mathematics, namely differential and integral calculus, even to this day some people occupy their students with such loose speculations, which make it difficult to understand and remember the ideas contained in them - they can perhaps be excused for the fact that the principles of the said calculations have not so far been included in elementary mathematics.

Leaving aside the advantage that flows from unifying derivations by means of differentiation and integration, let us compare the solution of a problem achieved in this way with the so-called elementary solution of the same problem, and we shall be convinced that the latter, although it encompasses the essence of the matter less, is usually more involved.

To the objection which some might make to us that uniformity of method does not contribute to the acquisition of sound judgment, that it does not therefore sharpen the quickness of reason, which is recommended by the elementary and mostly synthetic method, we reply that the teacher may sometimes recommend, or even command, the pupil to learn such elementary arguments, and may himself skilfully include some of the more important ones in his lectures; - in this way he himself will present suitable tasks for the pupils, and they will find an opportunity to experience their independence and to train themselves in independent inquiry.

I have not included here examples of the application of proportions, equations, and other theories, if only to leave the teacher free to choose appropriate topics from the reference books, a collection of examples which is not lacking today and which I myself intend to place in the hands of the readers of my work in time.

III. As to those parts in the elaboration of which my own studies more or less predominate, the attentive reader will easily recognise what is the fruit of my own labour. Considering, however, that it is impossible to expect everyone to devote so much time and patience to this purpose, I include in the preface some relevant hints.

The complete illustration of spatial quantities, the measurement of homogeneous quantities as a starting point in the theory of operations, the teaching of combinations, logarithms and exponentiations, the solving of phonetic equations by means of involution and the calculation of the numerical root, the first digits of which are known on the same basis; an explanation of the division and multiplication of spatial quantities with given directions, an outline of the construction of goniometric-logarithmic tables - these are the passages contained in the first volume, the evaluation of which I recommend to the gracious judgment of the attentive reader.

In the second volume, I venture to draw attention to the main principles of analytic geometry. There I refer in spatial structures to a system of axes inclined to each other at any angle; I have succeeded in giving to the formulas concerning them transparent forms by means of suitable symbols, that, like the usual ones referred to rectangular axes, they are easily accessible; on the other hand, they greatly facilitate analytic discussions by the fact that it is not necessary in each particular case to select suitable rectangular axes, while by the nature of the task another system of axes, usually not rectangular, turns out to be suitable (as, for example, in crystallography). I have also extended Cauchy's investigations concerning lines and surfaces of the second degree to a system of axes with any angles of inclination, and I have obtained in an incomparably simpler way analytic features for particular cases, which it is impossible to reach on the basis of the rectangular coordinate system; I have reconciled Fourier's method of separating the roots of an equation with the method of involution, and have both extended and simplified it.

As regards terminology, except for expressions for new concepts coined with all caution, I have retained all other terms in the sense in which they are commonly used in the works of our writers, and of which works the library of His Royal Majesty's Count Dzieduszycki in Łódź abundantly provided me.

Finally, I consider it a pleasant duty to express my heartfelt thanks to His Excellency Count Włodzimierz Dzieduszycki for facilitating the publication of this work. His constant zeal/readiness to make any sacrifices for the advancement of scientific literature, namely national literature, encouraged me to accelerate this work, providing me with abundant means to publish it. In noble disinterestedness, accepting an appropriate number of copies, at the price at which they will sell, he facilitated my payment of the advance to cover the costs of publication which were incurred, and thus enabled me to establish the price of the work as moderate as possible.

Lwów, 10 December 1861.

On some devices for plotting.

In order to solve a mathematical problem, we first try to find the relations between the data and the sought quantities, and we express these relations by equations: only then do we determine the unknown quantities by calculation.

However, solving equations is not always easy, and even equations containing only one unknown quantity can only be solved exactly if they do not contain an unknown to a degree higher than the 4th.

It is very advantageous to use geometric methods to solve equations, which is achieved by drawing directly the desired results. However, using only a ruler and compass for this purpose, as devices drawing only a straight line and a circle, one must limit oneself to equations of the first and second degree.

In order to enable progress in this field and to introduce the construction method to the same limit to which the methods of calculation led, that is to the solution of equations of the 3rd and 4th degree, the speaker invented other devices, namely a marked ruler, but otherwise free, and a marked ruler moving according to certain laws along certain lines. Finally, using a marked circle, moving in a certain special way, the speaker invented a construction method for solving even a multitude of transcendental equations, which could not be solved by calculation at all.

How far the mere marking of a simple ruler extends the importance of this instrument is proven by the fact that using it solves many tasks known in antiquity, but not solved so far, or at least solved only in an extremely intricate and laborious way.

One such task is the problem of dividing an angle into three parts, the famous "trisectio anguli"