1.1. From the Introduction.

There was a time - and who wouldn't remember works from that time when world history was neatly divided - and great care was taken to ensure that nothing from one section overlapped into another. There was a history of the Greeks and a history of the Romans, perhaps also a history of oriental empires, and when these three separate boxes were placed in one case, it was called the history of antiquity. The same was true for the history of the Middle Ages and the modern era. But the division was even stricter when we consider the history of individual branches of culture. It was almost self-evident that a history of religions should not concern itself with the history of art, that the latter had nothing in common with political history, and that the history of this or that science constituted a self-contained whole, surrounded by a Great Wall.

In what is now called world history, no one seems to consider such a fundamental separation possible anymore. The conviction has been reached that humanity resembles an organism whose individual members are in constant, if not always immediately apparent, interconnectedness. The certainty has become unavoidable that every movement of an individual member, sooner or later, more or less, is communicated to the others, and therefore it can only be considered and judged in context. Even the histories of individual cultures had to give way to a cultural history. The aim was to demonstrate - and the number of schools of thought attempting this grows daily - that the cultural life of nations is no more complete than their political life. This is actually self-evident, and yet it is still necessary to dispel some prejudices.

It remains to be proven that wherever peoples have met peacefully or hostilely and interacted for extended periods, their cultures have necessarily blended, but that the reverse conclusion can also be drawn: if a similarity exists among nations in this or that area of ​​intellectual development, it is usually not mere coincidence but rather the result of mutual influence or common origin. This latter line of reasoning can only be undertaken by specialised research, and indeed, men of broad knowledge have compared the deities of the various religions, the word forms of the individual languages, and the artistic styles of the buildings and sculptures scattered here and there for the stated purpose. If, however, I add to these three areas the field of literature, which actually falls under the purview of comparative linguistics, then the enumeration of those attempts is pretty much exhausted. For a number of years now, I have thought it must be possible to demonstrate other points of contact and fusion between peoples. For that road which led from people to people was not only traversed by priests, builders, sculptors, and poets; surely every social class must have walked upon it and left its mark. I could therefore well imagine that, for example, a comparative history of the culinary arts of the individual nations might be of interest, long before I learned by chance that a man who died in Donaueschingen - a researcher, if I'm not mistaken Mr Schuch, was diligently working on compiling such a list.

My own profession as a mathematician likewise assigned me a field of research that had hitherto been little or not at all explored. And yet, what could be more natural than that in the interactions between peoples, as well as in those between individuals, such conditions would inevitably arise that made mathematical education, at least of the simplest kind, partly necessary, partly presupposed. The investigations that had to be undertaken from this perspective of mine certainly revealed a not insignificant difficulty. The subject matter - the simplest propositions of measurement, the beginnings of arithmetic, the inseparable question of number systems, and above all, of numerical symbols - were such elementary matters that preliminary work on their history, such as I required, were found to be so scattered that collecting this material had to be left almost more to chance than to any organised activity. But even worse, the things whose history I wanted to study were, as I have already said, so indispensable to individual trade that simultaneous invention in different places was quite possible, and then all conclusions that might be drawn about the connections between peoples ceased. However, in contrast to these difficulties, there was also a great relief in that I could focus on details that can be understood far more reliably as the result of dissemination than of independent origin. Great ideas sometimes mature simultaneously in different minds, but minor computational advantages, specifically shaped signs, symbolisations that have no inherent basis - such things, if they occur in different places, can certainly be attributed to transmission. It is similar to those small ornaments of architecture, which, in terms of style, allow us to trace the propagation of art from place to place. Thus, for this purpose, I collected wherever useful material might appear, and during this collecting, the fundamental idea behind it became clearer and clearer to me, an idea that had initially guided me semi-consciously. I have already published some of what I found in this way in individual articles in my journal for mathematics and physics, which I co-edit with Professor Schloemilch and Dr Kahl. Nevertheless, I fear no reproach for revisiting what I have already presented here in a new context. While the actual subject of a historical work cannot be solely to present new, self-conceived ideas, the crucial factor by which a historical work is to be judged lies precisely in the context in which events appear ordered.

2.1. From the Foreword.

The volume presented here to its readers has grown out of the lectures on the history of mathematics that I have been striving to establish at Heidelberg University for a number of years. Former students will especially appreciate the validity of the statement that this volume has grown out of those lectures. As the material accumulated from semester to semester, without it seeming feasible to extend the now over one and a half years of lectures or to limit them to two hours per week, the need to treat now one section or that with particular emphasis, or to neglect it to a certain extent, became increasingly compelling. Thus, the desire gradually arose to offer a substitute in the form of a book for the sections covered more quickly. At the same time, however, the conviction arose that such a book still needed to be written, that it was also destined to address a far more significant need than that of the relatively small number of students.

In various publications I have released over the 25 years of my work in historical mathematics, I have been pleased to emphasise that the circle of contributors in this field is constantly expanding. Discovery followed discovery, creating entirely new foundations of which the authors of older histories of mathematics, especially Montucla, whose great work, though admittedly flawed in many ways, remains unsurpassed, had no inkling. As knowledge increased, so too did the difficulty of acquiring it. Not everyone is able to obtain the many smaller individual works containing the new information; not everyone even knows them all by title. It seemed necessary once again to provide a comprehensive compilation of everything that constitutes the current body of historical-mathematical sciences, so that mathematicians who wish to seek advice on a historical question can obtain it, as far as is possible today, so that it may become clearer where gaps and doubts still exist, and where independent historical research should begin, with the hope of not wasting its efforts. This, too, is the aim of the now completed volume.

A danger did threaten, however, and, I must admit, not entirely in vain. When a field of knowledge is being studied by many scholars, a comprehensive handbook can easily become outdated while it is still under the author's pen. I have attempted to avoid this danger by condensing the final draft of this volume into the shortest possible time. It was written with the expenditure of all my available labour between 1 November 1879, and 15 March 1880, and I consciously left no treatise unused that had come to my attention up to the latter date, as a glance at the appended notes will prove to the reader. Therefore, during the printing process, for the swift and careful execution of which I extend my warmest thanks to the publishing house, I categorically excluded any factual changes, reserving the right to address in the preface those matters that are of the most recent date, or at least have only recently come to my attention.

It is primarily the work of a French colleague that I must mention in this regard. Mr Paul Tannery has published treatises on the history of Greek mathematics, partly in the "Revue philosophique" and partly in the "Bulletin des sciences mathématiques et astronomiques," which absolutely deserve to be mentioned at various points in this volume. I have been informed of other works by him, which have not yet been published, through his kind letters, and the latter in particular would probably have significantly altered some of my conclusions had I been able to access them earlier. I am entitled to reveal that Mr Tannery examined all the approximate square roots in Archimedes and Heron of Alexandria individually and determined from them that my opinion that those values ​​did not arise according to the formula $√(a^{2} + r) = a + \Large\frac{r}{2a}\normalsize$ is erroneous, that, on the contrary, all the values ​​in question follow from this approximately correct equation, provided only that a few specific additional assumptions are introduced, which relate in particular to the decomposition of a fraction into unit fractions.

The essay "A quelle époque vivait Diophante? par M Paul Tannery" in the Bulletin of Science, Mathematics and Astronomy would also have had a transformative influence on my account, had I known of it earlier. In this eight-page treatise, the author proves that the Diophantus mentioned by Suidas, teacher of Labienus, was by no means the mathematician Diophantus, thus eliminating this support for the assumption that Diophantus lived in the second half of the fourth century BC. Furthermore, it is made probable that Diophantus already lived at the end of the third century BC, a likely contemporary of Pappus, and thus the difficulty that Metrodorus' epigram about Diophantus must chronologically pose disappears.

A comment in a letter from Mr Tannery prompts me to return to the question of Thymaridas's lifetime. In my Math. Beitr. Kulturl., pp. 97 and 380, note 175, I assumed as a matter of course that the inventor of the epanthem [method of solving simultaneous linear equations] was Thymaridas of Tarentum, whom Iamblichus named among the immediate pupils of Pythagoras. Mr Th Martin (Les signes numéraux et l'arithmétique chez les peuples de l'antiquité et du moyen-âge, p. 25) objected, arguing that I was justified in considering Thymaridas of Tarentum to be the correct one, since Iamblichus also mentions a Pythagorean, Thymaridas of Paros, whose age is entirely unknown. This objection seemed so justified to me that I did not return to the possibility that the epanthem had originated before Plato in my text. If, on the other hand, one truly wishes to place Thymaridas and his epanthem so far back, I will be the last to object, since my entire view of the originality of Greek algebra is only confirmed by this.

Due to its late publication date, the Easter program of the Eisenach Realgymnasium, "On the Boethius Question by Prof Dr Weissenborn," could not be considered. The results of my investigations would, incidentally, have been no different, since Mr Weissenborn was no more successful in this last treatise than in the earlier ones in converting me to his conviction of the spuriousness of Boethius' geometry.

So much for works that came to my attention during the printing of this volume. Other recent investigations relate to subjects that will only be addressed in the two following volumes. I intend, in fact, to in a second volume, the materials of which are already combined in a first version for the purposes of my lectures, to trace the development of mathematics from the year 1200 up to Leibniz, and as a third volume to treat Leibniz together with his immediate and indirect successors up to and including Lagrange.

Finally, it remains my pleasant duty to express my gratitude to the colleagues who enabled me to acquire a fairly complete knowledge of the subject matter. The reader will find their individual names listed in the footnotes. However, I could not express my gratitude there to two scholars: Professor E Windisch in Leipzig and Mr Thorbecke in Heidelberg, who proofread the Indian and Arabic sections, respectively, and ensured the correct spelling of the names and words in question. How necessary this was is sufficiently clear from the preceding sections, in which such names appear, albeit not frequently, but in an all the more misspelled form. I apologize for this, citing my lack of knowledge of the languages ​​in question and the diversity of the secondary sources from which I was obliged to draw.

Heidelberg, August 1880.

3.1. From the Foreword.

The second volume of my lectures on the history of mathematics followed the first more slowly than I had originally intended. More quickly than I had hoped, the appearance of the second volume was followed by the necessity of reprinting the first volume, and for this reason I had to interrupt work on the third volume, the first section of which, extending to the year 1700, has been written. In the 13 years that have now passed since the first printing of the first volume was made public, numerous investigations have been carried out, some by long-established collaborators in the field of history, others by newly emerging younger researchers. These investigations have clarified many previously doubtful matters, while also raising new doubts about what was once considered certain. It is perhaps not entirely unjustified if, along with the joy over such advances in the field of knowledge whose study has become my life's purpose, a certain pride is combined, if I imagine that my own work gave the impetus to the suddenly awakened, more vigorous activity. Be that as it may, the reader of this new edition will gain one conviction: that I have endeavoured to utilise everything made available to me over the last 13 years, in some cases right up to the time of printing itself, sometimes by simply adopting the new results, sometimes by merely mentioning them without subscribed to them.

If I may conclude with a request and a wish, the former is that opponents and supporters alike may seek to appreciate the new edition impartially; the latter is that once again new and ever more collaborators may be found to cultivate and expand the field. It is by no means exhausted yet, and work on it is still worthwhile

Heidelberg, December 1893.

3.2. Review by: T.
Revue des Études Grecques 7 (26) (1894), 247-248.

This volume contains the best and most complete history of Greek mathematics, and it is for this reason that it is of interest to our readers. The success achieved by Mr Cantor's work and evidenced by the need for a second edition of the first volume, thirteen years after its appearance, makes it unnecessary for us to praise it, and we can only wish the author to soon publish a third, which he may perhaps have to revise further. For although he has carefully revised his writing to eliminate some inevitable errors in such an extensive work, and although he has made numerous additions to mention the various works devoted since 1876 to the history of ancient mathematics, it has not been possible for him to discuss the question, very recently raised again by Diels, of the period in which Hero of Alexandria lived; whether this author should be placed in the 1st century AD and not around the year 100 BC, its role is obviously to be appreciated in a completely different way.

4.1. Review by: T J McC.
The Monist 7 (2) (1897), 314-317.

It would be impossible to do justice to this monumental work within the brief limits of a book review, even if the task were not rendered supererogatory by the high standing of the work and the acknowledged authority of its author. Cantor's Lectures on the History of Mathematics are the work of a man who has unswervingly devoted a life-time to this single task, who thirty-three years ago was well known for his important contributions to this subject, and who can now in the second edition of the first volume of his great work point with pride to the impulse and awakened interest which his endeavours have aroused in the historical studies of his science. He has had many predecessors, each of whom has distinguished himself in certain branches and by certain excellences - Montucla who excelled in lucidity, elegance, and popularity; Libri who seems to have united in an eminent degree all the qualities necessary to the makeup of a writer of a universal mathematical history, but whose work extends only to the period preceding Galileo in Italy; Hankel, whose contributions to the history of early mathematics are marked by much acumen; and several others. Nevertheless, it may safely be said that profundity, accuracy, and extensiveness of treatment have never before in any history of mathematics been so thoroughly and intimately united as in the three volumes constituting these Lectures of Moritz Cantor. The first volume embraces the period from earliest antiquity to the year 1200 A. D. and is now in its second edition, thoroughly revised and brought down to date (1894). The second volume embraces the time from 1200 to 1668 A. D. The third and last volume will comprise the time from 1668 to 1759, concluding with the first epoch-making papers of Lagrange in the Proceedings of the Turin Academy. The first two instalments only of this third volume have appeared (1894-1896), the third is still in preparation.

In the Introduction to Volume I., which contains 883 pages with a chart of ancient numerical characters, we have some brief philosophical considerations concerning the psychological origin of mathematical operations and the invention of numerical signs. As to the theory that the first numerical words originally denoted not numbers but definite objects, Prof Cantor remarks that philology has not succeeded in proving its position. Nor can he himself offer much to the solution of the problem. We are on sure ground, he says, only when we come to derivative numerical words. We have also some interesting remarks on the various systems of numbers, namely, the decimal, vigesimal, undecimal, sexagesimal systems, etc. The true history of mathematics, the author contends, begins only with the first written monuments and inscriptions which are presumably found in Egypt. 55 pages are devoted to the mathematics of Egypt, 31 to that of the Babylonians, 65 to that of the Indians, 29 to that of the Chinese, and 118 to that of the Arabs. The remaining three divisions of the first book are devoted to the mathematical achievements of the Greeks, which naturally take up the largest space, and to those of the Romans and of the early mediaeval monasteries. The researches of the ancient nations are extremely interesting, not only from the point of view of mathematical history but equally so from that of philosophy and psychology. Their insight and errors are of extreme importance, and it is both profitable and fascinating to witness the primitive operations of the human mind as employed upon this its surest and most fundamental subject. Of the Greeks the most interesting chapters are those relating to Pythagoras and Archimedes. Dr Cantor gave long ago, in his Mathematische Beiträge zum Kulturleben der Völker, 1863, a charming appreciation of the life and achievements of Pythagoras, only differing from the chapter on the great philosopher in the present work by being more popular and less exhaustive. In Archimedes we have the man who may be regarded as the incarnation of the mathematical genius of antiquity, and the chapter devoted to him shows at its best the precious heritage which he left to us. It is surprising to note to what a pitch the Indians advanced arithmetic and algebra, and also to follow the work of the Arabs. In fine, the entire first volume is a book which can be read and consulted by writers of average elementary mathematical attainments, and offers material from which all readers may draw profit and entertainment.

The year 1200 was an important one in the history of European Mathematics, and is fitly chosen as the beginning of the second volume. Christianity was then in possession of the art of arithmetic, as it had been recovered from its different ancient and Eastern sources. It was also in possession of the zero and of the no less important principle of the positional value of figures. Algebra, as far as equations of the first and second degree, had been compassed, the geometry of Euclid, the astronomy of Ptolemy, the writings of Theodosius, and of Menelaus, existed in Latin translations, and appositely to the right time came the right men who were destined to achieve great things in mathematical science, - Leonardo of Pisa and Jordanus Nemorarius.

"Leonardo," says Cantor, "was a practised arithmetician and geometer, an ingenious algebraist, conversant with the application of algebra to geometry, as well as a creative genius of high rank in the theory of numbers." Jordanus Nemorarius was a priest and member of a powerful order; he fell little short of Leonardo in point of mathematical ability, but by reason of his ecclesiastical position his influence was more powerful and decisive than that of the other who was a merchant. From these two great landmarks the second volume traces the history of mathematics through the early developments of algebra and geometry in England, France, Italy, and Germany, including Nicolaus of Cusa, Regiomontanus, Leonardo da Vinci, Luca Pacioli, Michael Stifel, etc., down to the researches on cubic equations by Cardano and Tartaglia, where the first instalment ends. The second instalment is devoted to the advances made in cyclometry and trigonometry by Vieta, Van Roomen, etc., to the researches on equations of the fourth degree by Bombelli, etc., Kepler's and Pascal's investigations in geometry, the rise of mechanics, logarithms, continued fractions, the theory of numbers, analytical geometry, and lastly to the germs of the infinitesimal calculus in Kepler, Cavalieri, and most notably of all in Fermat. The volume concludes with the year 1668-1669, a momentous epoch in the history of mathematics, for at that time Gottfried Wilhelm Leibnitz was publishing his Doctor's dissertation at Leipzig, and Isaac Newton had just been elected to the chair of Mathematics in Cambridge University England.

With this epoch the second volume begins. The period which follows is of all that of most import for modern mathematics, and its utterances are associated with the most interest for professional readers. The first instalment deals with the "geometrical character" of Leibnitz, with certain developments of commercial arithmetic, with the history of series as developed by Mercator, Brouncker, Gregory, Newton, Leibnitz, Halley, De Moivre, James Bernoulli, with continued fractions, the theory of curves, etc. We have also in this instalment a chapter on Newton and Leibnitz's first discoveries in the domain of the infinitesimal calculus, chapters on Leibnitz and on the brothers Bernoulli both preceding and during their famous strife. The great controversy concerning the priority of invention of the differential calculus between the followers of Newton and Leibnitz, - a controversy which excited the mathematical world for more than twenty-five years, and which was really not definitively settled until the present century, - takes up a good part of the second and latest instalment of Cantor's third volume. There is now, of course, little to be said upon the subject of this controversy, and Cantor does not claim to add much to its elucidation, except to point out an omission made in the copying of a letter by Leibnitz to Wallis of the word hodie, which might easily have led to certain suspicions in the English mind as to Leibnitz's fair dealings. His conclusion is that now that both great inquirers have received their just share of the credit owing to them for their discoveries, a careful and unprejudiced examination of the controversy unfortunately shows that the conduct of the matter reflected no little discredit upon all parties concerned. The last instalment closes with the developments of the calculus, of Algebra, and of analytical and projective geometry down to the year 1726. The work of Euler and his period remains.

Cantor's history now comprises 2218 pages. The final instalment, reaching to the year 1759, and which is yet to appear, will certainly not increase its bulk to much over 2600 pages, leaving the vast material from the date of Lagrange's first memoirs on to be elaborated by another hand. The history will thus hardly exceed in size some of its predecessors, but it will contain proportionately more material, from its being almost exclusively devoted to the solider scientific aspects of its sub-ject and not so much to biographical and personal details, which served so greatly to swell the work of Montucla. In fine, it is far and away the concisest, yet most comprehensive and authoritative treatment of the subject that we have. As such, it is the indispensable adjunct of every mathematical worker and absolutely necessary in every mathematical library.

5.1. Note.

In what follows cameralists are mentioned several times. Cameralists were 17th- and 18th-century German-speaking public administrators, civil servants, and economists who shaped state policy to maximize the financial power and authority of their rulers.

5.2. From the Foreword.

For many years, I have given a two-hour lecture every winter at the University of Heidelberg for cameralists, which is announced under the title Political Arithmetic. Often asked whether I could recommend a book on this subject, I have always had to answer in the negative, not as if no work on political arithmetic had been published, but because none contains everything I discuss in those lectures. I resolved to publish the lecture notes that had gradually accumulated, and while I was preparing them for publication, I came to the conclusion that the resulting little book might be able to satisfy a real need beyond the narrow circle of cameralist readers.

Nowadays, it will be necessary for almost everyone to understand something of the accounting methods of stock market transactions, which are limited to buying and selling, however dispensable, indeed however harmful, knowledge of those forms of business peculiar to the stock market game may prove to be under certain circumstances In this booklet, the reader will find information on one thing while deliberately avoiding the other, unless one considers the promissory note business a stock market game, which is more accurately described as a variant of the lottery. Municipal and state administration make it desirable to know something about taking out and repaying loans. These matters are addressed here. Insurance has moved to the forefront of modern life. I hardly believe that one can obtain such concise information on this topic elsewhere as here. For some of the subjects discussed, a reference to the Civil Code for the German Reich, which will soon come into force, is necessary. I am fulfilling a duty I consider pleasant by naming my son, attorney Dr Otto Cantor in Karlsruhe, as my expert advisor and collaborator in this area. Logarithmic arithmetic is not everyone's cup of tea. In an appendix, there are calculated tables of $1.0 p^{n}$ and $1.0 p^{-n}$ for the three currently most common interest rates of $3, 3\large\frac{1}{2}\normalsize$, and 4 percent over all years from $n=1$ to $n=100$, so that logarithms become almost entirely unnecessary. Such content perhaps justifies my above-mentioned conjecture.

Two conditions, however, had to be met if my book was to fulfil its expanded purpose. It had to be written legibly, and it had to be available at a relatively low price. In the latter respect, the publisher's cooperation has probably done everything possible. Deciding whether a book is legible is a matter for the reader, not the author I can only say that I have made every honest effort to remain understandable, even though the subject matter meant that mathematical formulas could not always be avoided. Interspersed examples serve to illustrate them, so that, I hope, they will not present an insurmountable obstacle for the non-mathematical reader. Moreover, there is a relatively loose connection between the individual chapters, so that one can quickly move on to easier topics by skimming over what is unclear. However, mathematical readers, such as secondary school teachers, who are able to work with their students on compound interest and probability problems, will have to excuse the relatively broad scope of my presentation as well as the not always perfectly rigorous proofs. Satisfying all readers is impossible, and therefore I felt compelled to attempt a compromise between the often conflicting interests of the reader.

Heidelberg, October 1898.

5.3. Review by: G Schanz.
FinanzArchiv / Public Finance Analysis 16 (1) (1899), 473-474.

"Political Arithmetic" has undergone many transformations since the days of Petty, who gave the name to the famous calculations of his friend Graunt (1683). With the advent of modern statistics, it seemed to be completely absorbed by it and to disappear as such. However, it has persisted despite this, albeit with much new content. If it has been able to do so, it is due to the distinct need, which exists in wide circles, for the handling of certain calculations and knowledge of the related relationships.

It can be left open whether this field deserves the name of an independent scientific discipline; its great usefulness cannot be doubted, and one must gratefully welcome the opportunity given at German universities to acquire this knowledge. The author, known for his history of mathematics, also gives a two-hour lecture every winter at the University of Heidelberg for cameralists under the name "Political Arithmetic," and the present booklet originated from these lectures. Cantor seeks to delimit the field of political arithmetic by designating as all tasks of political arithmetic the application of interest calculation, specifically simple and compound interest calculation, not connected or linked with the application of probability theory.

The presentation is divided into 9 chapters: 1. Simple interest; 2. Compound interest; 3. Fundamentals of probability theory; 4. Lottery loans; 5. Insurance; 6. Mortality tables; 7. Simple life insurance and immediate annuity; 8. Dividend calculation; 9. Less simple types of insurance based on mortality.

On the whole, the booklet is very skilfully written and exceptionally suitable for students.

I must, of course, make some reservations. Anyone who acquires the knowledge presented in the book wants, and should also be able, to apply it practically to a certain extent. However, the author by no means always provides this opportunity. This is particularly true of life insurance; the student sees the formulas, but the auxiliary tables for their application are missing; the two mortality tables provided are insufficient; a model of a truly useful presentation in this respect is the small book by Professor Spitzer, "Instructions for Calculating Annuities and Expectancies as well as Disability Pensions, Dowries, and Health Insurance."

In the presentation of the current account interest calculation, it is objectionable that the tiered method is not also shown, especially since it is generally required in court and only yields correct results when the interest rate differs between credit and debit balances, which seems to have been overlooked on page 11. It is also incorrect, as the author notes on pages 7 and 8, that English banks did not grant current account interest; he is probably confusing the English bank with the English banking houses. The explanation of the symbols behind the price is incomplete insofar as the expressions bz. G., bz. P. have been omitted (I refer him, among other things, to my article "Stock Exchange" in the Dictionary of Economics, p. 426); it is also incorrect that only stockbrokers are authorised to broker purchases on the stock exchange.

5.4. Review by: Otto Larsen.
Nyt tidsskrift for matematik 10 (A) (1899), 112-115.

In this book, the renowned historian of mathematics Professor Cantor has presented an account of business practices from everyday life, where it is necessary to possess a somewhat greater arithmetical insight than that which is usually imparted in schools, when one wants to be able to calculate one's chances of a certain investment or to exercise control over the management of one's money by others. It will be understood, then, that it is banking and insurance that the book deals with.

The first - and also the biggest - chapter is entitled simple interest. The reason why this has become so relatively comprehensive is that the author goes into not only business-related day counting and the use of the general divisor, which can and should also be included in the mathematics teaching for the real-life final exam, but also the drawing up of current accounts and an account of their individual items. However, the word current account is taken here only in its narrower meaning as a generally semi-annual statement of the balance between a bank and a person (a company, a corporation) who has his various money transactions carried out through the bank. In other words, it is the doctrine of securities that is presented here, and it is done in a fairly exhaustive and highly instructive way.

In the second chapter, compound interest is treated, in that the general and some more advanced formulas, including those for amortization, are derived and applied in well-chosen examples. Quite interesting here is a remark about the influence that a bank clause stating that no interest is paid on an amount less than 10 marks, and for larger amounts only on the highest multiple of 10 marks contained therein, can have on the growth of a capital: When the University of Heidelberg celebrated its 500th anniversary in 1886, 10 marks were deposited in the savings bank there and it was decided that this sum with accrued interest should be placed at the disposal of the university on its 1000th anniversary; if the annual interest were set at just 3%., the savings bank would at that time would have to pay 26218778 marks if the aforementioned clause did not exist, but, when this is taken into account, only (!) 14975273.80 marks. And the reduction is of course even more noticeable in relative terms, as long as the amount is smaller.

This is followed by a presentation of the basic features of probability theory, in which all the essentials are included, and after having introduced the concepts of mathematical and moral hope, of which the latter is only briefly mentioned, the author proceeds to the treatment of those investments of money which are more like lottery games, including such loans, by which the creditors' interest and instalment certificates are redeemed for different amounts, the size of which is determined each time for each individual by drawing lots, and the various types of insurance. Of these, those which have nothing to do with life and death are first mentioned, especially the fire insurance system. In the following section the principles for drawing up mortality tables are given, and it is shown which shortcomings are inherent in them due to their changing material and summary treatment; but like the previous author does not go into detail about what reservations one should therefore take when using them, so the whole presentation of the insurance system, which is continued in the last chapters with life insurance of all kinds, is the weakest point of the book. The attempt to treat, for example, the doctrine of premium payment in a satisfactory way and yet in such a way that it can be appropriated by people whose mathematical knowledge is limited to what is usually included before the introduction of irrational numbers, cannot be said to have succeeded, but is interesting in itself, and one also encounters several instructive considerations here. Nor has it been possible to avoid, according to the plan of the book, that a mathematician will find the presentation in several places smeared, the formulas somewhat ugly and some expressions not unobjectionable. Which will also contribute to the fact that the book here at home will hardly have the distribution that one would otherwise expect after the ever-increasing demand, given the ever-increasing demand to learn the subject matter it deals with, but it must be recognised that it quite naturally rests in legal terms on German laws and is particularly aimed at German conditions. Among other things, this seems to have influenced the chapter on the distribution of dividends by companies.

The book is, as one might expect, provided with numerous historical notes and is presented in a format that is as nice as the affordability has allowed.

6.1. Review by: Anon.
The Mathematical Gazette 4 (76) (1908), 399-400.

It is fifteen years since the second edition of the first volume of Cantor's Vorlesungen made its appearance, and last year a third edition was called for. After the completion of Vol. iii., which brought this monumental work up to 1758, the author felt that the time had come when the claims of advancing years were strong enough to force him to leave the completion of his magnum opus to others. He must have felt that he had not lived in vain and that an historical school worthy of German traditions had grown up around him, when, inspired by his example and enthusiasm, men were found able and eager to continue his great work under his direction. We are unfortunate in that the health of the aged savant has prevented him from giving that full personal supervision which is so essential in securing a general unity of treatment when the mathematical labours of half a century are divided among so many hands, however individually competent they may be. And when we say that no less than nine monographs by nine men constitute this fourth volume, it is not surprising to find, that although there is less diversity of treatment than might be expected, yet the volume is not what it would have been had Cantor been able to continue his colossal task. As it is, he had intended to append to this portion of the work a general treatment of the progress of mathematics during the latter part of the eighteenth century. We are, however, disappointed at the absence of what would of course have been a masterly survey of the development of ideas in that period, and greatly regret that the plan had to be abandoned. The greatest mathematical historiographer found himself compelled to be content with a chronological index. Even so, it is one of the most useful features in the book. His Überblick gives a mass of information in a couple of dozen pages, fortified with ample cross references to the sections of this volume. Dr Günther was entrusted with the purely historical section. Under this heading we have accounts of works dealing with the history of the subject, monographs on its various branches, dictionaries, and editions of the works of classical authors. Arithmetic, the Theory of Equations, and the Theory of Numbers fell to the lot of Fl Cajori. Netto of Giessen writes on Permutations and Combinations, the Theory of Probability, and on Imaginary Series. Elementary Geometry, Mensuration, etc., and the doctrine of parallels as it stood in this period are undertaken by Bobynin. The additions to Trigonometry made by Euler and his contemporaries, with what is called Polygonometry, Polyedrometry and Cyclometry are adequately dealt with by Braunmühl. Analytical Geometry of the Plane and Space is admirably handled by Kommerell, while Gino Loria is quite at home in his treatment of Perspective and the golden age of Descriptive Geometry. The calculus forms, as might be expected, one of the longest of the monographs, forming about one-fifth of the book. This was Vivanti's share in the work, while Differential Equations and the Calculus of Variations fell to C R Wallner. When we remember that we have now got to the days when most of the Bernoullis, Bezout, Condorcet, d'Alembert, Arbogast, Euler, Lagrange, Lambert, Laplace, Carnot, Legendre, Monge, etc., were alive, it may be surmised that the mathematician will find this volume as full of good things as the proverbial egg is of meat. And so it is.