1.1. Preface.

Modern Analytical Geometry excels the method of Des Cartes in the precision with which it deals with the Infinite and the Imaginary. So soon, therefore, as the student has become familiar with the meaning of equations and the significance of their combinations, as exemplified in the simplest Cartesian treatment of Conic Sections, it seems advisable that he should at once take up the modern methods rather than apply a less suitable treatment to researches for which these methods are especially adapted.

By this plan be will best obtain fixed and definite notions of what is signified by the words infinite and imaginary, and much light will be thereby thrown upon his knowledge of Algebra, while at the same time, his facility in that most important subject will be greatly increased by the wonderful variety of expedient in the combination of algebraical equations which the methods of modern analytical geometry present, or suggest.

With this view I have endeavoured, in the following pages, to make my subject intelligible to those whose knowledge of the processes of analysis may be very limited; and I have devoted especial care to the preparation of the chapters on Infinite and Imaginary space, so as to render them suitable for those whose ideas of geometry have as yet been confined to the region of the Real and the Finite.

I have sought to exhibit methods rather than results, - to furnish the student with the means of establishing properties for himself rather than to present him with a repertory of isolated propositions ready proved. Thus I have not hesitated in some cases to give a variety of investigations of the same theorem, when it seemed well so to compare different methods, and on the other hand interesting propositions have sometimes been placed among the exercises rather than inserted in the text, when they have not been required in illustration of any particular process or method of proof.

In compiling the prolegomenon, I have derived considerable assistance from a valuable paper which Professor Tait contributed five years ago to the Messenger of Mathematics. My thanks are due to Professor Tait for his kindness in placing that paper at my disposal for the purposes of the present work, as well as to other friends for their trouble in revising proofs and collecting examples illustrative of my subject from University and College Examination Papers.

Liverpool
15 September, 1866

2.1. Preface to the First Edition.

The following pages are a reproduction of lectures on Arithmetic, given in Queen's College, Liverpool, in the Michaelmas Term, 1866. Many of the students to whom the lectures were addressed were just entering upon the study of algebra, and it seemed well, while the greater part of their time was devoted to the somewhat mechanical solution of examples necessary to give them a practical facility in algebraical work, that their logical faculties should be meanwhile exercised in the thoughtful applications of the arithmetical art with which they were already familiar.

I had already discovered, that the usual method of treating questions of selection and arrangement was capable of modification and so great simplification, that the subject might be placed on a purely arithmetical basis; and I deemed that nothing would better serve to furnish the exercise which I desired for my classes, and to elicit and encourage a habit of exact reasoning, than to set before them, and establish as an application of arithmetic, the principles upon which such questions of "choice and chance" might be solved.

The success of my experiment has induced me to publish the present work, in the hope that the expositions already accepted by a limited audience may prove of service in a wider sphere, in conducing to a more thoughtful study of arithmetic than is common at present; extending the perception and recognition of the important truth. that arithmetic, or the art of counting, demands no more science than good and exact common sense.

In the first chapter I have set down and established as arithmetical rules all the principles usually required in estimating the choice which is open to us in making a selection or arrangement out of a number of given articles under given conditions. In the second chapter I have explained how different degrees of probability are expressed arithmetically, and how the principles of the preceding chapter are applied to the calculation of chances. These two chapters will prove intelligible to anyone who understands the first principles of arithmetic, provided he will consider each step as be goes on; not content with the mere statement of any rule, but careful to follow the explanations given end to recognise the reason of each successive principle.

For the sake of mathematical students I have added, as an appendix, a new treatment of permutations and combinations with algebraical symbols. In my experience as a teacher I have found the proofs here set forth more intelligible to younger students than those given in the text books in common use.

Liverpool
1st February, 1867.

2.2. Preface to the Second Edition of 1870.

In this second edition I have enlarged the appendices so as to meet the wants of advanced students. I have also added a collection of upwards of one hundred miscellaneous examples, which I think will add very much to the utility of the book.

It should be observed that the two chapters headed respectively Choice and Chance are simply arithmetic, and ought not to be beyond the comprehension of the ordinary reader who has never seen an algebraical symbol. But while expressly written for unscientific readers, they have been found very helpful to the young mathematician, when he was about to read in his algebra the hitherto difficult and embarrassing chapters on permutations and combinations, or on probability.

The appendices are addressed entirely to algebraical students. In the first appendix the usual theorems respecting permutations and combinations are established by new proofs, the same reasoning which was pursued with as little technicality as possible in the body of the work, being here expressed in algebraical language.

In the second and third appendices, which are newly added in this edition, a series of propositions are given which are not usually found in textbooks of algebra. But I can see no reason why examples of such simple propositions as the xiiith and xxvth should be excluded from elementary treatises in which more complex but essentially less important theorems generally find place.

The classification of a variety of propositions under the titles of Distribution and Derangement will contribute (it is hoped) to disentangle the confusion in which all questions involving selection or arrangement are commonly massed together, and will facilitate in some degree that precision of language and clearness of expression which ought always to be aimed at in mathematics.

In the fourth appendix I have exhibited the seeming paradox that a wager which is mathematically fair is mathematically disadvantageous to both contracting parties. And I have endeavoured to cast into a simple and intelligible form the principles upon which the difficulties of the celebrated Petersburg problem are explained.

St John's College
1 January 1870

3.1. Preface.

This Manual is intended to accompany oral teaching from the black-board. A. specimen of such teaching is given in the "Introductory Lesson."

My arrangement of the subject is distinguished by three points, two of which are, I believe, quite novel, while the third has a historical sanction, being, in fact, the temporary use, as a stepping-stone for the learner, of one of the stages by which the system of notation in present use was originally reached.

The three points are as follows:-

1. Instead of explaining - (minus) as a sign of operation, and afterwards showing that it may be a sign of affection as well, the reverse order is adopted.
   
   
2. Letters and symbols of quantity are not introduced until the first four rules and the use of brackets are thoroughly understood in their application to positive and negative numbers.
   
   
3. The notation of indices is not used until the pupil is familiar with such quantities as $xx, xxx$ &c.

The advantage is gained that the pupil has not to work in the dark with mysterious letters, the use of which he is not to understand until afterwards, but from the beginning he is really learning something and grasping some new ideas.

A good teacher will strive to excite in his pupil an enthusiasm for his subject. And even in the processes of elementary Algebra sources of pleasurable interest will be found, especially in the charms of symmetry and the excitement of rapid simplification. Therefore, in framing these exercises, I have had great regard to symmetry, and have largely provided for the gratification of that pleasure which we enjoy when we find that an expression, very complicated in appearance, is capable of being reduced to a very simple form.

I have not only put the most liberal construction on the Government requirement of "Algebra to Simple Equations inclusive," for Standard VI of the New Code, but I have also added exercises on parts of the subject which, though they come logically within the defined limits, are not understood to be included in the requirements of the Education Department. Thus the student will find in the book every variety of exercise that he can require until he is ready to approach Quadratic Equations, which may come into a sequel.

The different points which the teacher will need to explain and illustrate as the student proceeds are indicated in small type, and a brief syllabus is given of the propositions which constitute the Theory of Fractions.

4.1. Review in The Eagle.

The following is taken from The Eagle, Lent 1883, Vol XII (1883), 307.

Mr Whitworth has recently issued the Churchman's Almanac for Eight Centuries (Wells, Gardner, & Co., 2/6).

The year's Calendar admits of 70 variations. Easter may fall on any day from March 22 to April 25, and the year may be a Leap year or an ordinary year.

Mr Whitworth's book contains the 70 different Calendars of Sundays of the year, and above each Calendar a list of the years to which it is applicable from 1201 to 2000. Other tables are given indicating the page on which the proper Calendar of any particular year is to be found.

The use of these Calendars is two-fold. The simpler and less important is that of giving beforehand the Almanacs of the next century. In doing the same thing for the past the Almanacs are themselves a history and also an important aid in determining the chronology of history. The earlier dates refer to the Julian style, the later to the Gregorian, whilst the transition period, and especially the months of change. October 1582 at Rome and September 1752 in England, are carefully discussed. The value of the compilation as a handy guide to the determination of chronology will be sufficiently clear to anyone who knows how dependent we must often be for our dates upon the concurrence of epochs with certain days of the week, feasts, or celebrations, or one with another.

4.2. Review in Bibliographer.

The following is taken from Bibliographer 3 (4) (1883), 112.

This called the Churchman's Almanac, but it will be found of great value to the historian, and will save him a large amount of calculation. The Introduction contain a full account of the mode by which the change from the Julian to the Gregorian calendar made in England in 1752. Some curious instances of the differences between the two calendars are given. In some years the difference was imply that of eleven days, but in other years a much more serious difference occurred - the paschal full moon of one system being a lunation later than that of the other, so that a difference of four or even five weeks occurred in the keeping of Easter. For example, in 1701 the Julian Easter was on April 20th, old style, which was the same day as May 1st, new style. But the Gregorian Easter fell on March 27th, new style, or five week earlier. In the next year there is no such discrepancy, the Julian Easter Day being April 5th, 1702, old style, which is the same day April 16th, new style, on which the Gregorian Easter falls. The change of style is so great a puzzle to many, that such a help as this book holds out is sure to be welcomed.

5.1. From the Preface.

Only once in the following proofs have I made use of the Integral Calculus, and then only to extend to the case of a fractional index a proposition which is proved algebraically when the index is a positive integer.

The general Theorem on page 9 has a great fascination over me. At one time I thought that I could establish it almost intuitively from elementary considerations, but there appeared a want of rigour in the argument. The proof which I have substituted if somewhat cumbrous is rigorous; but I hope that some more expert thinker will show that the proposition can be deduced directly from general considerations.  I ask those who possess the fourth edition of Choice and Chance to regard the present tractate as an additional chapter of that book, to be incorporated in any future edition.