Notes

¹The assumption that this is possible is equivalent to the assumption of what is known as the Axiom of Archimedes.

²This is the point of view which was adopted in the first edition of this book.

³In these sections I have borrowed freely from Appendix I of Bromwich’s Infinite Series.

⁴It will be convenient to denote a section, corresponding to a rational number denoted by an English letter, by the corresponding Greek letter.

⁵There are also sections in which every number belongs to the lower or to the upper class. The reader may be tempted to ask why we do not regard these sections also as defining numbers, which we might call the real numbers positive and negative infinity. There is no logical objection to such a procedure, but it proves to be inconvenient in practice. The most natural definitions of addition and multiplication do not work in a satisfactory way. Moreover, for a beginner, the chief difficulty in the elements of analysis is that of learning to attach precise senses to phrases containing the word ‘infinity’; and experience seems to show that he is likely to be confused by any addition to their number.

⁶I.e. is included in but not identical with .

⁷See Ch. II, Misc. Exs. 22.

⁸I.e. there are two values of $x$ for which $a{x}^2+2bx+c=0$. If $b^2-ac<0$ there are no such values of $x$. The reader will remember that in books on elementary algebra the equation is said to have two ‘complex’ roots. The meaning to be attached to this statement will be explained in Ch. III. When $b^2=ac$ the equation has only one root. For the sake of uniformity it is generally said in this case to have ‘two equal’ roots, but this is a mere convention.

⁹The figure is drawn to suit the case in which $b$ and $c$ have the same and $a$ the opposite sign. The reader should draw figures for other cases.

¹⁰I have taken this construction from Klein’s Leçons sur certaines questions de géométrie élémentaire (French translation by J. Griess, Paris, 1896).

¹¹This supposition is merely a hypothesis adopted (i) because it suffices for the purposes of our geometry and (ii) because it provides us with convenient geometrical illustrations of analytical processes. As we use geometrical language only for purposes of illustration, it is not part of our business to study the foundations of geometry.

¹²A proof will be found in Ch. VII.

¹³See Hobson’s Trigonometry (3rd edition), pp. 305 et seq., or the same writer’s Squaring the Circle (Cambridge, 1913).

¹⁴The discussion which follows is in many ways similar to that of § 6. We have not attempted to avoid a certain amount of repetition. The idea of a ‘section,’ first brought into prominence in Dedekind’s famous pamphlet Stetigkeit und irrationale Zahlen, is one which can, and indeed must, be grasped by every reader of this book, even if he be one of those who prefer to omit the discussion of the notion of an irrational number contained in §§ 6–12.

¹⁵There were three in § 6.

¹⁶This was not the case in § 6.

¹⁷The reader will hardly require to be reminded that this course is adopted solely for reasons of linguistic convenience.

¹⁸This clause is of course unnecessary if $\xi$ does not itself belong to $S$.

¹⁹I borrow this instructive example from Prof. H. S. Carslaw’s Introduction to the Calculus.

²⁰If $B=0$, $y$ does not occur in the equation. We must then regard $y$ as a function of $x$ defined for one value only of $x$, viz. $x=-C/A$, and then having all values.

²¹Polar coordinates are sometimes defined so that $r$ may be positive or negative. In this case two pairs of coordinates—e.g.  and —correspond to the same point. The distinction between the two systems may be illustrated by means of the equation $l/r=1-ecos\theta$, where $l>0$, $e>1$. According to our definitions $r$ must be positive and therefore $cos\theta <1/e$: the equation represents one branch only of a hyperbola, the other having the equation $-l/r=1-ecos\theta$. With the system of coordinates which admits negative values of $r$, the equation represents the whole hyperbola.

²²It will be found convenient to take the scale of measurement along the axis of $y$ a good deal smaller than that along the axis of $x$, in order to prevent the figure becoming of an awkward size.

²³The definitions of the circular functions given in elementary trigonometry presuppose that any sector of a circle has associated with it a definite number called its area. How this assumption is justified will appear in Ch. VII.

²⁴See Chs. IV and V for explanations as to the precise meaning of this phrase.

²⁵We assume that the effects of the earth’s curvature may be neglected.

²⁶It is hardly necessary to caution the reader against confusing this use of the symbol  and that of Chap. II (Exs. XVI. and Misc. Exs.).

²⁷Strictly speaking we ought, by some similar difference of notation, to distinguish the actual length $x$ from the number $x$ which measures it. The reader will perhaps be inclined to consider such distinctions futile and pedantic. But increasing experience of mathematics will reveal to him the great importance of distinguishing clearly between things which, however intimately connected, are not the same. If cricket were a mathematical science, it would be very important to distinguish between the motion of the batsman between the wickets, the run which he scores, and the mark which is put down in the score-book.

²⁸The two preceding examples are taken from Willard Gibbs’ Vector Analysis.

²⁹The phrase ‘real number’ was introduced as an antithesis to ‘imaginary number’.

³⁰We shall sometimes write $x+iy$ instead of $x+yi$ for convenience in printing.

³¹See Appendix I.

³²It is evident that $\mid z\mid$ is identical with the polar coordinate $r$ of $P$, and that the other polar coordinate $\theta$ is one value of $amz$. This value is not necessarily the principal value, as defined below, for the polar coordinate of § 22 lies between $0$ and $2\pi$, and the principal value between $-\pi$ and $\pi$.

³³It will sometimes be convenient, for the sake of brevity, to denote $cos\theta +isin\theta$ by $Cis\theta$: in this notation, suggested by Profs. Harkness and Morley, De Moivre’s theorem is expressed by the equation .

³⁴The numbers $a+\sqrt{b}$, $a-\sqrt{b}$, where $a$, $b$ are rational, are sometimes said to be ‘conjugate’.

³⁵We suppose that as we go round the triangle in the direction $ABC$ we leave it on our left.

³⁶In the last case $N$ depends on the time, and convict $x$, where $x$ has a definite value, is a different individual at different moments of time. Thus if we take different moments of time into consideration we have a simple example of a function of two variables, defined for a certain range of values of $t$, viz. from the time of the establishment of Dartmoor prison to the time of its abandonment, and for a certain number of positive integral values of $x$, this number varying with $t$.

³⁷Here and henceforward we shall use in the sense of Chap. II, i.e. as the greatest integer not greater than $x$.

³⁸See Bromwich’s Infinite Series, p. 485.

³⁹There is a certain ambiguity in this phrase which the reader will do well to notice. When one says ‘such and such a theorem is almost obvious’ one may mean one or other of two things. One may mean ‘it is difficult to doubt the truth of the theorem’, ‘the theorem is such as common-sense instinctively accepts’, as it accepts, for example, the truth of the propositions ‘$2+2=4$’ or ‘the base-angles of an isosceles triangle are equal’. That a theorem is ‘obvious’ in this sense does not prove that it is true, since the most confident of the intuitive judgments of common sense are often found to be mistaken; and even if the theorem is true, the fact that it is also ‘obvious’ is no reason for not proving it, if a proof can be found. The object of mathematics is to prove that certain premises imply certain conclusions; and the fact that the conclusions may be as ‘obvious’ as the premises never detracts from the necessity, and often not even from the interest of the proof. But sometimes (as for example here) we mean by ‘this is almost obvious’ something quite different from this. We mean ‘a moment’s reflection should not only convince the reader of the truth of what is stated, but should also suggest to him the general lines of a rigorous proof’. And often, when a statement is ‘obvious’ in this sense, one may well omit the proof, not because the proof is in any sense unnecessary, but because it is a waste of time and space to state in detail what the reader can easily supply for himself.

⁴⁰We naturally suppose that neither $a_0$ nor $b_0$ is zero.

⁴¹This will certainly be the case as soon as $PQ/2^n<\epsilon$.

⁴²These examples are particularly important and several of them will be made use of later in the text. They should therefore be studied very carefully.

⁴³The binomial theorem for a positive integral exponent, which is what is used here, is a theorem of elementary algebra. The other cases of the theorem belong to the theory of infinite series, and will be considered later.

⁴⁴The reader should be warned that the words ‘divergent’ and ‘oscillatory’ are used differently by different writers. The use of the words here agrees with that of Bromwich’s Infinite Series. In Hobson’s Theory of Functions of a Real Variable a series is said to oscillate only if it oscillates finitely, series which oscillate infinitely being classed as ‘divergent’. Many foreign writers use ‘divergent’ as meaning merely ‘not convergent’.

⁴⁵All the results of Exs. XXIX may be extended, with suitable modifications, to decimals in any scale of notation. For a fuller discussion see Bromwich, Infinite Series, Appendix I.

⁴⁶An infinite aggregate of numbers does not necessarily possess a least member. The set consisting of the numbers

for example, has no least member.

⁴⁷A number of simple proofs of this result are given by Hardy and Littlewood, “Some Problems of Diophantine Approximation”, Acta Mathematica, vol. xxxvii.

⁴⁸A few proofs given in Ch. VIII can be simplified by the use of the principle.

⁴⁹A proof that , and that therefore each function tends to the limit $e$, will be found in Chrystal’s Algebra, vol. ii, p. 78. We shall however prove this in Ch. IX by a different method.

⁵⁰Exs. 8–12 are taken from Bromwich’s Infinite Series.

⁵¹Thus $\sqrt{x}$ stands in this chapter for the one-valued function $+\sqrt{x}$ and not (as in § 26) for the two-valued function whose values are $+\sqrt{x}$ and $-\sqrt{x}$.

⁵²We shall sometimes find it convenient to write $+\infty$, $x\to +\infty$, instead of $\infty$, $x\to \infty$, .

⁵³In the corresponding definition of § 62, we postulated that for all values of $n$, and not merely when $n\geqq n_0$. But then the two hypotheses would have been equivalent; for if when $n\geqq n_0$, then for all values of $n$, where $K\prime$ is the greatest of , , …, and $K$. Here the matter is not quite so simple, as there are infinitely many values of $x$ less than $x_0$.

⁵⁴For some further discussion of the notion of a function bounded in an interval see § 102.

⁵⁵Thus in Def. A of § 93 we make a statement about values of $y$ such that $0<y\leqq y_0$, the first of these inequalities being inserted expressly in order to exclude the value $y=0$.

⁵⁶In the examples which follow it is to be assumed that limits as $x\to 0$ are required, unless (as in Exs. 19, 22) the contrary is explicitly stated.

⁵⁷The proofs of the inequalities which are used here depend on certain properties of the area of a sector of a circle which are usually taken as geometrically intuitive; for example, that the area of the sector is greater than that of the triangle inscribed in the sector. The justification of these assumptions must be postponed to Ch. VII.

⁵⁸If $\beta =b$ we must replace this interval by , and $\beta +\eta$ by $\beta$, throughout the argument which follows.

⁵⁹See § 104.

⁶⁰The word overlap is used in its obvious sense: two intervals overlap if they have points in common which are not end points of either. Thus and  overlap. A pair of intervals such as and  may be said to abut.

⁶¹That is to say ‘in and not at an end of’.

⁶²The reader should draw a figure to illustrate the definition.

⁶³We leave out of account the exceptional case (which we have still to examine) in which the curve is supposed to have a tangent perpendicular to $OX$: apart from this possibility the two forms of the question stated above are equivalent.

⁶⁴See, e.g., Chrystal’s Algebra, vol. i, pp. 151 et seq.

⁶⁵In these examples $m$ is a rational number and $a$, $b$, …, $\alpha$, $\beta$ … have such values that the functions which involve them are real.

⁶⁶A function which is continuous but has no derivative may have maxima and minima. We are of course assuming the existence of the derivative.

⁶⁷The maximum is , the minimum , of which the latter is the greater.

⁶⁸See § 119 for the rule for determining the ambiguous sign.

⁶⁹See, for example, Chrystal’s Algebra, vol. i, pp. 151–9.

⁷⁰See the author’s tract “The integration of functions of a single variable” (Cambridge Tracts in Mathematics, No. 2, second edition, 1915). This does not often happen in practice.

⁷¹The method of integration explained here fails if $a/A=b/B$; but then the integral may be reduced by the substitution $ax+b=t$. For further information concerning the integration of algebraical functions see Stolz, Grundzüge der Differential-und-integralrechnung, vol. i, pp. 331 et seq.; Bromwich, Elementary Integrals (Bowes and Bowes, 1911). An alternative method of reduction has been given by Sir G. Greenhill: see his A Chapter in the Integral Calculus, pp. 12 et seq., and the author’s tract quoted on p. 1537.

⁷²See the author’s tract quoted on p. 1537.

⁷³It is in fact sufficient to suppose that  exists. See R. H. Fowler, “The elementary differential geometry of plane curves” (Cambridge Tracts in Mathematics, No. 20, p. 104).

⁷⁴A much fuller discussion of the theory of curvature will be found in Mr Fowler’s tract referred to on p. 1537.

⁷⁵The new points which arise when we consider functions of several variables are illustrated sufficiently when there are two variables only. The generalisations of our theorems for three or more variables are in general of an obvious character.

⁷⁶Of course the fact that $\Delta x=\delta x$ is due merely to the particular value of $\Delta r$ that we have chosen (viz. $P{P}_2$). Any other choice would give us values of $\Delta x$, $\Delta r$ proportional to those used here.

⁷⁷Or with $\mid \delta x\mid +\mid \delta y\mid$ or $\sqrt[]{\delta x^2+\delta y^2}$.

⁷⁸The argument which follows is modelled on that given in Goursat’s Cours d’Analyse (second edition), vol. i, pp. 171 et seq.; but Goursat’s treatment is much more general.

⁷⁹The $s$ and the $S$ do not in general correspond to the same mode of subdivision.

⁸⁰All functions mentioned in these equations are of course continuous, as the definite integral has been defined for continuous functions only.

⁸¹Exs. 9–13 are taken from Prof. Gibson’s Elementary Treatise on the Calculus.

⁸²The method used in § 147 can also be modified so as to obtain these alternative forms of the remainder.

⁸³The corresponding inequality for a real integral was proved in Ex. LXV. 14.

⁸⁴In this and the following examples the reader is to assume the continuity of all the derivatives which occur.

⁸⁵In connection with Exs. 33–35, 38, and 40 see a paper by Dr Bromwich in vol. xxxv of the Messenger of Mathematics.

⁸⁶It is of course a matter of indifference whether we denote our series by $u_1+u_2+\dots$ (as in Ch. IV) or by $u_0+u_1+\dots$ (as here). Later in this chapter we shall be concerned with series of the type $a_0+a_{1}x+a_2{x}^2+\dots$: for these the latter notation is clearly more convenient. We shall therefore adopt this as our standard notation. But we shall not adhere to it systematically, and we shall suppose that $u_1$ is the first term whenever this course is more convenient. It is more convenient, for example, when dealing with the series $1+\frac{1}{2}+\frac{1}{3}+\dots$, to suppose that $u_n=1/n$ and that the series begins with $u_1$, than to suppose that and that the series begins with $u_0$. This remark applies, e.g., to Ex. LXVIII. 4.

⁸⁷Here and in what follows ‘positive’ is to be regarded as including zero.

⁸⁸The last part of this theorem was not actually stated in § 77, but the reader will have no difficulty in supplying the proof.

⁸⁹We shall use $r$ in this chapter to denote a number which is always positive or zero.

⁹⁰It will be proved in Ch. IX (Ex. LXXXVII. 36) that if $v_{n+1}/v_n\to l$ then $v_n^{1/n}\to l$. That the converse is not true may be seen by supposing that $v_n=1$ when $n$ is odd and $v_n=2$ when $n$ is even.

⁹¹This theorem seems to have first been stated explicitly by Dirichlet in 1837. It was no doubt known to earlier writers, and in particular to Cauchy.

⁹²In Exs. 2–4 the series considered are of course series of positive terms.

⁹³Five terms suffice to give the sum of $\sum n^{-12}$ correctly to $7$ places of decimals, whereas some $10,000,000$ are needed to give an equally good approximation to $\sum n^{-2}$. A large number of numerical results of this character will be found in Appendix III (compiled by Mr J. Jackson) to the author’s tract ‘Orders of Infinity’ (Cambridge Math. Tracts, No. 12).

⁹⁴This theorem was discovered by Abel but forgotten, and rediscovered by Pringsheim.

⁹⁵The test was discovered by Maclaurin and rediscovered by Cauchy, to whom it is usually attributed.

⁹⁶See Bromwich, Infinite Series, pp. 225 et seq.; Hobson, Plane Trigonometry (3rd edition), pp. 268 et seq.

⁹⁷For such a proof see the author’s tract quoted on p. 1537.

⁹⁸For fuller information as to ‘scales of infinity’ see the author’s tract ‘Orders of Infinity’, Camb. Math. Tracts, No. 12.

⁹⁹The exponential function was introduced by inverting the equation $y=logx$ into $x=e^y$; and we have accordingly, up to the present, used $y$ as the independent and $x$ as the dependent variable in discussing its properties. We shall now revert to the more natural plan of taking $x$ as the independent variable, except when it is necessary to consider a pair of equations of the type $y=logx$, $x=e^y$ simultaneously, or when there is some other special reason to the contrary.

¹⁰⁰See for example Chrystal’s Algebra, vol. i, ch. XXI. The value of ${log}_{e}10$ is $2.302\dots$ and that of its reciprocal $.434\dots$.

¹⁰¹‘Hyperbolic cosine’: for an explanation of this phrase see Hobson’s Trigonometry, ch. XVI.

¹⁰²The phrase ‘very large’ is of course not used here in the technical sense explained in Ch. IV. It means ‘a good deal larger than the roots of such equations as usually occur in elementary mathematics’. The phrase ‘a little greater than’ must be interpreted similarly.

¹⁰³See the footnote to p. 1539.

¹⁰⁴See Appendix II for some further remarks on this subject.

¹⁰⁵The formula for