CHAPTER 11
LIMITS AND CONTINUITY
Differential calculus was developed without any explicit definition of either limits or continuity, but with an intuitive assumption that both could in some sense be taken for granted. Widespread use of the calculus during the eighteenth century led to more careful consideration of such matters, but it was not until the early nineteenth century that Bolzano and Cauchy arrived at what are more or less the modem definitions. In this chapter we trace the history of both ideas up to the early 1820s.
11.1 LIMITS
11.1.1 Wallis's 'less than any assignable', 1656
The first writer to work with the concept of a limit in something like the modern sense was Wallis, who in his Arithmetica infinitorum in 1656 repeatedly claimed that two quantities whose difference could be made less than any assignable quantity could ultimately be considered equal (see, for example, 3.2.3). In 1656 Wallis stated this as a self-evident fact, but thirty years later, in his Treatise of algebra, he attempted to justify it by appealing to Euclidean ratio theory. In the ElementsBook V (Definition V) Euclid had stated a special property of homogeneous magnitudes (that is, magnitudes of the same kind): given any pair of such quantities, the smaller of them, however tiny, can always be multiplied to exceed the greater. Wallis argued the converse, namely, that if a quantity is (or becomes) so small that it cannot be made to exceed a larger
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quantity, no matter many times it is multiplied, it must be regarded as 'no quantity' or nodiing:1
And whatever is so little or nothing in any kind, as that it cannot by Multiplication, become so great or greater than any proposed Quantity of that kind, is (as to that kind of Quantity,) None at all
Wallis then went on to claim something rather stronger: if a difference between two quantities is less than any assignable quantity, then by definition it cannot be multipLied to exceed some given quantity, and therefore by the previous argument it is nothing, and the two original quantities are equal. Again, Wallis claimed Euclid as his authority:2
.. .he [Euclid] takes this for a Foundation of his Process in such Cases: That those Magnitudes (or Quantities,) whose Difference may be proved to be Less than any Assignable are equal. For if unequal, their Difference, how small soever, may be so Multiplied, as to become Greater than either of them: And if not so, then it is nothing.
Though he attributed his arguments to Euclid, Wallis was stretching them considerably further than Euclid or any other Greek author had ever done. The first proposition of Book X of the Elements makes the following claim: if from a given quantity there is repeatedly subtracted a half (or more), then what remains will eventually be less than any preassigned quantity. This was crucial to the method of exhaustion; it enables one to prove, for instance, that the space between a circle and an inscribed polygon can be made as small as one pleases by repeatedly doubling the number of sides of the polygon. Nowhere, however, did Euclid or any other Greek mathematician claim that this steadily dimmishing quantity could be considered non-existent, or zero. Instead, Proposition X.1 was used in proofs by double contradiction to show, for example, that the space inside a circle was neither greater nor less than some predetermined quantity (see 1.2.3).
Wallis's insight may not have had the classical authority he claimed for it but, like several of his ideas in the Arithmetica infinitorum, it was put to particularly good use by Newton.
11.1.2 Newton's first and last ratios, 1687
In the Principia in 1687 Newton gave Wallis's idea of'ultimate equality' the status of a proposition, indeed he made it the opening Lemma of Book I, Section I (see 5.1.2).
At the very end of Section I, Newton introduced the Latin word limes, in the everyday sense of a boundary which may not be crossed, just as Barrow had done in 1660 (see 1.2.1). He used 'limes' in a similar sense again in the final sentence when he spoke of quantities decreasing sine limite, that is, without end, or mdefinitely. Newton also observed that a quantity may approach such a boundary as closely as one pleases; by Lemma I this was equivalent to 'ultimate equality'.
Newton's idea of a limit
from Newton, Principia mathematica, 1687,1,35-36
í 35 3
coatenta. Frajmiú vers bac Lemmata uterfugererrr tidiumdedi-cendi perplexai dm-Mftrarioato, íbóreisslérum Geametracurn, adabfurdum. Contractiores enim redduntur demonftrationes per methodum hidtviflbilinm. Sed <^uouiam .durior eft mdiviflhilium Hypotheíis; & propterea Methodus ilia xiiinus GeomKrica cen-fetur, malui demonit'rationes rerum fequentium ad ukinus quan-titatum evaneictntium liimmas & rationcs, primal'q; Dalccntium, id eft, adlimires vumiriarnm ■& rationum deducere, Sc propterea íimituŕn illorum dľmoiíftraťiorirs qua potuibreuitate ptamittcrr. His enim idem praiftatar quod per methodum indivifibilium,' $c principiis demouftratis jam ratios utemur. Proinde in fequenti-bus, fiquando qnantitates tanquam ex particnlis -conftantes confi-deravero, vel fi.pro reétís ufurpaverodineoks curvas, riolim in-divifibilia fed évaneícentla divrfibilia, non fornums & ratbnes partium determinatarum, fed uimmarum Sc rationuftidiinitaK'iein-per intelligi, vimq; talium demonftrarioiinm ad .methodum pne-cedentium Lemmatum femper revocari.
Ob/ečKo eft, quod quantitatum evanefcentiura nulla fit ultima proportio; quippe quse, antequam evanuerunt, noneftultima, u-bi evanuerunt, nulla eft. Sed Sc eodern argumento seque conten-di poffet nullam effe corporis ad cerium locum pergentis veloci-tatem ultimam. Hanc enim, antequam corpus attingit locum, non effe ultimam, ubi atrigit, ntillam effe. Et refponfio facilis eft. Per velociratem ultimam intelligi earn, qua corpus rnovetur neq; antequam attingit locum ultimum & motus ceffat, neq; poftea, fed tunc cum attingit, id eft illam ipfarn velocitatem quacum corpus "attingit locam ultimum & quacum motus ceffat. Et (imilker per ultimam rationem quantitatum evancfcentium inrelligendam effe rationem quantitatum non antequam evanefccnr, non poftea, fed quacum evanefcunt. Pariter & ratio prima nafcentium eft nuio quacumnafcuntur. Et fumma prima fk ultima eft quacum jefte'( vel ?tigeri& minui J) incipiunt& ce(iar,t. Extar fimEsquem velocitas in line motus attingere poteft, non autem tranfgredi.
F i Hax
1. Wallis 1685, 281.
2. Wallis 1685,282.
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[  36 ]
Hscc eft velocitas ultima. Et par eft ratio limlris quantitatum 8c proportiosum omnium incij ientium 8cceflantium. Cumq; hie limes fit certus £c derimtus, ťroblema eft vere Geometricum eun-drm determirmre. Gcometrica vero omnia in aliis Geometricis determinandis ac demonftrandis legitime ufurpantur.
Contend! etiam poteft, quod fi dentur ultima; quantitatum e-Vanefc?ntium rationes, dabuntur & uitimae magnitudines; & iic quantitas omnis conflabir ex indivifibilibus, contra quam Enchdei de.incornrrienfurabilibus, in libro de'eimo Elementorum, demon-■ ftravit. ..Verum hrrc Objeflio fzKx innititur bypothefi. Ultimo rationes i\lx' quibúlcum quantitatesevanefcunt, revera non I'unt rationes quantitatum ultimarum, fed Hmices ad quos quantitatum fine limite decrefcentium rationes fem per appropinquant, Stquas --propius afTequi polfunt quam pro data quavis differentia, nun--sjúain vertvrxanfgredi, neq; prius attingere quam quantitates di-■minuuntur in infinitum. Res clarius inrelligetur in infinite magnis. Si quantitates dux quarum data eft differentia augeantur in infinitum, dabitut harum ultima ratio, nimirum ratio atqualitatis, nec tarnen ideo dabuntur quar.titate*s ultima; feu maxim« quarum ifta eft ratio. Igitur in íěquentibus., fiquando facili rerum ima-ginationi confulens, dixero quantitates quam minimas, vel eva-nefeentes vel ultimas, cave inwlligas quantitates magnitudine de-terminatas, fed cogita femper diuiinuendas fine limite.
TRANSLATION
I have put forward these lemmas at the beginiring, in order to avoid the tedium of composing intricate demonstrations by contradiction in the maimer of the ancient geometers. For the demonstrations are rendered more concise by the method of indivisibles. But since the hypothesis of indivisibles is cruder, and that method therefore judged less geometrical, I have preferred to deduce the demonstrations of what follows by means of first or last sums and ratios of nascent or vanishing quantities, that is, to limits of sums and ratios, and therefore to put forward demonstrations of those limits as briefly as I could. For the same can be shown by these as by the method of indivisibles, and the principles having been demonstrated, we may now more safely use them. Consequently in what follows, whenever I have considered quantities as if consisting of particles, or if f have used little curved lines for straight lines, I do not mean indivisibles but vanishing divisibles, and there should always be understood not sums and ratios of the known parts but the limits of sums and ratios, and the vaHdity of such demonstrations is always to be based on the method of the preceding lemmas.
The objection is that the ultimate ratio of vanishing quantities might not exist; since before they vanish, it is not ultimate; and where they have vanished, it is non-existent. But by the same argument it could equally be contended that the ultimate velocity
of a body arriving at a certain place does not exist. For in this case, before the body reaches the place, the velocity is not ultimate; where it reaches it, it does not exist. And the answer is easy. By the vdtimate velocity is to be understood that with which the body moves, not before it reaches the final place and the motion ceases, nor after, but as it reaches it; that is, that same velocity with which the body reaches the final place and with which the motion ceases. And similarly by the ultimate ratio of vamshing quantities there must be understood the ratio of quantities not before they vanish, nor after, but with which they vanish. And equally the first ratio of nascent quantities is the ratio with which they originate. And the first or ultimate sum is that with which they begin or cease to be (according as they are increasing or decreasing). There exists a limit which at the end of the motion the velocity may attain, but not exceed. [36] This is the ultimate velocity. And likewise for the limiting ratio of all quantities and proportions beginning or ceasing to be. And since this limit is fixed and definite, the problem is to determine it correctly geometrically. Indeed anything geometric can legitimately be used to determine or demonstrate other things geometrically.
It may also be contended that if ultimate ratios of vanishing quantities are given, so are the ultimate magnitudes; and thus every quantity will consist of indivisibles, contrary to what Euclidpioved of incommensurables in the tenth book of the Elements. But this objection is based on a false hypothesis. Those ultimate ratios with which quantities vanish, are not actually ratios of ultimate quantities, but limits to which the ratios of quantities decreasing without limit always approach, and which they may attain more closely than by any given difference, but never exceed, nor attain before the quantities are infinitely diminished. This may be more clearly understood for the infinitely large. If two quantities, whose difference is given, are infinitely increased, their mtimate ratio will be given, namely the ratio of equality, but nevertheless there will not thereby be given the ultimate or greatest quantities of which this is the ratio. Therefore whenever in what follows, to make things easier to imagine, I speak of quantities as the smallest, or vanishing, or ultimate, avoid thinking of quantities of finite magnitude, but always consider that they are to be decreased without limit.
11.1.3 Maclaurin's definition of a Limit, 1742
Maclaurin, writing some sixty years after Newton, continued to use the word 'limit' in much the same sense, as a bound that may be approached as closely as one wishes. Stung by the criticisms of Berkeley and others (see 10.2.2) he took great pains to show that limits were well defined, but his words 'it is manifest ...' did nothing to avoid or disguise the fundamental problem of neglecting 0 after dividing by it.
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Maclaurin's definition of a limit
from Maclaurin, A treatise of fluxions, 1742,1, §502-§503
j02. But however fafe and convenient this method may be, fome will always fcruple to admit infinitely little quantities, and infinite orders of infinite fimals, into, a fcience that boafts of the snoft evident and accurate principles as well .as of the moil rigid demonftrations; and therefore we chofe to eftablifh fb extenfive and ufeful a doctrine in the preceeding cHapters on more unex-ceptifinable fojiulata. In order to avoid fuch fuppofkions, Sir Isaac Newton confiders tbe limultancous increments of the flowing quantities as finite, and then inveftigates the ratio which is the limit of the various proportions which thofe increments bear to each other, while he fuppofes them to decreafe together till they vanifh; which ratio is the fame with the ratio of the fluxions by what was fhewn.in art. 66, 6j and 68, In order to difcover this limit, he firit determines the ratio of the increments in general, and reduces it to the moft fimple terms fo as that (generally fpeaking) a pare at leaft of each term may be independent of the
value
Chap. XII.       Of the limits of Ratios. 411
value of the increments themfelves ; then by fuppofing the increments to decreafe till they vanifh, the limit readily appears.
503. For example, let a be an invariable quantity, x a flowing quantity, and 0 any increment of x;. then the fimultaneous increments of xx and ax will be 2x0 + 00 and aor which are in the fame ratio to each other as ax -(- 0 is to a. This ratio of ix -f- 0 to a continually decreafes while 0 decreafes-, and is always greater than the ratio of ix to a while 0 is any real increment, but it is manifeff. that it continually approaches to the ratio of 2x to a as its limit ■ whence it follows that the fluxion, of xx is. to the fluxion of ax as 2x is to a. If x be (uppoied to flow uniformly, ax will likewife flow uniformly, but xx with a motion continually accelerated : The motion with which ax flows may be meafured by ao, but the motion with which xx flows is not to be meafured by its increment 2x0 + 00, (by ax. 1.) but by the part 2x0 only, which is generated in confe-quenceof that motion - and.the part 00 is to be rejected becaufe iris generated in confequence. only of the acceleration o£ the motion with which the variable fquare flows, while 0 the increment of its fide is generated : And the ratio of 2x0 to ao is that of ax to a, which was found to be the limit of the ratio of the increments 2x0 -f- 00 and ao.
11.1.4 D'Alembert's definition of a limit, 1765
When d'Alembert wrote and edited the mathematical sections of the great Encyclopedie of Denis Diderot, published between 1751 and 1765, he provided new and useful definitions of many recent mathematical concepts. His definition of 'limit' in Volume IX was close to Newton's idea of a limit as a bound that could be approached as closely as one chose, and because d'Alembert, like Newton, worked with examples that were primarily geometric, there was still no obvious need to consider quantities that might oscillate from one side of a limit to the other.
D'Alembert's definition of a limit
from Diderot and d'Alembert, Encyclopedic 1751-65, LX, 542
TRANSLATION
LIMIT (Mathematics). One says that a magnitude is the limit of another magnitude, when the second may approach the first more closely than by a given quantity, as small as one wishes, moreover without the magnitude which approaches being allowed ever to surpass the magnitude that it approaches; so that the difference between such a quantity and its limit is absolutely unassignable.
For example, suppose we have two polygons, one inscribed in a circle and the other circumscribed; it is clear that one may increase the number of sides as much as one wishes, and in that case each polygon will approach ever more closely to the circumference of the circle; the perimeter of the inscribed polygon will increase and that of the circumscribed polygon will decrease, but the perimeter or edge of the first will never surpass the length of the circumference, and that of the second will never be smaller than that same circumference; the circumference of the circle is therefore the limit of the increase of the first polygon and of the decrease of the second.
1. If two magnitudes are the limit of the same quantity, the two magnitudes will be equal to each other.
2. Suppose A x B is the product of two magnitudes A, B. Let us suppose that C is the limit of the magnitude A, and D the limit of the quantity B; I say that C x D, the product of the limits, will necessarily be the limit of A x B, the product of the magnitudes A, B.
These two propositions, which one will find demonstrated exactly in the Institutions de Geometrie, serve as principles for demonstrating rigorously that one has the area of a circle from multiplying its seimckcumference by its radius. See the work cited, p. 331 and following in the second volume.
The theory of limits is the foundation of the true justification of the differential calculus. See differential, fluxion, exhaustion, infinite. Strictly speaking, the limit never coincides, or never becomes equal to the quantity of which it is the hmit,
9
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but the latter approaches it ever more closely, and may differ from it as little as one
wishes. The circle, for example, is the limit of the inscribed and circumscribed polygons;
for strictly it never coincides with them, although they may approach it indefinitely.
This notion may serve to clarify several mathematical propositions. For example, one
says that the sum of a decreasing geometric progression in which the first term is a and aa
the second b, is--; this value is never strictly the sum of the progression, it is the
a — b
limit of that sum, that is to say, the quantity which it may approach as closely as one wishes, without ever arriving at it exactly. For if e is the last term in the progression,
aa — be aa
the exact value of the sum is -, which is always less than-because even
a — b a — b
in a decreasing geometric progression, the last term e is never 0; but as this term
continually approaches zero, without ever arriving at it, it is clear that zero is its limit, aa — be aa
and that consequently the limit of-is-, supposing e = 0, that is to say, on
a — b     a — b
putting in place of e its limit. See sequence or series, progression, etc.
11.1.5 Caudiy's definition of a limit, 1821
Cauchy's definition of a limit, first given in his Cours ďanalyse in 1821, imitated that of d'Alembert and combined the same basic ideas: the existence of a fixed value, and the possibility of approaching it as closely as one wishes. The same definition was repeated, with further examples, at the beginning of his Résumé des legons in 1823.
Cauchy established the concept of a limit as the starting point of textbook expositions of analysis but in most respects his definition was no clearer than Newton's 150 years earlier, for there was still no precise discussion of what it meant to approach a fixed value '^definitely', nor of whether a variable quantity might actually attain or even at times surpass its limit. Cauchy offered the well worn illustration of a circle and polygons, but also produced a newr and more interesting example, of an irrational number approached by rationals; he did not yet suggest, however, that a hmit could be approached from both sides simultaneously.
Cauchy's definition of a limit, 1821
from Cauchy, Cours ď analyse, 1821,4-5
On nomine q u an ti té variable celie que l'on con-siděre comme clevant rccevoir successivement plu-sieürs valeurs différentes les unes des autres. On designe une semblable quantité par une lettre prise ordinairement parmi les demiěres de {'alphabet. On appelte au contraire quantité constante, et on designe ordinairement par une des premieres Iettres de l'alpliabet toute quantité qui reeoit une valeur fixe et détenninée. Lorsque les valeurs successivement attribuěes á une méme variable s'approchent indéfiniment ďune valeur fixe, de maniere ä finir par en différer aussi peu que ťon voudra, cette deruiěre est appelée la limite de toutes les autres. Ainsi, par exemple, un nombre irrationnel est la limite des diverses fractions qui en fournissent des valeurs de plus en plus approchées. En geometrie, la surface du cercle est la limite vers laquelle convergent les surfaces des polygones inscrits, tandis que Ie nombre de leurs côtés croit de plus en plus; &c....
Lorsque les valeurs numériques successives d'une méme variable décroissent indéfiniment, de maniere ä s'abaisser au-dessous de tout nombre donne, cette variable devient ce qu'oŕi nomme un infinimentpetit ou une quantité infiniment petite. Une variable de cette espéce a zero pour limite.
Lorsque les  valeurs numériques successives
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P RELIMINAIR ES. 5
d'une meine variable croisseiit de. plus en plus, de TJoatiiere á selever au-dessus de tout nouibre donne, on dit-que cette variable a pour limite ^inßnijiositif , indiqué par le signe oo , s'il s'agit d'une variable positive, et l'inßni négatif, indiqué par Ja notation — oo , s'il s'agit d'une variable negative. Les infinis posilif et négatif sont désignés conjoiuteruent sous le nom de quantités inßnies.
TRANSLATION
One calls a variable quantity one that is considered to take successively several values different from each other. One denotes such a quantity by a letter usually taken from amongst the last in the alphabet. On the other hand one calls a constant quantity every quantity that takes a fixed and known value, and one usually denotes it by one of the first letters of the alphabet. When the values successively attributed to the same variable approach indefinitely to a fixed value, in such a way as to end by differing from it as little as one wishes, this last is called the limit of all the others. Thus, for example, an irrational number is the limit of various fractions that furnish values more and more closely approaching it. In geometry, the area of a circle is the limit towards which converge the areas of inscribed polygons, when the number of their sides increases more and more; etc....
When the successive numerical values of the same variable decrease indefinitely, in such a way as to fall below every given number, this variable becomes what one calls an infinitesimal or an infinitely small quantity. A variable of this land has zero for its limit.
When the successive numerical values [5] of the same variable increase more and more, in such a way as to rise above every given number, one says that this variable has for its limit positive infinity, indicated by the sign oo, if one is dealing with a positive variable, and negative infinity, indicated by the notation -co, if one is dealing with a negative variable. Positive and negative infinities are known jointly under the name of infinite quantities.