"Mr. Edison and His Critics" (The Sun, 1880)

The following story comes from the New York Sun, 24 January 1880:

MR. EDISON AND HIS CRITICS.

THE ONE FEATURE THAT HE SAYS IS NOW TO BE PERFECTED

Intimating that He Laughs Best who Laughs Last–A Schedule of the Hours his Lamps have Burned–A Customer's Experience

"When Sir Humphrey Davy, the famous English chemist, was apprised of the project of forcing carbiretted hydrogen gas through a system of pipes for purposes of illumination, he laughed in derision. Nevertheless," continued Mr. Edison, "illumination by coal gas proved to be a great success. My project for the subdivision of the lectric light is treated in like manner by all those persons who are profoundly ignorant of the system which I am day by day perfecting. It is a singular fact that persons conversant with the subject, after inspecting my laboratory, are ready to allow that I am all right as far as I have gone. I ask no more."

Mr. Edison insists that the only question now is of the perfect formation of the glass globes of his lamps. This, he says, will soon be brought about. Notwithstanding the occasional unfavorable reports of Mr. Edison's experiments, as published, the company of capitalists who are backing him seem not to lose confudence in the inventor's ability to do all that he has promised.

The proprietor of one of the private dwellings in Menlo Park, that are illumined by the Edison lights, said last evening: "The lights seems almost perfect. They give us absolutely no trouble. When we retire we turn the little screws attached to each and the glow instantly ceases. Often during the day I turn the lights up. So far, nearly three weeks, they have not failed. The so-called scientific persons in New York and others who are continually condemning this plan for lighting as impracticable have a great surprise in store for them. They are sure to see their entire city lighted by these same jets."

A number of new lamps were set up yesterday in the Menlo Park laboratory. To the ordinary observer their intensity does not differ from that of those that have been burning for weeks. Examinations of the carbon contained in these latter, made by means of the photometer and galvanometer show their resistance to be unimpaired. That is to say, the amount of combustion that has taken place cannot be measured.

It is thought in Menlo Park that some of the light that were stolen during the first week of the public exhibition have fallen into the hands of Western electricians. Should similar lamps be manufactured and used, even without the improved dynamo machines, their introducers will be vigorously prosecuted by the managers of Mr. Edison's Electric Light Company.

A means of preventing the glass tubes containing the incandescent lights from cracking Mr. Edison says he has discovered, and will at once put in practice. At the point where the platinum wires pierce the tubes, a compound is applied having a fixed alkali for a base, and a conchoidal fracture. Several of these, he says, have been burning for eighty hours without even the sign of a crack.

The following are the number of hours that each of the lights now set up at Menlo Park had been glowing up to last evening. It was made by Mr. Edison himself from the book kept by one of his assistants:

597, 485, 465, 515, 508, 503, 503, 517, 487, 532, 190, 200, 200, 430, 390, 400, 367, 376, 370, 370, 373, 352, 332, 332, 332, 332, 417, 326, 336, 283, 286, 294, 294, 300, 300, 200, 283, 301, 367, 215, 381, 238, 194, 193, 194, 194, 136.

This includes not only the lights in the laboratory, but also those glowing publicly in the streets and in the private houses.

Mr. Edison exhibited last evening a series of elaborate drawings which comprise the entire plans for the station that he says he will shortly establish in his city. The machinery is to be placed in a building of 25x100 feet. In the cellar is to be five engines of 250-horse power each, making in all 1,250-horse power. The dynamo machines are to be in the second and third stories. Using the small horseshoes instead of those now in use, with the same resistance—100 ohms—Mr. Edison says he finds that he can obtain eleven and one-half instead of eight lights to the horse power. In this manner he believes he can generate 13,750 lights, each having a power of eighteen candles. He says, firther, that in the daytime, when his lights are not used, he can hire out the power of his engines at great advantage. By means of copper insulated wires he can transmit the power generated, and distributed it among the manufactories within a radius of half a mile, although never more than 50-horse power to any one building. An agent whom he has employed to inquire among the manufacterers in the vicinity of his proposed station has just, he says, reported even more favorably than he anticipated. Mr. Edison went on to say that although he could only deliver at the end of half a mile 65 per cent. of his 1,250-horse power, nevertheless, the profit would be enormous. It stands to reason, he argues, that you can run five 250-horse power engines cheaper than you can run forty-eight 25-horse power engines.

Mr. Edison added that he is now placing his light in a large steamship in course of construction by John Roach at Chester, Pennsylvania. The ship belongs to the Oregon Steam Navigation Company. Lights of three candle power are to be placed in each stateroom. The electricity will be generated by means of a small dynamo machine connected with the donkey engines.

A report from Baltimore says that a day or two ago the best bid for stock of the Edison Electric Company was $\$$500 a share, against reported sales three weeks ago at $\$$4,000.

A Paris despatch to the Telegram from M. Wilfred De Fonvielle, editor of L'Electricité, says that the Count de Moncel's attack on Edison in Le Temps attracted no attention whatever: the de Moncel is not a member of the Institute, as is supposed, and that in reality he is what is known as a membre libre, without pay and without vote, enjoying only secondary privileges. Moncel is said to be editor of the paper connected with Würdemann's [Richard Werdermann's?] lamp, which is specially intended for lighting large spaces, and which will be set aside by Mr. Edison's discoveries.

Mr. Jean Baptiste Dumas, the well-known scientist, said to a Paris correspondent of the Telegram: "I own that on first reading of Edison's discoveries, I was incredulous, but I now see no real impossibility in the thing. I do not doubt that the carbonized horseshoes may be held by the platinum clamps, for I have obtained in my own laboratory carbonized Bristol board of a certain cohesiveness. I could only render it incandescent, however, for the space of a few centimetres [the centimetre is about thirty-nine one-hundredths of an inch] by using the Bunsen pile of twenty elements. I am led to conclude that Edison must employ considerable force, and it is difficult for me to understand how he can long render incandescent a carbonized substance with such conductivity as the drawings represent."

M. Ninndet [?] Breguet, nephew to the great electrical manufacturer, said: "I found nothing very new or astonishing in the discovery of Mr. Edison, except the manner of fabricating the carbon horseshoes. I find the electrical magnetic  machines employed by him excessive, and that consequently there must be a loss of magnetic force in the generation of the electricity.["]

M. Fontaine, President of the Syndicate d'Electricité, of which all Paris electricians are members, and editor of the Revue Industrielle said: "I have read everything published on the subject. My conviction is that by the employment of the carbonized horseshoe, Mr. Edison has made an important advance in the application of the electric light. I believe that this result has been obtained by a certain dexterity of manipulation, of which Mr. Edison has preserved the secret, as, for instance, the judicious employment of the electric current during the process of carbonization."

"This is no Keely motor business out here, and people are at last waking up to what I'm doing," said Mr. Edison, when he spoke of the favorable criticisms of the French electricians. "Dumas, too, is an authority. He is mistaken in reference to the 'high electrical resistance in the incandescent substance,' however. It is exactly the quality we want, for it is the enormous resistance of the kind of carbon that I use that allows of the practical subdivision of the electric light."

The comment in brackets, translating centimetres into inches, was part of the original article. The other square brackets are mine.

"Gas Light or Electric Light" (1878)

The following story ran on 31 October 1878 in the Watertown re-union:

GAS LIGHT OR ELECTRIC LIGHT

The gas companies and their expert advisers declare that they are not at all afraid that there is any prospect of the substitution of electric light for gas light. Prof. Morton, a distinguished authority, has given them encouragement by publically averring that no practicable process of dividing and subdividing the force generated by the battery has yet been discovered so that the electric light can be made useful for houses and offices. In such subdivision there is not even the prospect of anything being done, says the learned Professor.

The Franklin Institute of Philadelphia, according to a speaker at last week's meeting of the American Gaslight Association, after expending money in experiments with the electric light, failed to discover in it any elements of practicability. The Permanent Exhibition at Philadelphia tried it, and it was a failure. It was either too intense or too irregular. Other speakers told similar stories of the ill adaptedness of the new light to ordinary and general uses, and the conclusion was that illuminating gas is not only good enough for all our needs, but will also continue to furnish us our artificial light for an indefinite period of time to come.

Meanwhile, however, as here in Canada, and in England, gas stocks have declined heavily, and they obstinately refuse to rebound. The public do not agree with the gas men. They believe we are going to have the electric light, that the difficulties in the way of its distribution have been or will be surmounted, and that it is destined to be the light of the future.

Mr. Edison also agrees with them and says he has already done what the advocates of gas say has not been done, and what they do not believe can be done.—That the electric light can be profitably be used for the illumination of large open spaces has been proven beyond question. It may now be seen on the Boulevard de l'Opera and at the Arc de Triomphe in Paris, and passing from the brilliant light it gives, that made by the neighboring gas lamps seems feeble in comparison.—We have also had opportunities of observing it in New York, and in London and in other parts of England it has been successfully experimented with in the streets, on the seashore, and in large buildings.

But can the electric light be distributed through the shops and dwellings of a city as gas is now distributed? Can it be furnished at a cheaper rate than gas, and can its intense glare be subdued and the light made constant and uniform—in fact, can it be substituted for gas light? To all these questions Mr. Edison answers, Yes. "The subdivision of the light," he told a reported of the Sun last week," is all right. The only thing to be accurately determined is its economy. I am already positive it will be cheaper than gas but have not determined how much cheaper."

The articles of incorporation of The Edison Electric Light Company were filed last week, and the company proposes to light the city, public buildings, and private residences with electric lights. We shall, therefore, before long have an opportunity to see for ourselves whether the gas men are right or wrong about the new light; and they will be wise to forego prophesying until the experiment has been tried.

Despite the reference to Canada in the article, Watertown is in fact in upstate New York, although very close to the Canadian border.

I think the "battery" referred to in the article is the power plant, not a portable power cell.

"Edison's Progress" (1880)

The following newspiece was reported in the Brockport republican on 23 September 1880, apparently citing a story from the New York Sun:

Edison's Progress

Mr. Edion's hundred--horse power engine is getting its finishing touches in the Porter Allen Engine Company's shops, Philadelphia, and in about three weeks it will be in position in Menlo Park. The engine will make 600 revolutions a minute. The cylinder is nine inches in diameter. With this pondrous machine Mr. Edison expects to produce the power that will enable him to light up the 800 lamps for which the workmen have just laid eight and one-half miles of mains.

Forty–five men have begun work in the nearly completed factory where the lamps are made, and more will be added in short time. They will turn off about 700 completed lamps a day for the present, but Mr. Edison expects to increase the number to 1,200.

Mr. Edison's new test of the light will be made about Oct. 1. He says that there is no defect in thermostruction of the lamps; that the carbon tips of Japanese bamboo are perfect and that the new trials of the light is not to be an experiment to see if it will burn, but is to test, on a large scale, its actual economy as compared with gas.

Mr. Edison has not experimented with the light for the past six months, but his time has been occupied in preparing for what he terms the commercial introduction of the light. Every detail as to wire, glass, wood, iron and other materials, and the best and most economical ways for fashioning them, he has gone over hundreds of times, his idea being, he says, to have the light a perfect success to begin with, and to leave no improvement to be desired. He does not want his invention looked upon in a few years as a crude effort in a new field, like the first sewing machine and the prioneer reaper.

Of the early introduction of his light in the city the inventor speaks with positiveness. Spread on his table are working diagrams of the two districts in in which it is proposed to begin work. The first district is as follows: From Peck slip through Ferry street to Spruce, to Nassau, to Wall, to South, thence to the place of beginning. The second district is as follows: Through Mail street to Park place, to West, to Rector, to Nassau, to Spruce.

There will be one station in each district where the power will be generated. This station will occupy a room about 25x100 feet in size. The building may be a cheap structure in the rear of other buildings, so that the item of rent or purchase money will not be considerable. Canvassers are now at work in the two districts getting in the names of tenants who will try the new light. They report that the owners of large buildings are very ready to have the experiment tried. As has been heretofore accounced, the purpose of the Electric Light Company is to sell power as well as light. Mr. Edison says they expected to sell more power than light, and perhaps eventually to make enough on power to be able to give the light away. Quarter-horse, half-horse, one-horse and five-horse and seven-horse power machines, he anticipates, will be in great demand for keeping ventilators in motion, swinging fans in restaurants running sewing machines and turning lathes, and so on.

Since Mr. Edison has considered his light as successful commercially, he has given its practical use by all sorts of people a good deal of attention. He says the popular prejudices and customs are not the least difficult things with which he has to deal. His aim has been, therefore, to make the new light as nearly as possible like gas in its operation. His was of making a practical test is to call in a servant girl from a neighbouring house, or a laborer from a field, and pointing to a lamp say: "Light that; now turn it off; light it again." Every difficulty in the way of the awkward experimenter is carefully noted by the inventor, who at once sets to work to obviate it. A single illustration will show how closely he has studied these minor matters. In making the first lamps he had the thumbscrew, which corresponds to that of a gas fixture, turn in the opposite direction. Everybody who took hold of it for the first time tried in vain to twist it the wrong way. Mr Edison took the hint, and the new lamps are lighted precisely like those of gas fixtures. In outward appearance, too, they resemble has lamps as closely as they can be imitated.

Should the Menlo Park trial, soon to be made, demonstrate all that is expected, Mr. Edison says that the company will be selling the light to consumers in New York by the first of January, 1881, at about fifty cents per 1,000 candles. The ordinary standard of measurement for the new illuminating power. Three thousand candles gave the light of 1,000 cubic feet of has, so that the equivalent of 1.000 feet of has in the electric light will cost $1.50. The bamboo burners are calculated to last in steady use for six months. Extra ones will be furnished to consumers put up in boxes, with sockets complete, for fifty cents each. The figures named by Mr. Edison are subject to change, but he says the only change will be to lower prices as the use of the light is extended.

Mr. Edison's electric locomotive which has been running over the track at his factory at forty odd miles an hour is dismantled at present. He is fixing up an experimental freight locomotive, and he is going to test it on steep grades.

Should Mr. Edison's inventions have the commercial success that he anticipates Menlo Park will be threatened with the loss of its name and fame. The company associated with the inventor are already looking for factory sites in Metuchen and Rahway, where better facultures will be available.—N. Y. Sun.

Chrystal: "On Some Fundamental Principles in the Theory of Probability" (1891)

I've been trying to get my hands on the following paper:

George Chrystal (University of Edinburgh): "On Some Fundamental Principles in the Theory of Probability." Transactions of the Actuarial Society of Edinburgh, Volume 2, January 1891, pages 420–439.

So far, no luck. The Cambridge Journals database has a copy, but it's behind a paywall, and my university doesn't have a subscription.

However, a number of other sources quote extensively from the paper, so I've been able to piece together an understanding of what it looks like.

Posterior Frequencies

It seems that Chrystal's main beef is with the use of Bayes' rule to update the probability of a certain set of hypotheses whose long-term frequencies are already given in advance. His reasoning seems to be that this conflates our subjective degree of belief (which may indeed change) with the objective frequency (which, by assumption, cannot).

This philosophical distinction is nicely presented in the following quote. It comes from an 1894 review (also available in book form on the Internet Archive) by somebody called G. F. Hardy, not to be confused with G. H. Hardy.

"There is," says Professor Chrystal, "in Laplace's view, a confusion between two senses of the word 'Probability', which although distinct are often more or less associated in point of fact. In common speech we say that a single event is more or less 'probable', and by this word we indicate our own mental attitude towards the event, an attitude that may be well or ill justified by facts. When an actuary says that the probability that a man of 20 will live to be 60 is $\frac{59}{97}$, he is not, strictly speaking, referring to any one event at all, but merely making an assertion to the effect that out of any considerable number of men of 20 years of age about $\frac{59}{97}$ will reach the age of 60. No one knows better than an actuary that this statement is a fact, established (under certain circumstances, and with certain limitations), by experience, and that it has nothing whatever to do with the mental attitude of anyone. Everyone will admit that we could never arrive at this result by analyzing the event of a man of 20 reaching or not reaching the age of 60 into cases regarding each of which we should be equally undecided,—mentally suspended, as it were, like Buridan's ass between the equal bundles of hay." (Hardy, p. 316)

It's not clear whether the emphases are in the original, but I'm guessing not.

Hardy does not give a page reference, but proper reference seems to be page 423. I got that figure from the manuscript of a 1893 presentation by a certain John Govan, F.F.A. (whose name, location, and date fit the proselytizing businessman John George Govan).

Govan's rendition of the quote occurs on his page 212. He doesn't use the emphases.

The Burial of Bayes

Several sources also report that Chrystal summarizes his discussion with the following tirade:

… both from the point of view of practical common-sense, and from the point of view of logic, the so-called laws of Inverse Probability are a useless appendage to the first principles of probability, if indeed they be not a flat contradiction of those very principles.

This is cited by a number of authors, including E. T. Whittaker, F.R.S. (in a footnote to a 1920 presentation, p. 165), Andrew I. Dale (1999, p. 485) and Sharon McGrayne (2011, p. 37).

Dale reports that this quote is found on page 438. Whittaker apparently reports it as page 421 (but that would be at odds with Dale's description of the quote as a occurring near the conclusion of the essay). McGrayne doesn't give a page number.

According to Whitaker, the quote continues as follows:

The laws of Inverse Probability being dead, should be decently buried out of sight, and not embalmed in text-books and examination papers.

McGrayne further reports the following conclusion:

The indiscretions of great men should be quietly allowed to be forgotten.

Checking the relevant footnote of McGrayne's book (note 9 of ch. 3, p. 260), this turns out to be a recycled quote from Anders Hald's A History of Mathematical Statistics, page 275. That book doesn't have a Google Books preview, and it's not in my library.

Sort-of-Long-Term Frequencies

Hardy's review continues to quote Chrystal's discussion of how probability is to be defined:

"The notion of probability is always attached to a class or series of events, which usually have more or less of other attributes in common, but are always distinguished by this mark, that certain phases of them, although not predicable with the smallest certainty in any individual case, are predicable with more or less uniformity in a certain proportion of cases in the long run. The fundamental features of this series are statistical uniformity combined with irregularity of every conceivable kind in the individual instance. The number of the events in the series must be large. Its extension both as to space and time is arbitrary, and in certain ideal cases infinite. It is in this last respect alone that probability has anything to do with our mental attitude; we may choose our standpoint, and this determines the probability to which our knowledge may make a better or worse approximation. As the series is varied the probability alters. . . .  We are thus led to the following abstract definition of the probability or chance or an event. If, on taking any very large number, N out of a series of cases in which an event A is in question, A happens on pN occasions, the probability of the event A is said to be p." (Hardy, p. 317)

Again, no page number is given.

Posterior Priors

The examples in Chrystal's paper seem all to be of the same kind: He describes a set-up in which certain a priori frequencies are given, and he then tells us that no amount of evidence should be able to change those frequencies; the only mental operation we can perform is to exclude logically impossible cases, not to compute posterior probabilities.

Govan thus quotes him as discussing a situation in which you draw two white balls from a bag of black and white balls. Then:

"Any one," says Professor Chrystal, "who knows the definition of mathematical probability, and who considers this question apart from the Inverse Rule, will not hesitate for a moment to say that the chance is $\frac{1}{2}$; that is to say, that the third ball is just as likely to be white as black. For there are four possible constitutions of the bag . . . each of which we are told occurs equally often in the long run, and among those cases there are two . . . in which there are two white balls, and among these the case in which there are three white occurs in the long run, just as often as the case in which there are only two." (Govan, p. 208)

According to the text of Govan's discussion, this quote must be on or around page 435 of Chrystal's text.

Another very similar example is attributed to Chrystal's page 437:

"A bag contains five balls which are known to be either all black or all white—and both these are equally probable. A white ball is dropped into the bag, and then a ball is drawn out at random and found to be white. What is now the chance that the original balls were all white?" Professor Chrystal asserts that the chance is precisely what it was before, viz. $\frac{1}{2}$. (Govan, p. 208–209)

"The ball drawn out," says Professor Chrystal, "may have been the one we put in, it may not; and this is all that any one can say." (Govan, p. 209)

Note that this is quite upside-down compared to how we usually think about frequentism: Here, Chrystal tells us to ignore the likelihoods and put all our confidence in the priors. We are used to thinking about frequentists as doing the exact opposite.

The Essential Tension

Govan objects to Chrystal's principle of rejecting the evidence:

… let us say we have two bags before us, one containing six white balls, the other five black balls and one white. There is nothing to indicate which is which. We draw from one of the bags chosen at random a ball which proves to be white. It is difficult to believe that any man in the possession of his faculties, say if his life depended on his guessing aright from which bag the ball had come, would hesitate to guess the former. Even according to Professor Chrystal he would be right 6 times out of 7 in the long run. Yet, again according to Professor Chrystal he would would be just as likely to be wrong as to be right. (Govan, p. 209)

Although they are talking past each other, this is certainly the core of the issue: The distinction between optimal, adaptive gambling behavior and fixed, objective frequencies.

A note on the Kolmogorov-Dynkin extension theorem

I've been somewhat confused about Kolmogorov's extension theorem, the uniqueness theorem for stochastic processes, but this note by Cozma Shalizi cleared up things a little bit.

Here's the set-up: We have a universe of values, $\Omega$, and this gives rise to a universe of sample paths, $\Omega^\mathbb{N}$. This universe of sample paths can be equipped with a probability measure. Such a measure maps bundles of sample paths, $B \subseteq \Omega^\mathbb{N}$, onto probabilities.

The measure has to respect the usual restrictions, including countable additivity. However, it doesn't have to (and in fact can't) accept all bundles as inputs.

The set of measurable bundles is defined inductively on the basis of one-dimensional base cases and some recursive operations. Specifically, a bundle is measurable if it is

1. the preimage under a coordinate projection of some measurable set $A \subseteq \Omega$;
2. constructed out of other measurable bundles by means of complementation, finite intersection, or countable union.

These rules define a system $S$.

We could also define a smaller system, $S^{-}$, by only allowing finite unions. A bundle in this smaller system corresponds to a proposition about a finite-dimensional projection of the stochastic process. If we take the $\sigma$-algebra generated by the smaller system, we recover the full system, $\sigma(S^{-})=S$.

The content of the Daniell-Kolmogorov exention theorem can now be stated as follows: Suppose that two measures $P$ and $Q$ assign the same values to all bundles in $S^{-}$; then they assign the same values to all bundles in $S$. This is equivalent to saying that if two stochastic processes have the same distribution under any finite-dimensional projection, then they have the same distribution.

Not surprisingly, the proof involves some facts about the limit behavior of measures on increasing sets. Specifically, when a series of sets $B_1, B_2, B_3, \ldots$ converges to some limit set $B$ from below, then the measures $\mu(B_1), \mu(B_2), \mu(B_3), \ldots$ must converge to the limit $\mu(B)$. This implies the uniqueness of the infinite-dimensional extension of a set of finite-dimensional projections.

Here is the proof as given by Shalizi: Let $W$ be the set of bundles under which $P$ and $Q$ agree. Then by assumption, $S^{-} \subseteq W \subseteq S$. We want to show that $\sigma(S^{-}) = S  \subseteq W$. We do this by showing that $W$ is a Dynkin system, and that $S^{-}$ is closed under finite intersection, since this will allow us to use Dynkin's $\lambda$-$\pi$ theorem.

A Dynkin system is a system of sets which contains the whole universe, is closed under complementation, and closed under disjoint, countable unions (or equivalently: increasing, countable unions). The set agreement system $W$ satisfies these three properties, since

1. $P(\Omega^{\mathbb{N}})=Q(\Omega^{\mathbb{N}})=1$, so $P$ and $Q$ agree on the bundle consisting of all sample paths;
2. For any bundle $B\in W$, $$P(B^c)=1-P(B)=1-Q(B)=Q(B^c),$$ so if $P$ and $Q$ agree on on a bundle, they also agree on its complement;
3. Finally, if $P$ and $Q$ agree on every bundle in an increasing series of sets $B_1, B_2, B_3, \ldots $, then the two series $$P(B_1), P(B_2), P(B_3), \ldots$$ $$Q(B_1), Q(B_2), Q(B_3), \ldots$$must have the same limit. Hence, whenever $P$ and $Q$ agree on a list of increasing bundles, they thus also agree on the set-theoretic limit of that series.

Thus, $W$ is a Dynkin system. This means that it is large enough to contain all the additional sets we put into the system $S^{-}$ by closing it under $\sigma$-operations. In other words, $\sigma(S^{-}) \subseteq W$. We thus have both $S \subseteq W$ and $W \subseteq S$ (which is true by definition), and thus $W=S$.

Here's an intuitive version of that proof:

The system $S$ on which $P$ and $Q$ are defined must itself be defined by certain recursive operations: Every set in $S$ is constructed from a finite-dimensional base case using only well-behaved operations.

The properties of probability measures guarantee that we can match each of these constructions, operation for operation and step by step, by a corresponding axiom which enforces a certain unique result. Because of this parallelism between the construction of the sample space and the propagation of necessity, no measurable set in $S$ actually has any wiggle room when the bases cases are identical.