In the opening lines of ENGR, Dr Watson remarks that the case "gave my friend fewer openings for those deductive methods of reasoning by which he achieved such remarkable results." [ENGR 274] This is undoubtedly true, if for no other reason because Holmes is only directly involved in one quarter of the story. However, this does not mean that the case is devoid of interest for the student of Holmesian reasoning. On the contrary, it contains several illustrations of the fact that the detective often has to be content with probabilities rather than certainties.

I have argued elsewhere that the most important form of reasoning used by Holmes is abductive, rather than deductive, which means that it aims at finding the best explanation of the observed facts, instead of drawing logically valid conclusions from these facts. But in order to give an abductive explanation the certainty of a logical deduction, it must be shown that it is the only possible explanation that fits the facts. Hence Holmes’s constant insistence on the importance of "eliminating the impossible" in favor of the truth, "however improbable".

However, in many situations it is not possible to eliminate all except one of the conceivable explanations, simply because there are too many alternatives and not enough facts to allow us to distinguish between them with absolute certainty. In such cases, the detective has to resort to probabilistic reasoning and be content with finding the most probable explanation, without eliminating all the alternatives.

There is a subtle but important difference between this form of reasoning and plain guessing, which is brought out in Holmes’s response to Dr. Mortimer in the following passage from HOUN: " ‘We are coming now rather into the region of guesswork,’ said Dr. Mortimer. ‘Say, rather, into the region where we balance probabilities and choose the most likely. It is the scientific use of the imagination, but we have always some material basis on which to start our speculation.’ " [HOUN 687]

There are several small inferences in ENGR that illustrate the nature of probabilistic reasoning, and we will have occasion to return to some of them below, but the most interesting one, also from a literary point of view, is without question the episode where Holmes baffles his company by explaining that they are all wrong concerning the location of the scene of the crime, which is really to be found in the centre of the circle drawn on Inspector Bradstreet’s map.

Let us begin by considering the facts to be explained. As far as Bradstreet, Hatherley and Watson are concerned, the essential fact is that there was a twelve-mile drive starting from Eyford station, from which they all draw the conclusion that the destination is to be found on the circumference of a circle with a radius of twelve miles and centered on Eyford. In other words, they make the erroneous assumption that the drive took place in a straight line.

For Holmes, however, there are other facts that have to be explained (and the best explanation is the one which accounts for all the facts). First of all, there is the fact that the horse was fresh and glossy. Secondly, there is the fact that when Hatherley became conscious again after fainting he found himself very close to Eyford station. And the most probable explanation for these facts is that the twelve-mile drive was in fact six miles away and six miles back. This is all the more plausible as the villains undoubtedly wanted to mislead Hatherley with respect to the exact location of the house.

Note, however, that it is far from being the only conceivable explanation (and therefore not a logically valid inference). The fact that the horse was fresh could be explained by assuming that the old horse had been replaced by a new one before going back, or that the man posing as Colonel Lysander Stark had stayed at the station all evening waiting for Hatherley. And the fact that Hatherley woke up near Eyford station could be explained – as Hatherley did himself at first – by assuming that the gang had transported him back twelve miles. At the end of day, however, these explanations are far less probable than the one advanced by Holmes, which also accounts for all the facts together. In all its simplicity, this illustrates the essence of the Holmesian method:

There are other illustrations of the principles of probabilistic reasoning in ENGR, although not as spectacular as the episode just considered. First of all, we have the inference made by Holmes concerning the fate of Mr. Jeremiah Hayling, the hydraulic engineer that went missing about a year earlier. The relevant facts here seem to be that Hayling left his lodgings at ten o’clock at night, never to be heard of again, and that, when Hatherley was attacked, Elise shouted: "Fritz! Fritz! --- remember your promise after the last time. You said it should not be again. He will be silent! Oh, he will be silent!" [ENGR 283] From these facts Holmes concludes that Hayling had fallen victim of the same gang as Hatherley, only with a more fatal outcome. Again, this is far from being the only conceivable explanation, but it is clearly the most probable one in view of the given facts.

Other examples include the inference that the villains were coiners, and that it was Hatherley’s oil lamp that put the house on fire. In both cases, other explanations are certainly imaginable, although they would appear very far-fetched.

Taken together, these examples of Holmesian reasoning are perhaps less impressive than usual, partly because they do not form a connected chain of inferences, and partly because they do not lead to the apprehension of the criminals. But in each one of them, we see the mark of the really great detective, which possesses not only the capacity to observe and to reason logically, but also a gift for "the scientific use of the imagination", which consists in balancing probabilities. Incidentally, this is why detection can never be a pure science but must always contain an element of art, which in turn means that a great detective must also be a great artist. And what could be more true of Mr. Sherlock Holmes?