The tables that the Royal Navy issued to its divers in 1907 were calculated, not found. The goats had settled two things: that a body can be taken suddenly to half its absolute pressure, and that the slow parts of a goat are full after about three hours. Everything else in the tables was arithmetic done on a model of the body that nobody had seen, and the paper of 1908 shows the arithmetic in one worked example. This piece follows that example, sets out what the two tables allowed, and gives the case against them that Leonard Hill made in 1912 and the revisions that followed.

The model

The body, in the model, is a set of parts, each with its own blood supply and its own capacity for nitrogen, and each therefore filling and emptying at its own speed. Blood leaving the lungs is saturated at once. A part fills along a curve: half of the way to full in one unit of time, three-quarters in two, seven-eighths in three. The paper’s own calculation for an averaged body, from the blood volume of a man and the extra nitrogen that fat dissolves, gave a half-time of about twenty-three minutes. The kidneys, Hill and Greenwood had shown, filled about ten times faster than that. The tunnel men’s experience, that a three-hour shift was safer than an eight-hour one, showed that something filled far slower. From the goats, allowing for their faster circulation, the authors put the slowest part of a man at a half-time of seventy-five minutes, so that he would be practically full in five hours; and they chose five parts to stand for the whole, with half-times of five, ten, twenty, forty and seventy-five minutes.

Emptying follows the same curve as filling, so long as no bubbles form. And bubbles, the goats had shown, do not form so long as the nitrogen pressure in a part is not much more than twice the pressure of the air being breathed. The rule for a safe ascent is then a sentence: at every moment, no part of the body may be more than about twice as saturated as the air.

The example

The paper takes the hardest case it can, a diver completely saturated at 213 feet of sea water, which is thirty-five and a half fathoms, an excess pressure of six and a half atmospheres and an absolute pressure of seven and a half, and asks how he may be brought up as fast as the rule allows.

In absolute terms his depth is 213 feet of water plus the 33 feet that stand for the atmosphere, 246 feet. Half of that is 123, and 123 feet of absolute pressure is 90 feet of actual depth. So “it would be safe to come up at once to a depth corresponding to 123 feet of absolute water pressure, i.e. to 90 feet of actual depth. Consequently the first stage would be a rapid ascent of 123 feet.”

At 90 feet he must wait until the slowest part of him has lost enough nitrogen that it will still be within the ratio at the next stop, 80 feet. At 80 feet the air is at 113 feet absolute, so the part may be saturated to 226 feet; it is saturated to 246; it must lose 20 feet of its excess, which is 16 per cent of the difference between where it is and where the air now is, 123 feet. A part with a half-time of seventy-five minutes loses 16 per cent of an excess in about nineteen minutes. “Hence a delay of 19 minutes would be necessary at 90 feet before coming up to 80 feet.” The same sum at 80 feet gives 21 minutes; at 70, 23; at 60, 26; at 50, 30; at 40, 35; at 30, 42; at 20, 51; at 10, 62. The stops lengthen as the surface nears, because the difference between the body and the air, which drives the nitrogen out, shrinks with every stage. The total is 309 minutes, “or more than five hours, to reach surface.”

The paper notes what the figure shows and what it costs. The time needed does not grow in proportion to the depth: fifteen feet more depth at 50 feet costs forty-five minutes more ascent, the same fifteen feet at 200 feet costs fifteen minutes. The last stop could be halved without serious risk, which would save half an hour and leave the slowest part saturated to an excess of 1.8 atmospheres, “a safe enough limit perhaps, but leaving no great margin to spare.” A uniform ascent in the same 309 minutes would bring the diver to the surface with his slow parts at an excess of 2.1 atmospheres; a uniform ascent in ten hours would still leave more than one. “Increased safety at the end is only secured in combination with useless delay at the beginning.”

Five hours is not a dive. So the second half of the arithmetic is to keep the diver from filling. A man going down at a uniform rate takes up about half as much nitrogen as if he had spent the same time on the bottom, and so does a man coming up at a uniform rate; both are “all to the bad.” The remedy is to go down as fast as the ears allow, which an experienced diver can do at 198 feet in two minutes, and to come straight up to the first stop. The second worked example is the goats’ own: fifteen minutes at 75 pounds, six atmospheres absolute, reached in six minutes, so eighteen minutes of virtual exposure. The first stage is from 6.1 atmospheres to 2.8, which is from 168 feet to 60 in four minutes; then two minutes at 60 feet, three at 40, five at 30, seven at 20 and ten at 10, thirty-one minutes in all. By the model this leaves the twenty-minute part saturated to 1.4 atmospheres at the surface, slightly beyond what was calculated to be desirable; in thirty-four decompressions of eighteen goats there were five cases of bends and nothing worse; so the table cut the allowed stay to fifteen minutes from the surface.

The two tables

Table I, “Stoppages during the ascent of a diver after ordinary limits of time from surface,” runs in rows of six feet of depth from the surface to 204 feet. For each row it gives the pressure in pounds, the longest time allowed from leaving the surface to beginning the ascent, the approximate time to the first stop, and the minutes to be spent at 60, 50, 40, 30, 20 and 10 feet. Up to 36 feet there are no stops and no limit. The limits shrink with depth so that the diver, whatever his depth, is back on the surface within about half an hour of starting up. Near the surface the ratio is allowed to be a little exceeded in the fastest parts, “which saturate and desaturate very rapidly, and, as already explained, give rise to no danger.” At every stop the diver is to keep his arms and legs moving.

Table II is for the diver who has overstayed Table I, because he was fouled or the tide held him or the work could not be left. It gives stops for any length of stay. Its times are long, and the paper says they are shorter than the model asks: “the time allowed for decompression after very prolonged exposures is somewhat curtailed, but not so much as to permit of risk of more serious symptoms than ‘bends,’” because a diver who is cold and exhausted cannot be kept hanging on a rope for the five hours the arithmetic would like. For a man of heavy build the times should be a third longer, and such a man, especially over forty-five, “ought not to expose themselves to the risk of a prolonged stay in very deep water.” The diver who died on the torpedo boat, in the entry, was forty-nine and heavy, and was brought up over two and a half hours after two and a half hours at 150 feet; the paper’s conclusion from his death was that the second table’s stops were “none too long.”

Two more rules are in the paper. A diver who goes down twice at short intervals has not emptied his slow parts, and the two exposures should be added together for the second half of his stops. And at the deepest depths there is a second limit that has nothing to do with bubbles: oxygen at a partial pressure of about two atmospheres damages the lungs in hours, and the paper’s one trial of it, seven goats at 81 pounds in air enriched to 36 per cent oxygen, killed one of them with pneumonia in three hours. Up to 40 or perhaps 50 fathoms on air, “provided the stay is not long,” the authors thought there was no immediate risk.

Hill’s objection

Leonard Hill at the London Hospital had been working on compressed air as long as Haldane, with Major Greenwood as his subject and co-worker, and his book of 1912, Caisson Sickness and the Physiology of Work in Compressed Air, is where the Admiralty’s work met its first sustained criticism. Hill’s preface gives the committee its due, “the important contributions of John Haldane to this subject,” and then its objections.

The first is about evidence. “The mathematical calculations on which the Admiralty table of stage decompression is founded have never been published, and so cannot be critically examined. The table is built very largely on theoretical data concerning the circulation of the blood, which cannot be regarded as fixed in the varying states of activity of a man’s body.” The five parts were a convenience; nothing in a body corresponds to them; and the rate at which any real part fills or empties depends on how hard the man is working, which the model holds constant.

The second is about practice. Hill’s own experiments on himself and Greenwood had shown that exercise during decompression, which speeds the blood and the breathing, lets a man come out faster, and that oxygen breathed in the last stages helps. “The engineer considers the long periods fixed by the Admiralty Committee unpractical. He can safely shorten them if he trains his men to exercise themselves during decompression.” For tunnels and caissons, where men worked for hours at pressures of two or three atmospheres, he judged the Admiralty’s periods “about twice as long as necessary,” and he thought that the superiority of stages over a uniform ascent was “not so marked as the Admiralty Committee maintained,” though “the stage method can be made fairly safe, and is the best one to use for caissons.” On the goats themselves he was sharp in one place: many of the bends recorded were “classified as slight or doubtful,” and bends in a goat, he argued, were a poor statistic to compare methods by, since they marked the slow parts that the serious symptoms did not come from.

Haldane’s reply, in 1922, was experience. Damant had tried shortening the last stops with a recompression chamber standing ready; the divers began to get bends, the bends recurred, and the chamber went back to being for emergencies. “It appears that the times cannot be cut down without risk of trouble, unless the divers are placed in the chamber as a matter of routine after each dive.”

The revisions

The model was adopted, and then corrected by the people who used it most. The United States Navy tested the tables from 1912 and published its own in 1915. In the 1930s its Experimental Diving Unit, under Hawkins, Schilling and Hansen, ran enough dives to find that one ratio did not fit all the parts: the fast parts could stand more than two to one, and the slow parts less. The tables of 1937, by Yarbrough, gave each part its own allowed ratio. In the 1950s and 1960s the parts were given longer half-times, out to two hours and beyond, and Robert Workman replaced the ratios with a set of maximum allowed pressures for each part at each depth, which he called M-values; later tables added more parts, and other models began from bubbles instead of from dissolved gas. All of them keep Haldane’s two instruments, the half-time and the limit of supersaturation, and all of them are tuned, as his was, by counting who gets the bends.

What the goats had tested was the halving rule at six atmospheres and the comparison of stages against a uniform ascent after one exposure. What they had not tested was the five parts, because there is nothing to test; the parts are a way of doing the sum. The sum gave a table that kept the Royal Navy’s divers out of trouble for a generation, and the table has been changed in every decade since by people measuring where it was wrong. That is what a model is for.

Sources

• A. E. Boycott, G. C. C. Damant and J. S. Haldane, “The Prevention of Compressed-Air Illness,” Journal of Hygiene 8 (3), 1908, Part I sections A to D and Appendix IV. archive.org
• Leonard Hill, Caisson Sickness and the Physiology of Work in Compressed Air, 1912, preface and chapter on the theory and practice of decompression. archive.org
• J. S. Haldane, Respiration, 1922, chapter XIII, for the shortened stops and the reply. archive.org
• George D. Stillson, Report on Deep Diving Tests, 1915, for the American adoption. archive.org
• Wikipedia, “History of decompression research and development,” for the revisions of the 1930s onward. en.wikipedia.org