Did Haldane really use his "2:1"?

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DidHaldanereallyusehis„2:1“?
Contents
................................................................................................................................................. 1
Did Haldane really use his „2:1“? ......................................................................................... 2
Preamble: ............................................................................................................................. 2
Introduction: .......................................................................................................................... 2
Rumors: ................................................................................................................................ 5
1935 .................................................................................................................................. 5
1945 .................................................................................................................................. 5
1969 .................................................................................................................................. 5
1984 .................................................................................................................................. 5
1992 .................................................................................................................................. 6
Methods: ............................................................................................................................... 6
Results: ................................................................................................................................. 7
Haldane 2.0: ......................................................................................................................... 8
Discussion & Conclusion: ..................................................................................................... 8
References and Literature for further reading, source of photographies: ........................... 10
Preamble:
The following is along the traditional lines of a serious scientific paper, i.e.:
- Introduction
- Methods
- Results
- Discussion
- References
The contents, however, may be regarded as something winkingly …
Introduction:
The famous scottish physiologist John Scott Haldane (1860 – 1936) and his co-workers
published in 1908 a ground-breaking paper [1]. This is generally considered and accepted as
the foundation of staged decompression diving. The paper featured two diving tables in the
attachment. The first of these tables became very successfull. In deed, Table I (loc. cit. p.
442 ) was adopted for regular use in the Royal Navy ca. a year (p.367) before and was so
wildly successfull, that Haldane later on wrote ([2], p. 350):

The work of these 3 gentlemen and their 85 goats and hundreds of smaller animals ([1], p.
379) is still the basis of most modern decompression tables, desktop deco-software and
diver-carried dive computers: more than 800 dives with goats and more than 300 with the
smaller animals have been analysed carefully. In the refs. [3] – [7] is ample material. The
basic assumptions to calculate the tables have been:
 the model of a human body can be divided into 5 theoretical compartments with
halftimes from 5 to 75 min
 inertgas uptake and -elimination is exponential with the above halftimes and thus:
 symmetrical, provided no inertgas bubbles have been formed upon pressure
reduction
 these compartments can withstand a certain supersaturation. Below this threshold
value the risk of contracting decompression sickness is relatively low
 this particular threshold, the tolerated supersaturation, for these 5 compartments
follows a „2:1“rule, that is: ([1], p. 424):
In the course of time and extensive usage it became clear, however, that there are
operational shortcomings. Table I was overly conservative for short bounce dives and Table
II was grossly inadequate for long and deep dives. This is discussed as well in [3], [7], [8 
13] and the references therein. As well Haldane had caveats and mental preservations; one
is there, [1] on page 368:

And another one in [1], p. 357 is this very clearly stated footnote:
It seems that these caveats have been forgotten, especially by designers of modern
decompression software. Or, as Robert Henry Davis put it already that time [15], p. 8:
„In spite of this characteristic caution, later writers have mistakenly credited him with
asserting that rapid decompression from any pressure „2n“ to pressure „n“ is safe.“ And, on
this particular page further down: „ …10 atm, Haldane’s two to one law no longer held good,
a finding for which he had prepared us.“
The basic shortcomings have been alleviated with:
 increased number of compartments, i.e.: 9, 14, 16, 20 or more
 greater spectrum of halftimes. i.e. from 2.5 min to 900; and
 variable supersaturations, i.e.: from ca. 3.4 : 1 to 1.1 : 1.
But despite these operational shortcomings there have been rumors about the internal
design of the tables, i.e. if the calculations really follow the „2:1“ principle; that is, what was
stated on p. 355:
(Historical note: a lot of these calculations have been done by his son JBS Haldane, called
Jack. Despite beeing a child, he co-authored already earlier papers with his father; he was
considered a genius. As a reward he was allowed to take part in the diving experiments in
open waters ([1], p. 436):

Rumors:
Since then, decompression researchers and divers have been speculating about this “2:1”.
One explicit statement is from H. V. Hempleman in ref. [8], p. 233:
"One further important fact to be noted is that although the Haldane decompression
ratio of 2:1 is much discussed by everyone studying this subject it was not used by
Haldane for his calculations!”
1935
Already earlier this millenium Hawkings, Shilling and Hansen from the United States Navy
(USN) had been writing in:
(loc. cit: p. 333):
„ Actually we find when we calculate many of Haldane’s tables (1922) that he often
goes to a ratio of 2.1 or even 2.3 to 1, but this is well within the safety factor.”
1945
In a publication from 1945 about „explosive decompression“ with more than 400 animals
(mice, rats, guinea pigs and rabbits ) we find the following statement:
„ … it would seem that Haldane deliberately abbreviated the final equilibration period
by 10 – 20 min…”
(Source: J. Physiol. (1945) A STUDY OF THE EFFECTS OF RAPID 'DECOMPRESSION' IN
CERTAIN ANIMALS BY P. EGGLETON, S. R. ELSDEN, J. FEGLER AND C. 0. HEBB)
1969
In a research paper from the Submarine Base Groton, Conn. (NSMRI Report No. 580, 1969:
Decompression Patterns developed by an interdependent Electric Analog, Gary P. Todd) we
find on p.4 (citation):
“ … the original tables varied from 3,4 : 1 to 1,2 : 1 …“ .
1984
A very similiar, but more detailed, really in-depth analysis of 2 profiles is there [8] on the
pages 233 – 242! It looks like that the supersaturation ratios, used to build the tables, are
varying from 2,5 : 1 to 1,7 : 1, as well during all the deco stages! In clear words: from deco
stage to deco stage these ratios have been changed from „2:1“ to something differently!

1992
Last, but not least: my friend Karl Huggins put in his famous deco workbook (ref. [9], p. 2-6):
Methods:
We have been curious if these rumors and, sometimes harsh comments, have been
assessed correctly. You could do it yourself with virtually any desktop deco software there is,
provided it allows for a certain flexibility described below. We set out to check with our public
free-ware version of DIVE.
The allowed / tolerated compartment supersaturation follows a simple linear relationship (see
the graph above), as in all well-known decompression-models based on compartment
perfusion:
P
compartment, tolerated = (Pambient / b) + a
This is the formula found for eg. in Bühlmanns books ([13], p. 117); a very similar equation in
the notation from Bob Workman looks like that [14]:
M = M
0 + ∆ M * diving depth
By combining the Bühlmann formula with the Haldane law and compare the 2 linear
equations on the right hand-side you have now:
P
compartment, tolerated = (Pambient / b) + a = Pambient * 2
 a = 0.0 , b = 0.5
With DIVE Version 3 this check is now easy and straightforward:
take the coefficients matrix and modify it, according to Haldanes specifications; i.e.:
overwrite the first half-times (TAU) with the ones, Haldane used (5, 10, 20, 40 & 75 min.),
then put a = 0.0 and as well b = 0.50, as you may infer from the graph above or from the

formulas. In order to ensure that DIVE Version 3 does not go nuts with these things, you
have to copy to last line (compartment #5) and fill up the matrix up to #16 with the identical
values:
(Ignore the values HI = LO = 1.0! These are reserved for the so-called „Variable Gradient
Method“, the VGM, which we do not use now. And, to be honest, nobody used except some
strange divecomputers, obsolete since long …)
But to reach at a meaningful comparison, you have to adapt as well the following values:
 Geometric diving depth / depth of decompression stages: from feet to m
 Rate of ascent; Haldane put it to ca. 30 feet / 1 min
 Density of seawater: most deco softwares use freshwater density or another cryptic,
average value
 Inertgas content: deco software uses normally fN2 = 0.79 or similar value +/- 2 % for
air diving. Haldane et al. did not! They used instead fN2 = 1.00! Regular desktop deco
software may not allow fN2 = 1.00 because this would imply a relatively unhealthy
mixture without O2.
To get a grip on that one you could use something like an „inverted EAD“ (inverted
equivalent air depth). From your EAN / Nitrox courses your are familiar with the
regular EAD concept. This one here is just the inverse in the metric version:
(depth
Haldane + 10.0 ) / 0.79 -10.0 = depth calculation
Results:
As just one paradigm, let’s take the following dive from Table II to 132 – 144 feet for 90 min.
With the above adaptions in mind the results for the calculated inertgas partialpressures pN2
in the leading compartments (the one with the highest N2 pressures) looks like that:

Procedure Leading
Compartment pN2 [Bar] Ratio
on reaching the 12 m stage, after the 10 min stop # 3 4.1390 1.88136
on reaching the 9 m stage, after the 10 min stop # 4 3.6671 1.93005
on reaching the 6 m stage, after the 20 min stop # 4 3.1195 1.94960
on reaching the 3 m stage, after the 30 min stop # 5 2.5801 1.98460
on reaching the surface, after the 35 min stop # 5 2.2058 2.20580
(extending the last stop to 52 min 2.0150)
For this dive, the deeper stops are more conservative and giving thus a reduced
supersaturation, e.g. the first ratio of 1.88 gives something like a „Gradient Factor“ of 94 %,
aka „GF Lo = 0.94“; but the surfacing value is ca. 2.2 : 1 instead of 2.0 : 1. If the last stop at
the 10 feet stage is extended from 35 to ca. 52 min, then a nearly „2:1“ would have been
reached.
Haldane2.0:
A couple of other dives end up in ratios for the deeper stops with ca. 1.6. This would give
sometimes a GF Lo of 80%. So Gradient Factors are already in place since then! As well
things like „accelerated deco“ (p. 354, 371, 376), „EAN36“ (p. 379), linear decompression
only for saturation dives (p. 366), more and longer compartment halftimes (p. 376) and
asymmetric de-saturation (p. 344, 350) have alreday been contemplated. So some diving
magazines sell you these things as the latest cry from the TEC-scene: but obviously it’s not.
It has been around now for more than 105 years, just the terminus technicus, Haldanes
wording, has been quite different.
Discussion&Conclusion:
Basically we could confirm the above cited rumors, but as well Haldane’s own caveat, which
was clearly pronounced on p. 355:

In [1], p. 367 the authors state:
As well, on p. 361:
And, on p. 374:
In summary, some of the last stops have been obviously truncated, but, as well, it is clear
from all these statements above copied from [1], that the tables have been designed clearly
with that in mind!
Given the mathematical / operational possibilities of the time and the depth / time intervals of
the tables, the „2:1“ ratio is clearly followed, say by approx. +/- 20 % in Table II and by
approx. +/- 10 % in Table I. This would be perfecly in-line with the accuracy of the available
measurement methods for depth and time, and, as well with the clumsyness of a diving
operation (dry suit + helmet + weights + boots + come home bottle + … + at approx. 100 kg;
lowering & lifting the diver with a stage, i.e. manually geared elevator [15], p. 236):

(This is even adressed in modern diving operations: the stage with the 2 divers, called „team
red“ and „team green“ is lowered in the water to ca. 20 feet, they turn around, make a bubble
check and only thereafter the bottom time starts.)
If you consider also the basic restraint of Table I, i.e. a maximum TTS (time-to-surface) of
about 30 min and the operational difficulties of implementing a new procedure to a military
organisation, which, normally, behaves sort of beef-witted …
In clear and easy retrospect we would not join the choir of bleating from our colleagues but
instead bow to these tables! Especially if you consider the lines of reasoning and the
audacious transformation of the results from the goats in a deco-chamber to real human
divers in water!
ReferencesandLiteratureforfurtherreading,sourceofphotographies:
A) General:
=========
[1] Boycott AE, Damant GCC, Haldane JS.
The prevention of compressed air illness. J Hyg. 1908; 8: 342–443.
[2] RESPIRATION
BY J. S. HALDANE M.D., LL.D., F.R.S. FELLOW OF NEW COLLEGE, OXFORD HON.
PROFESSOR, BIRMINGHAM UNIVERSITY
NEW HAVEN YALE UNIVERSITY PRESS LONDON : HUMPHREY MILFORD : OXFORD
UNIVERSITY PRESS MDCCCCXXVII
[3] The Future of Diving: 100 Years of Haldane and Beyond;
published by Smithsonian Institution Scholarly Press;
ISBN-13: 978-0-9788460-5-3, ISBN-10: 0-9788460-5-2
Especially the chapters:

 J. S. Haldane, the First Environmental Physiologist
Alf O. Brubakk and Michael A. Lang, p. 5 - 10.
 Haldane Still Rules! David J. Doolette, p. 29 - 32.
[4] / and, as well: [119] Goodman, Martin (2007) Suffer and Survive:
Gas Attacks, Miners' Canaries, Spacesuits and the Bends: the Extreme Life of Dr. J.S.
Haldane, Simon & Schuster, London, ISBN 978-1-4165-2230-0
[5] A BRIEF HISTORY OF DIVING AND DECOMPRESSION ILLNESS
Chris Acott, SPUMS Journal Volume 29 No.2 June 1999
[6] JS HALDANE, JBS HALDANE, L HILL AND A SIEBE: A BRIEF RESUME OF THEIR
LIVES. Chris Acott, SPUMS Journal Volume 29 No.3 September 1999
[7] TESTING JS HALDANE’S DECOMPRESSION MODEL
Chris Acott, SPUMS Journal Volume 30 No.1 March 2000
B) Special:
=========
(concerning "2:1")
[8] / and, as well: [158] Shilling, C. W. Carlston, C.B. Mathias, R.A (1984) The Physicians's
Guide to Diving Medicine, Plenum Press, N.Y., ISBN-13: 978-1-4612-9663-8
[9] Huggins, Karl E. (1992) The Dynamics of Decompression, Workbook, 1st. edition. Ann
Arbor, Michigan
[10] Hawkings, J.A., Shilling, C.W., Hansen, R.A. A suggested change in calculating
decompression tables for diving, U.S. Nav. Med. Bul 1935; 33: 327 - 338
[11] J. Physiol. (1945) A STUDY OF THE EFFECTS OF RAPID 'DECOMPRESSION' IN
CERTAIN ANIMALS BY P. EGGLETON, S. R. ELSDEN, J. FEGLER AND C. 0. HEBB
[12] NSMRI Report No. 580, 1969: Decompression Patterns developed by an interdependent
Electric Analog, Gary P. Todd
C) Perfusion Models:
=================
[13] / and, as well: [65] Bühlmann, Albert A., Völlm, Ernst B. (Mitarbeiter), Nussberger, P.
(2002): Tauchmedizin, 5. Auflage, Springer, ISBN 3-540-42979-4
[14] Workman, Robert D. "Calculation of Decompression Tables for Nitrogen-Oxygen and
Helium-Oxygen Dives," Research Report 6-65, U.S. Navy Experimental Diving Unit,
Washington, D.C. (26 May 1965)
-/- a summary of all needed formula for decompression calculations

D) Source of Photographies:
=======================
Frontispiece:
© ALBI (when he was young, ca. 45 years ago, useing his DRÄGER helmet, checking the
come-home bottle, which is integrated into the breast-weight)
[15] one diver on stage:
Robert Henry Davis: Deep Diving and Submarine Operations: A Manual for Deep Sea Divers
and compressed Air workers, p. 236
two divers on stage with Kirby-Morgan helmet: © USN, Monitor 2001 expedition;
NOAA Monitor Collection