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The Drake Equation and Extraterrestrial Life, a Brief Overview

Peter Warrington, Manchester Astronomical Society

original source |  fair use notice

Summary: A number of the implications regarding the possibility of extraterrestrial intelligence are discussed. The Drake equation is introduced and some values for the established parameters suggested, a further parameter is proposed.

"For all our feelings of self importance, we are only a kind of biological rust, clinging to the surface of our small planet, and weighing far less than the air that surrounds us"

"One of the distinctions and triumphs of the advance of science has been the deprovincialization of our world view"

-- Carl Sagan


A number of the implications regarding the possibility of extraterrestrial intelligence are discussed. The Drake equation is introduced and some values for the established parameters suggested, a further parameter is proposed. Some of the principle arguments for and against extraterrestrial life are discussed. It is acknowledged that we have already announced our existence to any nearby technological civilisations and that not to anticipate contact with an extraterrestrial intelligence would be unwise. It is suggested that extraterrestrial life is possibly quite abundant, however the prospect of life on Earth being unique is also recognised.

A note about astronomical distances

The scale of astronomical distances and other large numbers, such as the number of stars in a galaxy, is something which often causes confusion. The universe is so inconceivably enormous that this is hardly surprising.

If for example, the thickness of a sheet of paper, is used to represent the distance from the Earth to the Moon. At it's closest approach Venus, our nearest planetary neighbour, is about 45 million kilometres away, that distance would be represented by 112 pieces of paper. A sheaf about 370 pieces thick are necessary to represent the distance from the Earth to the Sun. To represent the distance to the next nearest star, a distance of around 4.5 light years, requires a stack of paper just over 11.5 kilometres high. (A light year is the distance light travels in a year, and to put that into perspective, light travels fast enough to get around the Earth about 7 times per second).

The Sun is an ordinary star, one of about 400 billion others in the galaxy which we refer to as the Milky Way. If each of those stars were represented by a piece of paper, that would be a stack 32,800 km high. This galaxy like many others, is a flat disc with a central region which is slightly thicker, it has spiral shaped arms extending outwards. (The whole thing is often likened to two fried eggs back to back, though the "yolks" would be a little too thick to accurately represent the central region). The galaxy is thought to be about 100,000 light years in diameter. To represent that diameter using the thickness of this page would require a pile of paper high enough to reach the Moon!

To represent the distance from our galaxy to another galaxy would need a pile of paper from here to the Sun and beyond. To begin to represent distances of wider significance, such as that to the edge of the observable universe, thought to be about 13 billion light years away, demands a pile of paper whose size can really only be appreciated by using astronomical distances to describe them. With this analogy in mind it is perhaps not surprising that comprehending scales and distances in astronomy causes much confusion.


Ever since man became aware of the cosmos in its widest sense, either as Ptolemy thought that we were at it's centre, or the contemporary view of us as a particularly small, and insignificant component of the universe, man has wondered about 'who' else might be out there.

Astronomy has taken a great many leaps forward since the Gallileo first turned a rudimentary telescope of the heavens, there are now optical telescopes capable of resolving incredibly fine detail, and techniques developed which permit the scrutiny of distant objects to the point where today's astronomers confidently predict the nature of the universe during the first two or three seconds of its existence some ten to twenty thousand million years ago. In recent years the existence a organic molecules including amino acids have been discovered amongst the stars, and in meteorites, plus an appreciation of how similar compounds may have been synthesised in the primordial environment which existing as the Earth began to evolve, these and an increasing knowledge of the dynamics of the universe has fuelled the eternal question of "are we alone"?

Since the possibility of the existence of life elsewhere in the universe was accepted as a distinct possibility, man has wondered at the possibility of contact and speculated on the consequences of contact with another intelligence. It can also be argued that searching for extraterrestrial life is a waste of time and resources, even pointless. Others have argued the opposite and won significant funding to undertake searches for indications of life elsewhere. Man has already announced his existence by virtue of the mass of radio transmissions emanating from planet Earth.

This report will summarise some of the significant aspects from the multitude of questions raised by the debate, "is there life elsewhere in the universe?", "how many advanced civilisations might there be?", or "are we alone?"

Historical perspective

"Do there exist many worlds, or is there but a single world?

This is one of the most noble and exalted questions in the study of Nature".

St. Albertus Magnus (circa 1260 AD) 15

The belief in 'the plurality of worlds', is something Tipler 12 comments is generally linked to three beliefs, the first of which Lovejoy 19 has called the "principle of plenitude" which asserts that what can exist must exist somewhere - if a world like ours exists so must others, since no "genuine potentiality of being can remain unfulfilled". This principle of plenitude has become the more recent 'principle of mediocrity', that is the existence of intelligent life here is nothing out of the ordinary. A further belief is that of the cosmos being infinite, and the third, that believers in extraterrestrial life have tended not to have a 'sense of history' and they have failed to recognise how their ideas have merely been modern representations of old concepts.

(This representation of the original concept is something which occurs throughout the history of the subject, "many of the arguments pro and con are re-invented as a new generation of debaters take up their pens", and "extraterrestrial life believers have always been willing to suspend the physics of their day" Tipler 12.) Amongst the ancient Greeks and Romans reference to the 'world' implied a central Earth with it's Moon and Sun plus five planets and the fixed stars. Consequently the concept of the plurality of worlds implied each 'world' had it's own universe with a central and inhibited Earth. Both the Greek philosophers, Aristotle 16 and Plato 20 were against the concept of life elsewhere, Aristotle because of his belief that the other planets were of a completely different substance and in the finite nature of the universe. Plato because he believed the Earth to be unique.

St Thomas Aquinas 21, a pupil of Albertus Magnus argued against both the plurality of worlds and the principle of plenitude. His logic was simple, if God had made other worlds they would be either similar or dissimilar to this one. If similar, then they would be in vain and this would not be consistent with divine wisdom. If dissimilar, none of them could contain all things and therefore none would be perfect, and an imperfect world could not be the work of a perfect Creator.

To many people the shear size of the universe is so unimaginably big that they believe there must be other 'people', other 'life' somewhere out there. Popular science fiction, and science fantasy, seems to exist on the belief that where ever you might look, there will be lifeforms.

Certainly since the Renaissance almost every major scientific advance has confirmed the view of our mediocrity. We are not the centre of the solar system, Earth is one of many planets, and is vastly older than the human species. The Sun is merely an ordinary star, in an obscure location in our galaxy, the Milky Way, along with about 400 billion other stars. The Milky Way is just one typical galaxy, with perhaps hundreds of billions of similar galaxies grouped in clusters throughout the universe. It has recently been suggested that these groups of galaxies exist in 'strings' of a number of such clusters.

On planet Earth, we humans appear to have emerged from a common evolutionary origin as all the plants and other animals. "We do not possess any uniquely valid locale, epoch, velocity, acceleration, or means of measuring space and time", wrote Sagan and Newman 9. It has been suggested by Morrison 4 that once interstellar travel becomes a viable practicality at least one 'ceremonial voyage' would be undertaken, simply because the facility exists. After that it is debatable if long distance journeys would occur, intelligent species preferring what might in the contemporary jargon be termed 'virtual travel'. It must be recognised that as technological civilisations develop and become 'hyper-developed' to the point where they possess knowledge, (and possibly understanding), thousands or hundreds of thousands of years in advance of our development, that they will simply not have an interest in such primitive organisms as we humans.

Biochemistry of the young Earth

Oparin and Holdane 7 in the 1930's proposed that the atmosphere of the newly evolving Earth was similar to those of the outer planets of our solar system. In essence that the atmosphere was not rich in oxygen, as it is now, but contained large amounts of hydrogen and compounds such as methane and ammonia. This led in two scientists, Millar and Urey 7 in 1953 to undertake the first experiment to investigate the chemical reactions which are thought to have occurred in oceans and atmosphere of the primitive Earth. This now often repeated experiment consisted of heating water in a closed system of flasks and pipes forcing the vapour through a mock atmosphere of methane ammonia and hydrogen. This 'atmosphere' was exposed to continuous electrical discharge which simulated the effect of natural lightening causing the gasses to interact. The products of these reactions were passed through a condenser and dissolved in the water which represented a primitive ocean. The experiment was allowed to run continuously for several days, and analysis of the 'ocean' demonstrated that many amino acids were formed.

Some years after this initial experiment was performed, a meteorite fell near Murchison in Australia, and its subsequent examination showed it to contain a number of the same amino acids as Millar and Urey had synthesised in their primitive ocean. This coincidence led to support for the theory that such compounds are easily associated with the young Earth. Within a decade it had been established that nucleic acid bases could be obtained by the reaction of components known to have been present in the primeval conditions of the early evolution of the planet. What is more, the small molecules which have been identified as components of the these reactions, e.g. water, ammonia, formaldehyde, hydrogen, cyanogen and cyanoacetylene, have now been shown to be present in abundance in interstellar dust clouds - the regions where new stars form, so amply providing the evidence for the emergence of the chemical building blocks of life.

The Drake / Green Bank equation

The search for extraterrestrial life is the initial step towards a dialogue or contact with extraterrestrial life. Whether we should attempt communication with extraterrestrial life is, to all intents and purposes no longer a consideration. Since the first radio transmissions by Marconi the existence of a technological species here on Earth has been broadcast far and wide. Any attempts at hiding our existence are now practically impossible. Those early radio signals are rippling through space more than 90 light years away. Today thousands of gigawatts of radio energy radiate daily from the Earth broadcasting and highlighting our existence like a galactic lighthouse. As a result we should realise any nearby technologies might detect our (albeit) unintentional transmissions and return a message announcing their existence or despatch a probe to investigate our circumstances and degree of benevolence, (or threat).

In 1961 there was a now renown conference held at the National Radio Astronomy Observatory in Green Bank, West Virginia 3, to discuss the question of a 'search for extraterrestrial life' (SETI). That gathering brought together a worldwide array of prominent astronomers and 'exobiologists'. The conference set out with the intention of attempting to quantify, by theoretical means, the number of technically advanced extraterrestrial intelligence's within the galaxy. The solution was an equation, now known as the Green Bank equation, though also widely referred to as the 'Drake equation' after Frank Drake the astronomer who proposed the core of the expression. The equation seeks to quantify the number, N, of technical civilisations in the galaxy.

The equation has, N = R* fp ne fl fi fc L

R* = mean rate of star formation in the milky way, our local galaxy.

fp = the fraction of those stars which form planetary systems.

ne = the number of planets in those systems which are ecologically suitable for lifeforms to evolve.

fl = the number of those planets on which lifeforms actually develop.

fi = the number of those which evolve to an intelligent form.

fc = the number of advanced intelligent lifeforms which develop the capability of interstellar radio communication.

L = the lifetime of those advanced technically advanced civilisations.

Values for some of these parameters are, of course, open to considerable disagreement, something to which we shall return later, however a set of values is widely quoted. Most of these have not altered to any significant degree since that conference in 1961.

They are; R* = 10/yr, fp = 0.5, ne = 2, fl = 1, fi fc = 0.01, and L = 10.

The mean rate of star formation in the milky way, our local galaxy, and it's stellar population is well understood and this figure of 10 each year is widely held to be reasonable. The current theories of star formation accept the formation of an accompanying accretion ring which is expected to form the basis of planetary bodies. Although this is not universally accepted it has become possible in recent years to measure slight gravitational perturbations in the proper motion of stars. (Proper motion is the actual movement, of a star rather than it's apparent movement.) It has been found that a large proportion, around 50%, of the stars close enough to be subjected to this investigation have companion objects which affect their movement. These companions, which are too small or too dark to see, range from objects with mass a little smaller than Jupiter to a few tens that planet whose mass is 1.899 x 10^27 kg. Of course the most definite indication of the formation of planets is that of our own solar system with the nine planets and their satellites. Clearly the probable existence of objects affecting the movement of distant stars does not guarantee a viable ecology for life to exist, however the evidence does imply the possibility. In view of the limited observational data it seems reasonable to regard the solar system as a typical model, this suggests fpne equals one.

Because of the rapidity of the origins of life becoming established on Earth, as evidenced by the fossil record and experiments which reproduce early Earth biochemical environments, the likelihood of life evolving seems high. This may also be supported by the fact that many organisms survive in niche environments, at great ocean depths, else in remarkably inhospitable climatic conditions. These considerations support the contention that the value for fl being at least one.

The values attributed to fi and fc are somewhat more contentious, aspects of this argument are unable to draw on significant evidence, however many researchers of the topic agree 0.01 to be a 'fairly conservative' estimate. This seems reasonable given that intelligence has evolved about halfway through the expected lifetime of the Earth and Sun system.

Perhaps by far the most contentious issue of the entire equation is that of the lifetime of technically advanced extraterrestrial intelligence's within the galaxy. It is interesting to note that the 1961 conference proposed the rather pessimistic figure of 10 years before humans became unable to meet that description. At the time, when the 'cold war' was at it's most fierce, it was anticipated that man might not be able to manage the long term effects of the nuclear weapons then being amassed around the globe. Additionally the spread of nuclear power was something which many feared. A further aspect might have been the beginnings of the population explosion and the pressures which that and other environmental factors would put on the Earth. With these considerations in mind it appeared the most acceptable value for L would be 10. That being the case, the Drake/Green Bank equation can used to calculate N, giving the answer, 1. From that it appeared the only technically advanced civilisation in the galaxy was here on Earth. A result which some researchers, notably Tipler 11, has not overlooked, the main thrust of his argument is discussed below. Brin 2 has suggested arguments and values associated with the parameters are too simplistic and proposes the galaxy is rather sparsely populated.

Now, more than thirty years later, the world has seen a number of major changes, some affecting the Drake/Green Bank equation, a discussion of man's ability to control nuclear proliferation or world population is outside the scope of this discussion. However one aspect of the equation must be modified, and, in this writer's opinion, a further parameter added. The value of L can now safely be increased from 10 to at least 34, and an increase to 50 would not seem unreasonable. Incorporating 34 as the value for L into the equation gives N as 3.4, clearly any increase from the initially proposed figure of 10 implies a rather more hospitable galaxy. If technically advanced civilisations were to exist and have lifetimes of a few thousand years then a galactic community appears a distinct possibility. The Green Bank conference suggested that if technically advanced civilisations could avoid self annihilation then their lifetimes might be, by comparison to terrestrial geological time scales, very long indeed. The conference went on to suggest that if 1% of developing galactic civilisations made peace with themselves then our galactic neighbours would be only a few hundred light years away. Other researchers and working groups, for example Sagan 8 , have examined the question and concluded there could be 106 technologically advanced extraterrestrial civilisations in the galaxy.

The present writer considers an additional parameter in the equation is justified. Although the object of the original equation was to determine the number of technically advanced extraterrestrial civilisations, the debate has been centred on the search and eventual communication. A consideration is necessary to take account of the need for two communicating communities to exist simultaneously, or at least the searching community and it's potential target community. Given the age of the universe, currently believed to be some 10 to 20 thousand million years, Hawkins 5, and the elapsed lifetime of our evolution, it seems reasonable to consider there to be only a small chance, possibly 1%, that we might establish any communication with another community. With this factor in place the possibility of a dialogue is more remote but might easily be higher depending on other values entered into the equation.

Is there anyone there?

Many of the population would claim that extraterrestrials have been here in the preceding centuries pointing to many references to visitation by 'angels' and 'gods' in ancient texts and a wide variety of artifacts said to bear 'the finger prints of the gods'. These widely heard stories often have little basis in fact and are at best dubious. That many of these anecdotes exist in various sacred texts or are of other religious significance is used by their proponents to add weight to their case. In fact they have not stood the test of scientific scrutiny.

A large portion of the populous believe aliens to be regular visitors to Earth, citing the numerous reported sightings of unidentified flying objects (UFOs) and even abduction of witnesses as their evidence. The popular myth surrounding the UFO enigma is widely believed to be visitation by, mostly, benevolent aliens. There are hundreds of thousands of reported incidents where people have reported what have been labelled by the press as 'flying saucers'; alien craft piloted by intelligent humanoids. However a number of researchers having examined the evidence have reached a contrary conclusion. Haines, et al 4, have shown much evidence to illuminate the psychology of these experiences, the fallibility of witness testimony and some of the many sociological aspects of these phenomena. He has pointed to a considerable consistency amongst the reported features and correlates these findings with deep rooted expectations held by the population. The present writer and a colleague 13, have forwarded other evidence which contradicts the popular interpretation. Rather than 'spacemen' we have proposed a hypothesis to support the likely existence of a previously unrecognised form of atmospheric phenomenon. We have also found other possibilities which show an apparent link with the more exotic experiences such as those examined by Haines. These investigations have been well received by others working in the field and are held to represent a distinct advance in the subject. However, these events, which occur throughout most cultures and historical periods continue to be reported. Despite the protestations of eye witnesses, and the unproven assertions by others, there is no evidence to support the hypothesis that Earth is being visited by extraterrestrial entities.

Addressing the question of extraterrestrial life from our current position it is impossible to answer the question of whether a search for other life will be successful. The best which can be achieved from our limited knowledge and assessments is that the possibility for life elsewhere exists.

It has been argued that life in the universe is abundant, and therefore the possibility of communication with another civilisation is simply a matter of time. Whilst others have an opposing view that Earth is unique and therefore life is unique, that is we are alone in the vastness of the cosmos. There is however an intriguing alternative argument, that is although the number of civilisations may be considerable, we are never the less alone in space by virtue of our lifetime not coinciding with that of others.

Although there have been, and still are a number of observatories contributing to a search for deliberate or unintentional signals arising from an extraterrestrial civilisation, none have been detected. These searches have always been somewhat contentious, and their cost and funding being a major hurdle. In the last few decades the scientific research programs associated with the search for extraterrestrial life have benefited from conventional scientific funding. However, in 1993 the Sky Survey and Targeted Search aspect of NASA's High Resolution Microwave Survey, (HRMS), has had government funding withdrawn in favour of sourcing finance from private funds 17 . This search, now called Project Phoenix, will require around $3 million each year to complete the survey of about a thousand nearby F, G and K group stars. (These groups of stars are the most stable and are thought to be the most likely to harbour a suitable ecosystem in which life might evolve.) In an attempt to limit costs without affecting the integrity of the search, cuts have been imposed on the supporting conferences and collaborative ventures originally proposed.

Might we be alone?

There are prominent opponents to those who argument for there being significant colonies of intelligent species spread through the galaxy. Tipler 11 for example has claimed that extraterrestrial intelligent beings do not exist. He maintains that those who support the possibility of extraterrestrial life are mainly astronomers, and those who have argued against are largely biologists. He has made his camp firmly with the latter. His main thrust is simply, if they did exist, then they would have made themselves known to us, or we would have found evidence of their existence in the form of one or more of their exploratory probes. He proposes that any intelligent community capable of interstellar communications would, as a matter of course, go on to develop a means of interstellar travel. He goes further to claim that an automatic consequence of interstellar travel would be the exploration and/or colonisation of the galaxy. Tipler assumes that a developing technological species will eventually develop a "self replicating universal constructor with intelligence comparable with the human level - such a machine should be developed with a century", (this written in 1979). He goes on, "such a machine combined with present day rocket technology would make it possible to explore and/or colonise the galaxy in less than 300 billion years".

The machine Tipler proposes is the "von Neumann machine" 14 , a space probe sent by an emerging intelligence, equipped with an initial route plan to an inviting location - another planetary body, perhaps an inhabited one. Once there the von Neumann machine would use it's on board equipment to explore the new found environment and report back. From there the plan is that either the probe surveys the heavens for a second staging post, refuels itself and departs to repeat the process over again. Or, once at the first landfall, the von Neumann machine replicates itself, using the local materials to construct a clone (or improved version of itself), and then despatches one of the second generation probes to a further attractive destination. The first might remain at the first port of call and merely act as a production facility to begin a colony of von Neumann machines. The inbuilt instructions for their survival would include the means to prospect for and extract or synthesise whatever materials were needed for the subsequent generation, be that by mining planetary material, extracting the necessary elements from the local solar system or devising alternatives to suit whatever is available. As the fleet of probes explore ever further afield, their own capabilities might develop, but a built in quality control system could maintain the integrity of the original design concept and devise improvements for self implementation.

In this way it is suggested the galaxy is populated with intelligent probes in addition to the purely biological species which evolve. Tipler envisages situations where 'natural' biological species coexist with a peer group of von Neumann machines. This notion, bizarre though it may seem, would appear to be an inherent effect of the introduction of von Neumann machines. Perhaps it is because of this reason that Sagan and Newman 9 are opposed to the concept. Their stated objections include the obvious opposition that; given time these hungry and promiscuous constructions will overrun the entire galaxy and beyond. In fact it seems equally obvious that 'natural' intelligent species will expend a considerable amount of effort in ensuring that von Neumann machines are not permitted to exist. Unless stopped, these inventions would take over. That is their designed purpose. These intelligent entities, as they must become, would be determined and capable of whatever was necessary to guarantee their own survival. The danger is that once established, to attempt to destroy them it might become a destructive spiral with no survivors.

There is at least one alternative explanation of why we have not been contacted. Communication with extraterrestrial intelligence's, by definition, requires a discourse. Some, (for example Deardorff 3), have argued the possible existence of an embargo on contact until emerging civilisations have shown themselves to be capable of peaceful coexistence in the cosmos. This compellingly simple reason might be well founded. At the Green Bank conference the lifetime of our existence was postulated to be rather short. The proliferation of weapons of mass destruction, including the possibility of anarchic states, even individuals obtaining the nuclear potential is as Sagan accurately states "generally regarded as unstable". Even without that threat it seems entirely reasonable that if any existing extraterrestrial intelligence were monitoring our developmental progress they would not intervene until our global intentions became clear.

Deardorff also suggests that an advanced intelligence would not be likely to establish contact in a sudden or inconsiderate manner. Rather that initial contact may be made subtle means, perhaps more akin to a guiding hand or benevolent manipulation. He suggests that before any contact of genuine benevolence a period of acceptance of the concept of their existence would be necessary.

The history of mankind is liberally punctuated with wars and man made catastrophes, the assumption of mediocrity does not include a status which warrants sudden rescue from cataclysmic disaster.

If indeed we are part of the galactic population, then man will need to earn his place in that community. If not, then it might be that our passing goes unnoticed.


It appears certain that Earth is not currently under the direct scrutiny of an extraterrestrial intelligence. A considerable degree of difference currently separates the two schools of opinion; whether life as we know it is a natural and commonplace effect of evolution in the universe, or, if we genuinely are alone in time and space. In time the astronomers and biologists might agree a common scenario. However, it is not the sole domain of the scientist, the ordinary man and woman in the street is entitled to share the controversy, which at present is based as much on personal interpretation and opinion as scientific facts. The best science available to us cannot settle the question, it is impossible to state which side is correct, it is as much personal belief as it is proven knowledge.

It seems inconceivable that we will ever prove ourselves to be alone in the universe. The debate will only be resolved if, and when, the proponents of the former argument announce the most momentous discovery in the history of mankind.

Bibliography and references

1 Bracewell, R, N, 1974, Galactic Club, W H Freeman

2 Brin, Glen D, 1983, The great silence: the controversy

concerning extraterrestrial intelligent life. Q J RAS v24 n3

3 Deardorff, James W, 1986, Possible extraterrestrial strategy

for Earth, Q J RAS v27 n1

4 Haines, Ed, 1979, UFO Phenomena and the Behavioural Scientist,

Scarecrow Press, New York

5 Hawkins, S W, 1988, A brief history of time, Bantam Press

6 IAU Commission 51 - Search for extraterrestrial life. A new IAU

commission, established August 1982, Q J RAS v24 n2 1983

7 Life in the universe, Scientific American Special issue Oct 1994

8 Sagan, Ed, 1973, Communication with extraterrestrial Life, MIT Press

9 Sagan, C, & Newman, W, I, Q J RAS v24 n2, 1983. The solipsist

approach to extraterrestrial intelligence

10 Shklovskii, I S, & Sagan, C, 1966, Intelligent life in the

universe, Holden Day

11 Tipler, Frank J. 1980, Extraterrestrial intelligent beings do

not exist, Q J RAS v21 n3, (also, 1981, also, Additional remarks

on ETI, Q J RAS v22 n3

12 Tipler, Frank J, Q J RAS v22 n2 1981, A brief history of the

extraterrestrial intelligence concept

13 Randles J, & Warrington, P, 1985, Science and the UFOs, Blackwell

14 Neumann, J von, 1966, Theory of self-reproducing automata, Edited

and completed by Burks, A W, University of Illinois Press

15 St Albertus Magnus, De Caelo et Mundo, Lib. I, Tract III, Cap. I;

quoteded in McColley, G, 1936, Annals of Science, v1 n 385

16 Aristotle, De Caelo, v1 5-9

17 Chapman-Reitschi, P A L, 1995, The privatised world of SETI, The

Observatory v115 n1126

18 Encyclopaedia Britannica, 1985, University of Chicago

19 Lovejoy, A O, 1936, The great chain of being, Harvard University Press

20 Plato, Timaeus, v31 n55

21 Thomas Aquinas, Commentarius in Libros Aristotelis, Book 1 chapters 9-21

Copyright © 1995 Peter Warrington

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