work.[81] The problem is the general relation which holds between the
angles of incidence and of refraction when a ray passes from air into
water or from air into glass. He groups a series of the angles with a
close approximation to the truth, but just misses the perception which
would have turned his excellent raw material into the finished product
of science. His brick does not quite fit its place in the building. His
formula _i_ (the angle of incidence) = _nr_ (the angle of refraction)
only fits the case of very small angles for which the sine is
negligible, though it had the deceptive advantage of including reflexion
as one case of refraction. He did not pursue the argument and make his
form completely general. Sin _i_ = _n_ sin _r_ escaped him, though he
had all the trigonometry of Hipparchus behind him, and it was left for
Snell and Descartes to take the simple but crucial step at the beginning
of the seventeenth century.
The case is interesting for more than one reason. It shows us what is a
general form, or law of nature in mathematical shape, and it also
illustrates the progress of science as it advances from the most
abstract conceptions of number and geometry, to more concrete phenomena
such as physics. The formula for refraction which Ptolemy helped to
shape, is geometrical in form. With him, as with the discoverer of the
right angle in a semicircle, the mind was working to find a general
ideal statement under which all similar occurrences might be grouped.
Observation, the collection of similar instances, measurement, are all
involved, and the general statement, law or form, when arrived at, is
found to link up other general truths and is then used as a
starting-point in dealing with similar cases in future. Progress in
science consists in extending this mental process to an ever-increasing
area of human experience. We shall see, as we go on, how in the concrete
sciences the growing complexity and change of detail make such
generalizations more and more difficult. The laws of pure geometry seem
to have more inherent necessity and the observations on which they were
originally founded have passed into the very texture of our minds. But
the work of building up, or, perhaps better, of organizing our
experience remains fundamentally the same. Man is throughout both
perceiving and making that structure of truth which is the framework of
progress.
Ptolemy's work brings us to the edge of the great break which occurred
in the growth of science between the Greek and the modern world. In the
interval, the period known as the Middle Ages, the leading minds in the
leading section of the human race were engaged in another part of the
great task of human improvement. For them the most incumbent task was
that of developing the spiritual consciousness of men for which the
Catholic Church provided an incomparable organization. But the interval
was not entirely blank on the scientific side. Our system of
arithmetical notation, including that invaluable item the cipher, took
shape during the Middle Ages at the hands of the Arabs, who appear to
have derived it in the main from India. Its value to science is an
excellent object-lesson on the importance of the details of form. Had
the Greeks possessed it, who can say how far they might have gone in
their applications of mathematics?
Yet in spite of this drawback the most permanent contribution of the
Greeks to science was in the very sphere of exact measurement where they
would have received the most assistance from a better system of
calculation had they possessed it. They founded and largely constructed
both plane and spherical geometry on the lines which best suit our
practical intelligence. They gave mankind the framework of astronomy by
determining the relative positions of the heavenly bodies, and they
perceived and correctly stated the elementary principles of equilibrium.
At all these points the immortal group of men who adopted the Copernican
theory at the Renascence, began again where the Greeks had left off. But
modern science starts with two capital improvements on the work of the
Greeks. Measurement there had been from the first, and the effort to
find the constant thing in the variable flux; and from the earliest days
of the Ionian sages the scientific mind had been endeavouring to frame
the simplest general hypothesis or form which would contain all the
facts. But the moderns advanced decisively, in method, by experimenting
and verifying their hypotheses, and in subject-matter, by applying their
method to phenomena of movement, which may theoretically include all
facts biological as well as physical. Galileo, the greatest founder of
modern science, perfectly exemplifies both these new departures.
It is, perhaps, the most instructive and encouraging thing in the whole
annals of progress to note how the men of the Renascence were able to
pick up the threads of the Greeks and continue their work. The texture
held good. Leonardo da Vinci, whose birth coincides with the invention
of the printing-press, is the most perfect reproduction in modern times
of the early Greek sophos, the man of universal interests and capacity.
He gave careful and admiring study to Archimedes, the greatest pure man
of science among the Greeks, the one man among them whose works,
including even his letters, have come down to us practically complete. A
little later, at the beginning of the sixteenth century, Copernicus
gained from the Pythagoreans the crude notion of the earth's movement
round a great central fire, and from it he elaborated the theory which
was to revolutionize thought. Another half-century later the works of
Archimedes were translated into Latin and for the first time printed.
They thus became well known before the time of Galileo, who also
carefully studied them. At the beginning of the seventeenth century
Galileo made the capital discoveries which established both the
Copernican theory and the science of dynamics. Galileo's death in 1642
coincides with the birth of Sir Isaac Newton.
Such is the sequence of the most influential names at the turning-point
of modern thought.
Galileo's work, his experiments with falling bodies and the revelations
of his telescope, carried the strategic lines of Greek science across
the frontiers of a New World, and Newton laid down the lines of
permanent occupation and organized the conquest. Organization, the
formation of a network of lines connected as a whole, and giving access
to different parts of the world of experience, is perhaps the best image
of the growth of science in the mind of mankind. It will be seen that it
does not imply any exhaustion of the field, nor any identification of
all knowledge with exact or systematic knowledge. The process is rather
one of gradual penetration, the linking up and extension of the area of
knowledge by well-defined and connected methods of thought. No
all-embracing plan thought out beforehand by the first founders of
science, or any of their successors, can be applied systematically to
the whole range of our experience. It has not been so in the past; still
less does it seem possible in the future. For the most part the
discoverer works on steadily in his own plot, occupying the nearest
places first, and observing here and there that one of his lines runs
into some one else's. Every now and then a greater and more
comprehensive mind appears, able to treat several systems as one whole,
to survey a larger area and extend that empire of the mind which, as
Bacon tells us, is nobler than any other.
Of such conquerors Newton was the greatest we have yet known, because he
brought together into one system more and further-reaching lines of
communication than any one else. He unified the forms of measurement
which had previously been treated as the separate subjects of geometry,
astronomy, and the newly-born science of dynamics. Celestial mechanics
embraces all three, and is a fresh and decisive proof of the commanding
influence of the heavenly bodies on human life and thought. Not by a
horoscope, but by continued and systematic thought, humanity was
unravelling its nature and destiny in the stars as well as in itself.
These are the two approaches to perfect knowledge which are converging
more and more closely in our own time. Newton's work was the longest
step yet taken on the mechanical side, and we must complete our notice
of it by the briefest possible reference to the later workers on the
same line, before turning to the sciences of life which began their more
systematic evolution with the discovery of Harvey, a contemporary of
Newton.
The seventeenth century, with Descartes' application of algebra to
geometry, and Newton's and Leibnitz's invention of the differential and
integral calculus, improved our methods of calculation to such a point
that summary methods of vastly greater comprehensiveness and elasticity
can be applied to any problem of which the elements can be measured. The
mere improvement in the method of describing the same things (cf. e.g. a
geometrical problem as written down by Archimedes with any modern
treatise) was in itself a revolution. But the new calculus went much
farther. It enabled us to represent, in symbols which may be dealt with
arithmetically, any form of regular movement.
As movement is universal, and the most obvious external manifestation of
life itself, the hopes of a mathematical treatment of all phenomena are
indefinitely enlarged, for all fresh laws or forms might conceivably be
expressed as differential equations. So to the vision of a Poincare the
human power of prediction appears to have no assignable theoretical
limit.
The seventeenth century which witnessed this momentous extension of
mathematical methods, also contains the cognate foundation of scientific
physics. Accurate measurement began to be applied to the phenomena of
light and heat, the expansion of gases, the various changes in the forms
of matter apart from life. The eighteenth century which continued this
work, is also and most notably marked by the establishment of a
scientific chemistry. In this again we see a further extension of
accurate measurement: another order of things different in quality began
to be treated by a quantitative analysis. Lavoisier's is the greatest
name. He gave a clear and logical classification of the chemical
elements then known, which served as useful a purpose in that science,
as classificatory systems in botany and zoology have done in those
cases. But the crucial step which established chemistry, a step also due
to Lavoisier, was making the test of weight decisive. 'The balance was
the _ultima ratio_ of his laboratory.' His first principle was that the
total weight of all the products of a chemical process must be exactly
equal to the total weight of the substances used. From this, and rightly
disregarding the supposed weight of heat, he could proceed to the
discovery of the accurate proportions of the elements in all the
compounds he was able to analyse.
Since then the process of mathematical synthesis in science has been
carried many stages further. The exponents of this aspect of scientific
progress, of whom we may take the late M. Henri Poincare as the leading
representative in our generation, are perfectly justified in treating
this gradual mathematical unification of knowledge with pride and
confidence. They have solid achievement on their side. It is through
science of this kind that the idea of universal order has gained its
sway in man's mind. The occasional attacks on scientific method, the
talk one sometimes hears of 'breaking the fetters of Cartesian
mechanics', seem to suggest that the great structure which Galileo,
Newton, and Descartes founded is comparable to the false Aristotelianism
which they destroyed. The suggestion is absurd: its chief excuse is the
desire to defend the autonomy of the sciences of life, about which we
have a word to say later on. But we must first complete our brief
mention of the greatest stages on the mechanical side, of which a full
and vivid account may be found in such a book as M. Poincare's _Science
et Hypothese_.
Early in the nineteenth century a trio of discoverers, a Frenchman, a
German, and an Englishman, established the theory of the conservation of
energy. To the labours of Sadi Carnot, Mayer, and Joule is due our
knowledge of the fact that heat which, as a supposed entity, had
disturbed the physics and chemistry of the earlier centuries, was itself
another form of mechanical energy and could be measured like the rest.
Later in the century another capital step in synthesis was taken by the
foundation of astrophysics, which rests on the identity of the physics
and chemistry of the heavenly bodies with those of the earth.
The known universe thus becomes still more one. Later researches again,
especially those of Maxwell, tend to the identification of light and
heat with electricity, and in the last stage matter as a whole seems to
be swallowed up in motion. It is found that similar equations will
express all kinds of motion; that all are really various forms of the
motion of something which the mind postulates as the thing in motion; we
have in each case to deal with wave-movements of different length. The
broad change, therefore, which has taken place since the mechanics of
Newton is the advance from the consideration of masses to that of
molecules of smaller and smaller size, and the truth of the former is
not thereby invalidated. Newton, Descartes, Fresnel, Carnot, Joule,
Mayer, Faraday, Helmholtz, Maxwell appear as one great succession of
unifiers. All have been engaged in the same work of consolidating
thought at the same time that they extended it. Their conceptions of
force, mass, matter, ether, atom, molecule have provisional validity as
the imagined objective substratum of our experience, and the fact that
we analyse these conceptions still further and sometimes discard them,
does not in any way invalidate the law or general form in which they
have enabled us to sum up our experience and predict the future.
But now we turn to the other side. In spite of the continued progress
noted on the mechanical side, it is true that the predominant scientific
interest changed in the nineteenth century from mechanics to biology,
from matter to life, from Newton to Darwin. Darwin was born in 1809, the
year in which Lamarck, who invented the term biology, published his
_Philosophie Zoologique_. The _Origin of Species_ appeared in 1858 after
the conservation of energy had been established, and the range and
influence of evolutionary biology have grown ever since.
Before anything can be said of the conclusions in this branch of science
one preliminary remark has to be made. From the philosophical point of
view the science of life includes all other, for man is a living animal,
and science is the work of his co-operating mind, one of the functions
of his living activity. What this involves on the philosophical side
does not concern us here, but it is necessary to indicate here the
nature of the contact between the two great divisions of science, the
mechanical and the biological, considered purely as sciences. For,
though we know that our consciousness as a function of life must in some
form come into the science of life, and is, in a sense, above it all, we
are yet able to draw conclusions, apparently of infinite scope, about
the behaviour of all living things around us and including ourselves,
just as we do about a stone or a star. And we are interested in this
chapter in seeing how this drawing of general conclusions keeps growing
with regard to the phenomena of life, just as it has grown with regard
to all other phenomena, and we have to consider what sort of difference
there is between the one class of generalizations and the other.
For those of us who are content to rest their conclusions on the
positively known, who, while not setting any limits to the possible
extension of knowledge, are not prepared to dogmatize about it, it is
still necessary to draw a line. A dualism remains, name and fact alike
abhorrent to the completely logical philosophic mind. On the one hand
the ordinary laws of physical science are constantly extending their
sphere; on the other, the fact of life still remains unexplained by
them, and becomes in itself more and more marvellous as we investigate
it. The general position remains much as Johannes Mueller expressed it
about the middle of the last century, himself sometimes described as the
central figure in the history of modern physiology. 'Though there
appears to be something in the phenomena of living beings which cannot
be explained by ordinary mechanical, physical, or chemical laws, much
may be so explained, and we may without fear push these explanations as
far as we can, so long as we keep to the solid ground of observation and
experiment.' Since this was written the double process has gone on
apace. The chemistry and physics of living matter are being sketched,
and biologists are more and more inclined to study the mechanical
expression of the facts of life. Mr. Bateson, for instance, tells us
that the greatest advance that we can foresee will be made 'when it is
possible to connect the geometrical phenomena of development with the
chemical'. The process of applying physical laws to life follows, it
would seem, the reverse order of their original development. First the
chemistry of organic matter was investigated, then the physical
attraction of their molecules, and now their geometry is in question.
So, says Professor Bateson, the 'geometrical symmetry of living things
is the key to a knowledge of their regularity and the forces which cause
it. In the symmetry of the dividing cell the basis of that resemblance
which we call Heredity is contained'.
But such work as this is still largely speculative and in the future. It
does not solve the secret of life. It does not affect the fact of
consciousness which we are free to conceive, if we will, as the other
side of what we call matter, evolving with it from the most rudimentary
forms into the highest known form in man, or still further into some
super-personal or universal form. This, however, is philosophy or
metaphysics. We are here concerned with the progress of science, in one
of its two great departments, i.e. knowledge about life and all its
known manifestations, which from Aristotle onwards have been subjected
to a scrutiny similar to that which has been given to the physical facts
of the universe and with results in many points similar also. But the
facts, although superficially more familiar, are infinitely more
complicated, and the scrutiny has only commenced in earnest some hundred
years ago. Considering the short space for this concentrated and
systematic study, the results are at least as wonderful as those
achieved by the physicists. Two or three points of suggestive analogy
between the courses of the two great branches of science may here be
mentioned.
We will put first the fundamental question on which, as we have seen, no
final answer has yet been reached: What is life, and is there any
evidence of life arising from the non-living? Now this baffling and
probably unanswerable question--unanswerable, that is, in terms which go
beyond the physical concomitants of life--has played the part in biology
which the alchemists' quest played in chemistry. It led by the way to a
host of positive discoveries. Aristotle, the father of biology, believed
in spontaneous generation. He was puzzled by the case of parasites,
especially in putrefying matter. Even Harvey, who made the first great
definite discovery about the mechanism of the body, agreed with
Aristotle in this error. It was left for the minute and careful
inquirers of the nineteenth century to dispose of the myth. It was only
after centuries of inquiry that the truth was established that life, as
we know it, only arises from life. But the whole course of the inquiry
had illuminated the nature of life and had brought together facts as to
living things of all kinds, plants and animals, great and small, which
show superficially the widest difference. Illumination by unification is
here the note, as clearly as in the mathematical-physical sciences. All
living things are found to be built up from cells and each cell to be an
organism, a being, that is, with certain qualities belonging to it as a
whole, which cannot be predicated of any collection of parts not an
organism. The cell is such an organism, just as the animal is an
organism, and among its qualities as an organism is the power of growth
by assimilating material different from itself. Yet, in spite of this
assimilation and constant change, it grows and decays as one whole and
reproduces its like.
Another point of analogy between the animate and the inanimate sphere is
that the process of study in both has been from the larger to the
smaller elements. The microscope has played at least as decisive a part
as the telescope, and it dates from about the same time, at the
beginning of the seventeenth century. Since then it has penetrated
farther and farther into the infinitesimal elements of life and matter,
and in each case there seems to be no assignable limit to our analysis.
The cell is broken up into physiological units to which almost every
investigator gives a new name. We are now confronted by the fascinating
theory of Arrhenius of an infinite universe filled with vital spores,
wafted about by radio-activity, and beginning their upward course of
evolution wherever they find a kindly soil on which to rest. To such a
vision the hopes and fears of mortal existence, catastrophes of nature
or of society, even the decay of man, seem transient and trivial, and
the infinities embrace.
A third point, perhaps the most important in the comparison, is the way
by which the order of science has entered into our notions of life,
through a great theory, the theory of evolution or the doctrine of
descent. In this we find a solid basis for the co-ordination of facts:
it was the rise of this theory in the hands of one thinker of
unconquerable patience and love of truth which has put the study of
biology in the pre-eminent position which it now holds. But it is
necessary to consider the evolution theory as something both older and
wider than Darwin's presentation of it. Darwin's work was to suggest a
_vera causa_ for a process which earlier philosophers had imagined
almost from the beginning of abstract thought. He observed and collected
a multitude of facts which made his explanations of the change of
species--within their limits--as convincing as they are plausible. But
the idea that species change, by slow and regular steps, was an old one,
and his particular explanations, natural and sexual selection, are seen
on further reflection to have only a limited scope.