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CHAPTER IX
HELLENISTIC SCIENCE AND MATHEMATICS
I.
MEDICINE AND SURGERY
IN the fifth century Greek medicine reached its
highest level in the work of Hippocrates and his followers. It was their
achievement to see in disease a natural evil to be combated by natural means.
They were influenced by the philosophy of their day, though sometimes they
resisted its claims, but in time the study of medicine assumed a rather
different character. The science of the third century BC developed under
two powerful influences, the genius of Aristotle and the endowment of
research. Aristotle had done much to separate science from philosophy
by his differentiation of the various branches of knowledge and
by restricting speculation to those subjects which admit it
best. Natural history in particular was based upon a large collection
of observed facts, with greater attention to detail than
previous thinkers had thought necessary. Most in fact of the
conditions required for the development of science were recognized,
and efforts were made to secure them, although experiment had by no
means attained its rightful position. Scope for work on the lines laid
down by Aristotle was provided by the endowment of research at Alexandria
and other centres. While mathematicians and astronomers were making
brilliant conquests of knowledge and speculation, the laborious diligence
of students of medicine had the opportunity of doing its special task in
the surroundings provided by the Ptolemies and similar patrons of
learning. There are no startling discoveries, no illuminations of genius.
On the other hand, there is a distinct advance in knowledge of detail,
a knowledge made possible by careful and patient observation. Human
anatomy in particular was studied with success, and the work done at
Alexandria compares very favourably with the crude information and
fanciful guess-work that disfigures so many of the treatises in the
Hippocratic Corpus.
In the medical school at Alexandria worked Herophilus
the anatomist and Erasistratus the physiologist. We know little
about their lives, except that they came from Asia Minor to work at Alexandria
during the first half of the third century, Erasistratus being slightly the
younger. Their works are entirely lost, but we can gather a great deal of
information from references in Galen, Soranus and Celsus. Modern scholars have been particularly successful in
reconstructing an account of the researches of Herophilus. The fact that
these two medical men came from Asia Minor suggests that Alexandrian
medicine may have been subjected to Eastern influences. Research, however,
cannot trace any such influence, nor yet any direct connection with native Egyptian
science, although we know this to have been by no means contemptible.
Herophilus was a careful student of Hippocrates, and
wrote commentaries on two of the Hippocratic treatises, maintaining
the humoral pathology in opposition to Erasistratus. He paid
great attention to pulsation, following in the steps of his teacher Praxagoras, who was the first physician to insist on the
importance of the pulse, although pulsation was known at least a century
before. He used drugs far more than the Hippocratics,
regarding them as invaluable helps in the cure of diseases. His
anatomical researches centred around the brain, nerves, liver, lungs, and
the organs of generation. The brain he regarded as the seat of
intelligence, connecting it with the nervous system, of which he was the
first to form any clear conception. It is interesting to observe that
Herophilus must have dissected animals, because he described the rete
mirabile, which exists at the base of the brain of animals, but not in
human beings. He distinguished ‘sensory’ from ‘motor’ nerves, and
contributed much to the knowledge of the ventricles of the brain. He
distinguished also the cerebrum from the cerebellum and paid
great attention to the cavity in the fourth ventricle. The other
anatomical researches of Herophilus are less striking, but he named the
duodenum and described at some length the uterus. Tradition asserts that
he conducted postmortem examinations, and certainly his work could not
have been done without dissecting the human subject. Recent research shows that he used
an ingenious instrument to measure the rate of pulses. Here we have
one of the earliest efforts—perhaps the first—in the direction of that
exact measurement which is the very soul of modern science.
Erasistratus rejected the humoral pathology,
substituting for it a doctrine of air or pneuma, which he considered to
be a vital factor in the physiological processes. It has been well said
that this attention paid to air, which goes back at least to
Alcmaeon , finally resulted in the discovery of oxygen and of the
part it plays in supporting life. Erasistratus, however, appears to have
regarded the action of pneuma as purely mechanical. He improved on
the work of Herophilus on the heart and the brain, and distinguished yet
more clearly the sensory from the motor nerves. It has even been
maintained that the discovery is due to him rather than to his
contemporary. In his practice Erasistratus rejected bleeding, substituting
for it a low diet, and he used medicines only of the mildest kind, in the
latter respect reverting to the Hippocratic tradition. He invented a
catheter, but was probably not the first to do so.
Improved anatomy naturally resulted in improved
surgery, and the great glory of the Alexandrian school was the invention
of new surgical instruments with increased skill in the use of
them. Both Herophilus and Erasistratus have been accused of practising vivisection
on human beings. The evidence comes from certain passages in Celsus and
Tertullian1, and is therefore by no means contemptible. The Ptolemies are
said to have permitted the vivisection of condemned criminals, and no
ancient authority doubts the truth of the assertion. Celsus felt that
there were ethical arguments on both sides of the question, although he
personally regarded the practice as barbarous. It is interesting to
observe that a similar charge was brought against certain princes and
surgeons of the sixteenth century, but the best historical opinion rejects
the stories as based on untrustworthy evidence. Recent historians
of ancient surgery do not believe that the Alexandrian surgeons
had recourse to human vivisection, and they regard the accusation
as a lie manufactured by those who were opposed to dissection of
any kind. Even if this view be adopted, it is clear that surgeons
against whom such charges were made must have attached great importance to
ocular examination of the internal organs; they were, in fact, the real
fathers of modern anatomy.
In spite of their comparative freedom from
speculation, Galen classes the two great Alexandrian anatomists with what he
names the ‘Rational,’ or, as it is more commonly called, the
Dogmatic school of medicine. The essence of Dogmatism is that it values
the study of causes of disease, and therefore calls in to aid
medicine the kindred sciences of anatomy and physiology. A more
complete reaction from speculation came about 280 BC, when two
pupils of Herophilus founded at Alexandria the Empiric school. Empiricism
discarded anatomy and physiology, and held that medicine was concerned
only with the cure of diseases and not with their cause. The whole duty of
the physician was to discover what treatment will banish any particular set of
symptoms. He was to be guided by personal observation, by instruction from
others and by analogy. Empiricism probably did much good by opposing
the speculative tendency, always a weakness in Greek medicine, and by
investigating the usefulness of many new drugs and remedies.
One Empiric deserves a special notice. Heracleides of
Tarentum wrote an apparently excellent work on materia medica, containing, as he said, nothing that he had not observed himself.
He pointed out the value of opium, both in poultices and in medicines. By
calling attention to the proper use of anodynes he prepared the way for
the discovery of anaesthetics. He was also an excellent surgeon, as well
as a commentator on several Hippocratic treatises. He earns from Galen the
commendation that he never preferred his party to the truth. Heracleides was
thought until quite recently to belong to the last thirty years of the
third century. Recent research working on new evidence makes it
extremely likely that he did not flourish until the early part of the
first century BC.
It is noteworthy that the history of medicine, and in
particular the critical study of the Hippocratic writings, continued
to flourish during the third century. Menon’s Iatrica is rather earlier, but a long series of writers, whose works have
perished, show that the new methods of research adopted at Alexandria
did not blind students to the importance of keeping in touch with
the work of previous investigators. These writers include
Herophilus, his pupil Bacchius, Philinus and Glaucias. Some of this interest was possibly
antiquarian, but there is no reason to doubt that there was felt a genuine
appreciation of the continuity of science, and of the truth that the past
cannot be forgotten without serious danger to the present and the future.
In the Hippocratic Corpus are two short works
which may perhaps belong to the third century BC. These are Precepts and Decorum, both dealing chiefly with what may be called
medical etiquette. Their date is indeed a vexed question, but
Precepts contains passages very similar in language to remarks
attributed by Diogenes Laertius to Epicurus, while the ideal physician
of Decorum bears strong resemblance to the Stoic Wise Man.
The Greek of both works shows signs of a comparatively late date, and
it is hard to resist the conclusion that they are later than 300 BC.
During the fifth century philosophy made a strong
effort to thrust its method of postulates upon the science of medicine,
and the Hippocratic work called Ancient Medicine attacks those who
would assert, for instance, that all diseases are caused by heat, and would
accordingly base treatment upon this assumption. So this attempt of philosophy
was resisted, because it sought to interfere with the true function of the
physician’s art. But when philosophers concentrated their attention on
moral questions, as happened in the third century, medicine did not try to
resist it, because an educated and enlightened morality can do nothing
but good to medical practice. In insisting upon the moral side of
a physician’s work these later philosophers contributed much to
the building up of a sound medical etiquette. This etiquette aimed
at raising the doctor’s ethical standard rather than at defending
the business interests of the profession.
Two short sentences, one from the sixth chapter of
Precepts, the other from the fifth chapter of Decorum, well sum up
the influence of the later ethical philosophy upon medicine:
Where love of man is, there also is love of the Art.
A physician who is a philosopher is a god indeed.
II.
BIOLOGY AND
BOTANY
Chemistry was to the Greeks rather a collection of
trade secrets than a body of knowledge, ‘not a science, but a mystery.’ It was
to be otherwise with biology and botany. A people that lived so much in
the open air and by the sea, and that was naturally so observant and curious,
had not failed to notice the ways of birds, beasts, fishes and all manner
of plants. Then came Aristotle, who brought system and method, building up
and classifying a great body of observations in a series of treatises of
which the crown is the de generatione animalium. Theophrastus, his pupil, whose
interests ranged from the arts of government to the arts of the
stone-cutter, did for plants, in his sober way, what his master’s greater
genius had done for animals. We do not know exactly when Theophrastus
wrote his botanical works, The History of Plants and The Causes
of Plants, but, as a garden was his hobby throughout his life, it is
not unreasonable to suppose that his researches continued well into the
third century BC. Without the speculative genius of his master, he was
both by nature and by education fully competent to apply to plant-life the
Aristotelian method. It is perhaps fortunate that Theophrastus was more of
a scientist than a philosopher, for it is his patient adherence to
observed fact that makes his treatises the best specimens of
ancient biological study.
Before Theophrastus, plants were studied chiefly for
their use, and his chief claim to distinction is that he raised botany to
the rank of a pure science. The conception may have been due
to Aristotle, but the working out of the idea was left to his
pupil. The number of facts collected by Theophrastus is very great,
and points to his having used, not only his own observations taken
in the garden of the Lyceum, but also data supplied by friends
in many parts of the world. Some five hundred different plants are mentioned,
and the knowledge displayed is often astonishing, when we remember that as
yet the microscope was not and chemistry was in its infancy. The classification
of plants is of course unsatisfactory, although the flowering plants are
distinguished from the flowerless. The organs of plants are carefully
described: root, stem, branch, leaf, flower, seed and fruit being
distinguished with notable accuracy if the means of observation at his
disposal are taken into account. Nomenclature receives great attention. The
non-technical terms karpos and perikarpion are used in a technical sense, and
the zoological word metra is applied
to describe the core of any stem. Authorities agree in
praising Theophrastus for his acute observation of the phenomena
of germination, and also for his clear conception of the
distribution of plants as dependent on climate and soil.
That dates are fertilized was known to the Greeks from
early times, and Theophrastus describes the process at some
length, comparing it to the caprification of figs. The true sexual nature
of plants, however, was not yet known. Species, plants and
flowers have sex ascribed to them, but generally in a metaphorical
sense, ‘male’ being equivalent to ‘barren’ and ‘female’ to ‘fertile.’
Dr E. L. Greene, in a thorough examination of Theophrastean botany, draws up a list of seventeen items
representing original contributions of Theophrastus to botanical
discovery. We cannot be certain that in all cases the originality is made
out, but the mere claim proves that in many respects Theophrastus was
a pioneer. None, however, of the seventeen items, nor yet the sum of
them, can be compared with the outstanding merit of having set forth a
scheme of botanical science, free from utilitarian bias and marking out
various departments for the guidance of all later investigators.
III.
MATHEMATICS CREATED A SCIENCE: THALES AND THE
PYTHAGOREANS
The contributions to philosophy of men such as
Pythagoras and Eudoxus have been described in earlier volumes of
this work, and allusion has been made to their attainments
in mathematics and science. In the history of Greek mathematics three
periods may be distinguished. The first includes the beginnings of the
science and the working out of the elements by Pythagoras and
the Pythagoreans. The second begins with the mathematicians, mostly
outside the Pythagorean school, who first carried geometry beyond the elements
by attacking. the famous problems of duplicating the cube, trisecting any
angle and squaring the circle, which do not yield to elementary methods,
that is, those of the straight line and circle. The third period is
associated with Alexandria and includes the age of Euclid ( 300 BC or
a little earlier), who taught and founded a school at Alexandria, and
his successors, Archimedes and Apollonius of Perga,
both of whom studied there. The fact that the greatest monuments
of Greek mathematics are the work of these three men makes
the present a proper occasion for a more connected survey.
It was through contact with the more ancient
civilizations of Egypt and Babylonia that the Ionian Greeks became
acquainted with certain elementary facts in geometry and astronomy.
Egyptian geometry was mensuration, arising out of the requirements of land
surveying and architecture. Chaldean astronomy included a wonderful amount
of knowledge accumulated as the result of long-continued observations,
coupled with measurement of angles, calculation of times, etc. Working on
these things, the Greek genius, by an inspiration unique in the history of
the world, first conceived the idea of a science. Tradition credits Thales
with the first proofs of a few elementary theorems in geometry, e.g. that the base angles in an isosceles triangle are equal. What was
wonderful was that the idea of proving such things should occur to any one
at all; as Kant once wrote, ‘ A light broke on the first man who
demonstrated the property of the isosceles triangle, whether his name was
Thales or what you will ’; ‘ from that time the safe way of a science
was struck and traced out for all time.’ Thales also reduced
theory to practice by calculating (1) the height of a pyramid, (2) the
distance of a ship from the shore. In astronomy he predicted the
solar eclipse of May 585 BC.
After Thales come Pythagoras and the Pythagoreans. The
Pythagoreans, says Aristotle, devoted themselves to mathematics and were
the first to advance that science as a study pursued for its own sake1.
They made geometry a part of liberal education: their quadrivium comprised arithmetic, geometry, sphaeric (astronomy)
and music. By arithmetic in this classification is meant, not the
arithmetic of daily life, but the theory of numbers in themselves. We have
seen that Pythagoras discovered that the musical intervals correspond to
certain arithmetical ratios between lengths of string at the same tension, 2 :
1 giving the octave, 3 : 2 the fifth and 4 : 3 the fourth. These
ratios are the same as those of 12 to 6, 8, 9 respectively, and 6:8
= 9:12, so that this proportion shows all three intervals.
The principle of proportion so established became a uniform
principle for all science and notably for medicine.
An easy transition from arithmetic to geometry, from
numbers to geometrical magnitudes, was through figured numbers, triangles,
squares, etc. marked out by dots. This revealed a law of formation. Three
dots were placed in contiguity to one dot so as to form a square, five
dots round two sides of that square gave the next square, and so on,
showing that the sum of any number of terms of the series of odd numbers
beginning with 1 is a square number; to add any odd number to the sum of
all the preceding odd numbers (including 1) made one square into the next
larger square; hence the odd numbers were called gnomons. If
the gnomon (odd number) so added is itself a square, we have
two square numbers the sum of which is also a square; and from this
is easily deduced the general formula (attributed to Pythagoras)
for finding three numbers the squares of two of which are
together equal to the square of the third. Any triangle with its sides in
the ratio of three such numbers is right-angled; hence the rule
is connected with the theorem of the square on the hypotenuse,
the proof of which Greek tradition uniformly ascribes to
Pythagoras. The comparison, again, of right-angled triangles having their
sides in the ratio of integral numbers with other right-angled
triangles led to the discovery of the irrational or incommensurable.
Not only did the Pythagoreans discover that the ratio of the hypotenuse of
an isosceles right-angled triangle to one of its other sides or, in other
words, of the diagonal of a square to its side, cannot be expressed as a
ratio between integral numbers, but they discovered a method of finding a
series of ratios between numbers approximating more and more closely, in
fact as nearly as we please, to the value of this ratio, which we write as
√2.
The Pythagoreans applied their theory of proportion to
geometry as well as arithmetic, and to them also is attributed the discovery of
the general method in geometry known as the ‘application of areas.’ The
latter in its most complete form is the geometrical equivalent of the
solution of any quadratic equation in algebra so far as it has real roots;
and this method, with that of proportions, constituted the main resource
of Greek geometry for the solution of problems. Another achievement was a
proof, by means of the properties of parallels, that the three angles of
any triangle are together equal to two right angles; the
Pythagoreans could also construct a regular pentagon, and they had
some knowledge of three of the five regular solids, the tetrahedron,
the cube and the dodecahedron.
Speaking generally, the Pythagorean geometry seems to
have covered the bulk of the subject-matter of Euclid’s Books 1, 2, 4, 6
(and probably 3). The Pythagorean theory of proportion however, being
numerical, only applied to commensurable magnitudes ; hence, when the
irrational was discovered, doubt was thrown on such proofs as depended on
proportions, and there are signs that it was considered necessary in
various cases to substitute other proofs not so depending.
In astronomy progress was no less rapid. Pythagoras
himself is said to have been the first to hold that the earth is spherical
in shape. He realised that the sun, moon and planets have a motion of
their own in a sense opposite to that of the daily rotation (this he may
have learnt from the Babylonians). Later Pythagoreans actually deposed the
earth from its place in the centre and made the earth, like the sun, moon
and planets, revolve in a circle about the ‘central fire’; this first step
towards the Copernican hypothesis is alternatively attributed to Philolaus and one Hicetas of
Syracuse. Nor were the Pythagoreans alone in this field; it was
Anaxagoras who first declared that the moon receives its light from the
sun, and Oenopides of Chios first discovered the
obliquity of the ecliptic.
IV.
SECOND STAGE: HIGHER PROBLEMS. THE IRRATIONAL AND
INFINITESIMALS
Hippocrates of Chios (middle of fifth century) belongs
to the second period above distinguished. Having lost his property, he was
reduced to earning a livelihood by teaching. The first known writer of
‘Elements,’ he also grappled with two of the higher problems. He is said
to have reduced the duplication of the cube to the equivalent problem of
finding two mean proportionals in continued
proportion between two given straight lines; it was in this form that the
problem was thereafter solved. He tried also to square the circle. He
first proved, it is said, that the areas of circles are as the squares on
their diameters; then, with great ingenuity, he squared certain figures
bounded by arcs of circles and called lunes on account of their shape,
hoping thereby to pass to the squaring of the circle. He squared three
different lunes and also the sum of a circle and a certain lune; but he
failed to square the circle itself because that particular lune was not
one of the three which he had squared.
Democritus, the forerunner of modern atomic theory,
was in the front rank of mathematicians. He was the first to discover
that the volume of a cone or a pyramid is one-third of that of
the cylinder or prism respectively which has the same base and
height. He wrote on ‘irrational straight lines and solids (atoms)’ and
on ‘contact with a circle and a sphere’; these titles, with some remarks
of his about contiguous sections of cones, suggest some inkling of
infinitesimals.
Hippias of Elis, the sophist, a man of the most varied
accomplishments, taught (among other things) geometry, astronomy and music. His
theoretical trisection of any angle was by means of a higher curve generated by
the concurrent motions (one a translation and one a rotation) of
two straight lines; the curve was afterwards called tetragonizousa (quadratrix)
because Dinostratus and Nicomedes used it later
to square the circle.
Archytas, the friend of Plato, gave the first known
solution of the problem of finding the two mean proportionals;
he used a striking construction in three dimensions determining a
certain point as the intersection of three surfaces, (1) a right cone, (2)
a right cylinder, (3) an anchor-ring or tore with inner diameter nil.
Theodorus of Cyrene, Plato’s teacher in mathematics,
is associated with Theaetetus in the dialogue of that name as having contributed
to the theory of the irrational by proving that, no less than √2,
the surds √3> √5--√17 are all incommensurable
with unity. Theodorus went no farther, but Theaetetus generalized the
proof in a theorem afterwards reproduced by Euclid in his Book 10.
Theaetetus further distinguished two kinds of compound irrational, the
‘binomial’ and the ‘apotome,’ as well as the irrational straight line
which Euclid calls ‘medial,’ and proved propositions about them. He also
wrote the first theoretical treatise on the five regular solids; one
scholium says that he was the first to discover two of the five, the
octahedron and the icosahedron. We may assume therefore that for the substance
of his Books 10 and 13 Euclid was greatly indebted to Theaetetus.
Though Zeno of Elea was apparently not a
mathematician, it is certain that the statement of his famous paradoxes
profoundly affected the later course of Greek geometry. The first two,
the Dichotomy and the Achilles, proceeded on the
mathematicians’ hypothesis of the divisibility of magnitudes ad infinitum,
and argued that, on this hypothesis, motion can never begin, and
that, even assuming that it can, a swifter motion cannot overtake
a slower; the last two, the Arrow and the Stadium,
proceeding on the opposite assumption that continuous magnitudes (lengths
and times) are made up of indivisible elements, argued the impossibility
of motion on this hypothesis. But Democritus stated a similar dilemma with
no less force. Given a cone with a circular base, if we take the very next
parallel circular section, ‘what,’ said Democritus, ‘ are we to say of the
two circles ? Are they equal or unequal? If unequal, they will make the
cone irregular, with indentations, like steps...; if they are equal, the
sections will be equal, and the cone will appear to have the property of
the cylinder, as being made up of equal, not unequal, circles, which is
very absurd.’
Zeno’s arguments were thought to be fatal to the first
attempts to square the circle such as that of Antiphon, which was nevertheless
on right lines. Antiphon’s idea was to take an equilateral triangle or a
square inscribed in a circle; then, by bisecting the arcs and drawing
chords, to double the number of sides in the inscribed figure, and to
continue this process until (as he said) the area of the circle was used
up by reason of the sides of the polygon becoming so small as to coincide
with the circumference of the circle. No, Zeno would say, this could never
happen because the process would never end. The difficulty was avoided by
what is known as the ‘ method of exhaustion ’ in the form given to it
by its discoverer, Eudoxus of Cnidus (408-353 BC); for this
method only requires that it shall be proved, e.g. in the case of the
circle and the inscribed polygons, that, by continuing the construction
of Antiphon, we can make the inscribed polygon approach equality with
the circle as nearly as we please.
The ‘method of exhaustion’ became the one classical
method of proving propositions about the content of figures bounded
by curved lines or surfaces, although, as we learn from
Archimedes’ Method the results to be proved might be divined in other
less rigorous ways. Eudoxus himself used the method in proving, for
the first time, the propositions discovered by Democritus about the
volumes of the cone and the pyramid.
By another epoch-making discovery Eudoxus established
geometry once more on a secure basis after the shock which it received when the
discovery of the irrational showed the old arithmetical theory of proportion to
be inadequate. This was the great theory of proportion applicable to all
magnitudes commensurable or incommensurable which is expounded in Euclid’s
Book v; and the famous definition on which the theory is based still holds
the field.
In astronomy Eudoxus is famous for his hypothesis of
concentric spheres devised to account for the apparent irregularities in the
motions of the sun, moon and planets. This was, no doubt, his answer to
Plato, who had set to all earnest students the problem of showing by what
combinations of uniform and ordered movements the apparent motions of the
heavenly bodies can be accounted for. Eudoxus’ solution was
purely theoretical; he imagined, for the sun and moon, a system of three spheres,
and for each of the planets a set of four spheres, concentric with the
earth, rotating about different axes, and one within the other. The poles
about which each of the inner spheres rotates are fixed on the next
enclosing sphere. In the case of the planets the effect of the two
innermost rotations is to make the planet (fixed on the equator of the
innermost sphere) describe, on the sphere of which the zodiac is a great
circle, a curve called the hippopede (horse-fetter),
an elongated figure of eight, which lies along and is bisected by the
zodiac circle. This marvel of geometrical ingenuity itself stamps Eudoxus as a
geometer of the very first rank.
The same challenge by Plato was also, in all
probability, the incentive to the important forward step in astronomy taken
by Heracleides of Pontus (c. 388-315 BC), who declared that the earth
rotates on its own axis once in 24 hours, and that Venus and Mercury
revolve about the sun like satellites.
The problem of doubling the cube, or finding the two
mean proportionals, continued to attract attention.
Eudoxus himself is said to have solved it by means of certain curves the
nature of which is not known. But his pupil Menaechmus is credited
with solutions by means of the intersection (1) of two parabolas, and (2)
of a parabola and a rectangular hyperbola; it is inferred that Menaechmus
discovered the three conic sections, as is also implied by the line in
Eratosthenes’ genuine epigram about the problem, which bids us ‘not to cut
the cone in the triads of Menaechmus.’ Menaechmus, we are told, also
contributed to make the whole of geometry more perfect. The story about
the king who wanted a short cut to geometry, told of Euclid and Ptolemy, is
also related of Menaechmus and Alexander in this form: ‘O king,’ said
Menaechmus, ‘for travelling about the country there are royal roads and
roads for common citizens; but in geometry there is one road for all.’
Aristotle was no doubt less of a mathematician than
Plato, but both contributed to the elucidation of the first principles of
mathematics. Plato discussed the hypotheses of mathematics and
gave certain definitions, e.g. of odd and even, ‘figure,’ and
‘straight line’; ‘points’ he apparently refused to recognize as a
separate genus, regarding a point as the ‘beginning of a line’ or ‘an
indivisible line,’ a view which Aristotle controverts. Aristotle’s views
on the objects of mathematics have already been mentioned. Aristotle also lays
down very clearly the distinctions between axioms (common to all sciences) and
definitions, hypotheses and postulates (which are different for
different sciences, since they relate to the subject-matter of the particular science).
He is clear, for instance, that a definition says nothing about the
existence or non-existence of the thing defined; its existence has to be
proved (save in the case of the unit and magnitude, which alone must be assumed
to exist). A postulate, he says, is something which, e.g., the
geometer assumes, for reasons known to himself, without demonstration
(though properly a subject for demonstration), and without any assent on
the part of the learner, or even against his opinion rather than
otherwise.
The Peripatetic school, in the person of Eudemus of Rhodes, produced the first History of
Geometry (in four Books) and the first History of Astronomy; a
few citations from these works are made by various commentators and other
writers, but the loss of the rest can never be sufficiently deplored.
V.
ALEXANDRIA: EUCLID, ARISTARCHUS,
ARCHIMEDES, APOLLONIUS OF PERGA
After the death of Alexander, Athens remained a centre
of study, especially in philosophy; but the headquarters of science and
most forms of literature was transferred to Alexandria as the result of
the enlightened patronage of the first Ptolemies. Ptolemy I Soter was himself a
historian, and he tried to attract philosophers and writers to Alexandria.
Unsuccessful in his invitations to Theophrastus and Stilpo,
he received Demetrius of Phalerum at his court and appointed Strato
of Lampsacus, the successor of Theophrastus in the Peripatetic school, as
well as Philetas of Cos, poet and grammarian, to instruct his son Philadelphus.
In his reign (305— 283 BC) Euclid studied and taught mathematics, and the
great Herophilus anatomy, physiology and medicine, at
Alexandria. Ptolemy II Philadelphus seems to have been especially fond
of natural science and natural history, while the third Ptolemy, Euergetes
I (246—221 BC), seems to have been interested in mathematics.
Strato, the most original thinker of the Peripatetic
school, was not only tutor to Ptolemy Philadelphus; he also taught Aristarchus
of Samos. His greatest work was in natural philosophy and earned him the
title of ‘the physicist.’ By his theory of the vacuum in particular he
seems to have greatly influenced two branches of science, medicine
(through Erasistratus) and pneumatics (through Ctesibius).
According to Strato there is in nature no such thing as continuous void,
but substances, including air and gases, are made up of molecules, and
vacuum exists in the tiny interstices which everywhere separate the
molecules, and there only. This is why air is compressible; and this is
why light is not only reflected from water but shines through it on the
bottom of a vessel. If there were no interstices of vacuum between
the molecules of water, and light had to force its way through
the water in a vessel full of it, the water would overflow; as it is,
the sun’s light is partly reflected from the molecules of the water
and partly passes through it by the path provided by the
interstices. The same constitution of things explains their expansion by
heat. Strato discussed the attraction exercised by a magnet, and
there is some reason to believe that it was he who first conceived
the idea of a connexion between this phenomenon and the
shock communicated by the ‘electric fish,’ the narke.
Two mathematicians claim notice before we come to
Euclid, though nearly contemporary with him, Autolycus of Pitane and Aristaeus ‘the elder.’ Autolycus had as a pupil, at Pitane and Sardes, Arcesilas of Pitane (c. 315-241 BC), who later founded the
New Academy. Autolycus’ treatises On the Moving Sphere and On
Risings and Settings are the only works of any Greek mathematician
earlier than Euclid which have come down to us entire. The work On the
Moving Sphere treats of the geometry of the sphere with special
reference to the heavenly sphere and the circles on it which are used in
astronomy, and is a tract of the same sort as Euclid’s Phaenomena,
though in form less purely astronomical. Autolycus’ work preceded
Euclid’s, for Euclid makes use of propositions contained in it; further,
whereas Autolycus speaks of the horizon as ‘the circle which defines (horizon)
the visible and the invisible portions of the sphere,’ Euclid uses for the
first time the single word horizon. The form of Autolycus’ propositions is
like that which Euclid’s practice has made classical, showing that the
Euclidean type of proposition with its formal divisions was not invented
by him but had already taken shape before his time.
Aristaeus ‘the elder’ is mentioned by Pappus as the
author of a work in five Books entitled Solid Loci, now lost but extant
in Pappus’time. ‘Solid loci’ meant loci which
were conic sections, and were so called to distinguish them from plane
loci which were straight lines and circles only. Euclid apparently used
this work in writing his own Conics; but, as the Conics is also
lost, we cannot judge of the extent of his obligation. A certain Aristaeus
(probably the same person) wrote a Comparison of the Five Figures (the regular solids) in which he proved, among other things, that ‘
the same circle circumscribes both the pentagon of the dodecahedron and
the triangle of the icosahedron inscribed in one and the same sphere.’ As
this interesting theorem is not in Euclid, it has been suggested that
Aristaeus’ work was not before Euclid when he was compiling Book 13, but
was written later.
Next to nothing is known of the life and personality
of Euclid, though the two stories of him are worthy of their subject;
these relate how he told Ptolemy that ‘there is no royal road to
geometry’, and how, when a pupil, after the first proposition, asked him
‘What shall I get by learning these things?’, Euclid called the slave and
said ‘Give him threepence, since he must needs make money by what he
learns.’
Proclus says that he was not much younger than
Hermotimus of Colophon (who continued the work of Eudoxus and Menaechmus) and
Philippus of Medma or Opus (Plato’s pupil), and that
he lived under the first Ptolemy (305-283 BC) and before Eratosthenes and
Archimedes. He taught and founded a school at Alexandria, where, in his
turn, Apollonius of Perga studied under his
successors. Euclid has, at least in this country, always been so
identified with his great work, the Elements, that there are
probably many persons to whom it has not occurred that Euclid was a man
and not a book. Greek writers after Archimedes generally spoke of him as
‘the writer of the Elements’ (Stoicheiotes)
without using his name, while the Arabs tried to make out that,
etymologically, Uclides or Icludes meant the ‘key of geometry.’ No book in the world except the Bible has had
such a reign. Accepted from the first as the authoritative treatise
on the subject, it superseded all other compilations of
Elements. Apollonius, it is true, discussed some of the first principles,
e.g. certain definitions, and suggested some alternative
constructions for problems, but his criticisms and suggestions seem to
have fallen flat. The famous fifth Postulate on which Euclid bases
the theory of parallels was early recognized as a crux;
Posidonius sought to evade the difficulty by a different definition of
parallels based on equidistance, and Geminus after him tried to prove
the Postulate. Elaborate commentaries were written by two
able mathematicians, Pappus and Heron of Alexandria, as well as
by Porphyry, Proclus and Simplicius. Theon of Alexandria
(fourth century A.D.) produced a new edition with alterations and
additions designed to make the language more uniform and the proofs easier
to follow. Boetius is said to have translated the Elements into
Latin. The book passed to Arabia in the eighth century. The first
mediaeval translations, by Athelhard of Bath and Gherard of
Cremona (twelfth century), were made from the Arabic, as was also the
translation by Campano (thirteenth century), the first to be put into
print (Erhard Ratdolt, Venice, 1482).
As Proclus says, what is most to be admired in the
book is the selection made by Euclid of the things (definitions, postulates
and axioms) to be taken for granted and of the particular
propositions which deserve to be called ‘elements’ as being the most
fundamental and of the widest application. The selection and arrangement, as
well as many of the proofs, are Euclid’s own; for the content of the
treatise he laid under contribution every source open to him. We can say,
for instance, generally that in Books 1, 2, 4, 6 and probably 3 he is
giving the substance of the Pythagorean geometry, with the contributions
of such men as Oenopides and Hippocrates of
Chios, while, from hints regarding current proofs of certain propositions
found in Aristotle and elsewhere, we can judge of the definite innovations
which Euclid himself made in methods of proof and otherwise, e.g. in the formulation
of Postulate 5. Books 7-9, on the elementary theory of numbers, including
the numerical theory of proportion, must also be largely Pythagorean. Book
5, containing the theory of proportion applicable to commensurable and
incommensurable magnitudes alike, is all due in substance to Eudoxus,
though the form and sequence are probably Euclid’s. Book x, on
irrational straight lines, is again in substance largely due to
Theaetetus, as is Book 13 on the five regular solids. Book 12 includes
the propositions about the solid content of pyramids and cones
which we know to have been first proved by Eudoxus, and these must be
connected with the other propositions in which the method of exhaustion is
used, namely the theorems that circles are to one another as the squares,
and spheres are to one another as the cubes, on their diameters. But here,
as everywhere, we see the master-hand in the formulation of the
propositions and the logical development of the subject.
Euclid wrote other works not confined to geometry but
covering all branches of mathematics as then known. Those which are extant
in Greek include:
(1) The Data, containing propositions of the
same purport as some in the Elements but in a form more directly
serviceable in problems of construction;
(2) The Optics, a kind of elementary treatise
on perspective;
(3) The Phaenomena, containing propositions in
spherical astronomy;
(4) A Sectio Canonis,
a book on the elements of music, which however is scarcely Euclid’s in its
present form, though probably based on a genuine treatise by him.
There were other works by Euclid which are now lost.
One belonged to elementary geometry and was called Pseudaria or Fallacies, the purpose of which was to put pupils on their
guard against possible pitfalls in geometrical reasoning. More important
were three treatises in higher geometry. For what we know of these we are
indebted to Pappus. The first is the Conics, which roughly covered
the ground of the first four Books of Apollonius’ Conics and, no
doubt, like the Elements, carried its subject to the limit of what was
then known. The second is the Surface-Loci. The nature of its contents is
not known. The loci in question may have been loci on surfaces, as the
Greek title implies, or loci which are surfaces. It seems clear that
conics came into the discussion, for Pappus gives, as a lemma for use with
the book, a proposition proving the fundamental property of each of the three
conics referred to what we call the focus and directrix, and it is a
natural inference that Euclid was aware of this property and assumed it
without proof, although it does not appear in Apollonius’ Conics.
The Porisms in three Books, is probably the
treatise the loss of which is the most to be deplored. Pappus speaks of
the contents of this work in considerable detail, though in a
very summary form, and he gives a number of lemmas to it. Important attempts
have been made to restore the treatise, notably by Robert Simson and
Michel Chasles; but we cannot be sure that any
of them have even come near to a solution of the problem. Certain it is
that the work belonged to higher geometry and was of considerable scope
(Pappus says that it contained 171 theorems).
Aristarchus of Samos, the ancient Copernicus, was
known as ‘the mathematician,’ to distinguish him from the host of
other persons of the same name. He was a pupil of Strato, and
an observation of the summer solstice made by him in 281—280 BC is recorded
by Ptolemy; on the other hand, his extant book On the Sizes and
Distances of the Sun and Moon and other works of his were written
before the Psammites or Sand-reckoner of Archimedes. We
gather therefore that Aristarchus lived from about 310 to 230 BC. Like his
master Strato, he wrote on vision, light and colours; but it is as
mathematician and astronomer that he is famous. He followed Heracleides of
Pontus in attributing to the earth the daily rotation about its axis, but
he went farther than Heracleides by suggesting that not only do the
planets Mercury and Venus revolve about the sun like satellites (as
Heracleides held), but the earth itself, as well as the planets, revolves
about the sun, a hypothesis amounting to an anticipation of
Copernicus. Our authority for this is Archimedes, who, in his Sand-reckoner, says
that Aristarchus brought out a book of hypotheses, one of which was that ‘
the fixed stars and the sun remain unmoved, and that the earth revolves
about the sun in the circumference of a circle, the sun lying in the
middle of the orbit.’ On the other hand, the extant treatise On the
Sizes and Distances of the Sun and Moon was written from the ordinary
geocentric standpoint. This work is a perfect specimen of Greek geometry
at its best. Starting from certain assumptions of matters of fact and
observation, which are naturally inaccurate as compared with modern
results, Aristarchus obtains, by clever and rigorous geometry, upper and
lower limits for the measures of (1) the diameter of the moon in terms of
its distance from the centre of the earth, (2) the distance of the
sun from the earth in terms of the distance of the moon from
the earth, (3) the diameters of the sun and moon respectively in
terms of that of the earth. He finds, for instance, that, on his assumptions, the
distance of the sun from the earth is between 18 and 20 times the distance
of the moon from the earth, a much better result than others quoted by
Archimedes, who says that Eudoxus made it nine times, and Pheidias, his
own father, twelve times. It was for want of trigonometry that Aristarchus
could not arrive at definite figures for what are really the
trigonometrical sines, cosines, and tangents of certain small angles, but
could only find upper and lower limits, just as Archimedes did for the
value of 77. Archimedes says that Aristarchus also discovered that the arc
subtended by the diameter of the sun at our eye is one seven hundred and
twentieth part of a complete circle, that is, half a degree. We are told
that Cleanthes the Stoic said that Aristarchus ought to be prosecuted for
impiety in putting the Hearth of the Universe in motion. As Cleanthes
became head of the Stoa in 261, it seems likely that the heliocentric
hypothesis was not put forward before that date. Apparently Seleucus, of
Seleuceia on the Tigris, (second century BC) alone accepted
Aristarchus’ hypothesis and tried to prove its truth; Hipparchus reverted
to the geocentric view.
Conon of Samos was the courtier-astronomer who gave
the name Coma Berenices to a group of stars in honour of
Berenice, the queen of Ptolemy III Euergetes (246-221 BC). The
story is that Berenice vowed to sacrifice her hair for the safe return
of her husband from his campaign against Syria to avenge the death of
his sister Berenice; this was soon after Euergetes’ accession. When
Euergetes actually returned, the queen’s tresses were duly cut off and
placed in the temple of Arsinoe-Aphrodite, but disappeared, whereupon Conon
declared that they had been caught up to the skies and placed as a
constellation in the triangle between Leo, Virgo and Ursa Major. Conon
must therefore have died after (say) 245 BC. On the other hand, he died
long before Archimedes wrote his book On Spirals; for in sending
this to Dositheus, Conon’s pupil, Archimedes says that many years
had passed since Conon’s death. Archimedes had been in the habit of
communicating his discoveries, before publication, to Conon whom he
admired as a mathematician and cherished as a friend. Conon spent a
considerable time in Italy where he made his astronomical observations,
after which he settled in Alexandria. It was probably there that he made
the acquaintance of Archimedes. Conon is said to have left seven Books on
astronomy dedicated to Euergetes; he also made a list of the solar
eclipses recorded as having occurred in Egypt. In mathematics
Conon tried to prove certain propositions about the number of
points (at most) in which a conic section may meet another conic or
the circumference of a circle; this we learn from Apollonius.
Archimedes of Syracuse, the most original genius among
the Greek mathematicians, was born about 287 and was killed, at the sack
of Syracuse in 212 BC, by a Roman soldier who found him intent on a
diagram drawn on the ground and was incensed when Archimedes said to him
‘Stand away, fellow, from my diagram.’ Well-known stories about him tell
how he declared ‘Give me a place to stand on and I will move the earth’;
and how for a long time he kept the Romans at bay by all sorts of
mechanical devices which he employed against them. The problem of moving a
great weight by a small force he is said to have illustrated, at the
request and to the amazement of Hiero, by using a system of
pulleys, which he worked himself, to draw up a huge ship with three
masts and fully loaded. It was, no doubt, when he was in Egypt that
he invented the Archimedean screw for pumping water. But he himself
thought nothing of these things, which were merely the ‘diversions of
geometry at play’; his whole soul was absorbed in pure mathematical
research. Hence it was that he caused to be engraved on his tomb a
representation of a cylinder circumscribing a sphere with the ratio 3 : 2
which the cylinder bears to the sphere, in respect both of its volume and
of its total surface, evidently regarding this as his greatest discovery.
This tomb Cicero, when quaestor in Sicily, found in a neglected state
and repaired; no trace of it now exists. Cicero also himself saw
an astronomical sphere constructed by Archimedes to imitate
the motions of the sun, moon and planets; according to Cicero
it represented the periods of the moon and the apparent motion of the
sun with such accuracy that it would even (over a short period) show the
eclipses of the sun and moon.
Heron, Pappus and Theon of Alexandria all cite works
of Archimedes which no longer survive. These works therefore still existed
at Alexandria as late as the third and fourth centuries A.D. But attention
came to be concentrated on two works only, the Measurement of a Circle and On the Sphere and Cylinder, which were at some time turned from
their original Doric into the ordinary language, with alterations designed
to make them easier for elementary pupils; the former, as we have it, is
not in its original form, and may only be a fragment. The following are
the works extant in Greek, arranged as nearly as may be in the order
of composition: On Plane Equilibriums, Book 1, Quadrature of
the Parabola, On Plane Equilibriums, Book 2, The Methodt (addressed to Eratosthenes), On the
Sphere and Cylinder, On Spirals, On Conoids and Spheroids, On
Floating Bodies, (formerly known only in the Latin translation by
William of Moerbeke but now in great part
recovered in Greek), Measurement of a Circle, the Sandreckoner,
the Stomachion (a fragment). The Method was discovered by Heiberg as recently as 1906 in a palimpsest at
Constantinople; the end of it is defective, but the beginning is there and
it is complete in all essentials. It was from the same MS. that Heiberg
was able partially to restore the Greek text of On Floating Bodies and the Stomachion (a sort of Chinese
puzzle).
A Book of Lemmas which has come to us through
the Arabic is not, in its present form, the work of Archimedes; but some
elegant propositions in it may go back to Archimedes, and notably
some theorems about figures called respectively salinon (salt-cellar) and shoe-maker's knife on account of their shape. The
famous Cattle Problem, purporting to have been sent by Archimedes to
the mathematicians at Alexandria in a letter to Eratosthenes, is
a difficult problem in indeterminate analysis leading to enormous numbers.
We have notices by Greek writers of works of
Archimedes now lost. (1) Pappus says that he investigated thirteen semiregular
solids, which are contained by equilateral and equiangular polygons but
not all of one kind; thus one figure has eight faces of which four are
equilateral triangles and four are regular hexagons, another has fourteen
faces of which eight are equilateral triangles and six are squares,
another has sixty-two faces of which twenty are equilateral triangles, thirty
are squares and twelve are regular pentagons, and so on (according to
Heron, Plato knew of two of the thirteen solids). (2) Theon of Alexandria
mentions a work by Archimedes called Catoptrica (theory of mirrors).
The details of Archimedes’ mathematics cannot be
entered on here; it must suffice to indicate the nature of his work. In
the geometrical treatises he stands, as it were, on the shoulders
of Eudoxus, measuring the content of curvilinear plane figures
and solids by methods which are a development of Eudoxus’ method of
exhaustion. If we may judge by the examples of the use of this method by
Euclid in his Book 12, Eudoxus approached the figure to be measured from
below only, inscribing figures more and more nearly approaching it and so
‘exhausting’ it. Archimedes approaches it both from above and from below,
circumscribing successive figures as well as inscribing them, and
showing that the final circumscribed and inscribed figures can be
compressed, as it were, as nearly as we please, into coincidence with the
figure to be measured and with one another. The desired result is, last of
all, established by a proof by reductio ad absurdum. In many
instances Archimedes’ procedure amounts to a genuine integration, as can
be seen by writing down the equivalent of his steps in modern analytical
notation; this is so with his investigation of the areas of a parabolic
segment and a spiral, the surface and volume of a sphere, and the volumes
of any segments of the conoids (paraboloids and hyperboloids of
revolution) and spheroids.
But the recently discovered Method shows that
this was not the way in which Archimedes made his first discoveries in this
field. Thus he first discovered that the area of a segment of a
parabola is four-thirds of that of the triangle with the same base and
height by a mechanical method which it is the object of the treatise
to expound. The procedure is to weigh countless elements of
the figure to be measured against the same number of elements
of another figure the mensuration of which is already known. The elements
are, in the case of plane areas, parallel straight lines in them and, in
the case of solids, parallel plane sections; and it is implied that the
figures are made up of the infinite number of parallel straight lines or
parallel plane sections in the figures respectively, which is very much the
view taken by Cavalieri. Hence Chasles could
truly say that Archimedes’ investigations ‘gave birth to the calculus of
the infinite conceived and brought to perfection successively by Kepler,
Cavalieri, Fermat, Leibniz and Newton.’ With regard, however, to the
mechanical method, Archimedes is careful to explain that, while it is useful in
suggesting results, it does not afford a conclusive proof of them; this
has later to be supplied by the rigorous method of exhaustion.
The Measurement of a Circle first proves
rigorously by exhaustion that the area of a circle is equal to that of a
right-angled triangle in which the perpendicular sides are respectively
equal to the circumference and the radius of the circle. Archimedes then finds
upper and lower limits to the value (
) of the ratio of the circumference
to the diameter by inscribing and circumscribing regular polygons of
96 sides, and approximating by sheer calculation to their contours
respectively. He assumes as known certain limits to the value of √3,
namely
> √3 >
> and finally arrives at the
well-known result 3
>
> 3
The Plane Equilibriums is the first known scientific
treatise on the first principles of mechanics. Premising a few postulates
suggested by common experience, Archimedes proves the principle of the
lever and then finds the centres of gravity of certain figures, a
parallelogram, a triangle, a trapezium, and finally (in the elaborate Book
2) of a parabolic segment and of any portion of such cut off between the
base and any straight line parallel to it.
In the Sand-reckoner Archimedes undertakes to
prove that, notwithstanding the inadequacy of the ordinary Greek
numerical notation to express large numbers, he can, by a certain
system developed by him in a separate work called Principles,
express any number however large, as e.g. the number of grains of sand
which, on any reasonable hypothesis as to the size of the universe,
the universe would contain. The system is one of orders and
periods based on powers of myriad-myriads (
), and would suffice to express
any number up to the 80,000 million-millionth power of ten! The number of
grains of sand required to fill up a space equal to the universe is, on
the hypothesis taken, proved to be less than “10,000,000 units of the
eighth order of numbers,” which is in fact
.
Lastly, Archimedes invented the whole science of
hydrostatics. In the treatise On Floating Bodies, after an assumption
about uniform pressure in a fluid, he first proves that the surface of
a fluid at rest is a sphere concentric with the earth.
Propositions following prove that, if a solid floats in a fluid, the
weight of the solid is equal to that of the fluid displaced, and if a
solid heavier than a fluid is weighed in it, it will be lighter than its
true weight by the weight of the fluid displaced. Then, after an
assumption that a body which is forced upwards in a fluid is forced
upwards along the perpendicular to the surface of the fluid drawn
through the centre of gravity of the body, Archimedes investigates
the positions of rest and stability of a segment of a sphere floating
in a fluid either side up, but with its base entirely above or
entirely below the surface. Book 11 goes still farther and
investigates fully all the positions of rest and stability of a right segment
of a paraboloid similarly floating in different cases according (1) to
the relation between the length of the axis of the solid and the parameter
of the generating parabola and (2) to the specific gravity of the solid in
relation to the fluid.
A younger contemporary of Archimedes was Eratosthenes
of Cyrene, with whom, as we have seen, Archimedes had relations. The other
activities of this many-sided man have already been described; it remains
to say a word about his mathematics. He devised a mechanical solution of
the problem of finding two, or any number of, mean proportionals in continued proportion between two straight lines. A bronze representation of
the contrivance (mesolabon, mean-finder) was
attached to the pillar which Eratosthenes erected in recognition of his
obligation to Ptolemy Euergetes, and on which were inscribed a
diagram and short proof of the solution, with an epigram.
Eratosthenes wrote a book entitled Platonicus which was used by Theon of Smyrna in his work on the mathematics which it
was necessary for the student of Plato to know; the Platonicus seems to have begun with the story of the problem of Delos and to have
contained disquisitions on proportion, the fundamental definitions
of geometry and the principles of music. Another work by Eratosthenes On
Means, in two Books, is included by Pappus, along with works by
Euclid, Aristaeus and Apollonius, in the ‘ Treasury of Analysis.’ His
device known as the sieve, for finding prime numbers, is not very
recondite. In astronomy he wrote a descriptive poem Hermes, fragments of
which remain, and another work which may have been the basis of the
Pseudo-Eratosthenic Catasterismi.
A school of mechanicians also flourished under the
first Ptolemies. Ctesibius, the earlier of two
persons of that name, probably lived under Philadelphus and Euergetes I.
This Ctesibius constructed engines of war and
pneumatic and hydraulic machines, among which Vitruvius mentions a
force-pump, a water-clock, etc. Philo of Byzantium, perhaps a generation
or so later than Ctesibius, wrote a
Mechanical Collection in at least nine Books of which the fourth (Belopoeica) and a further fragment survive
in Greek, while a portion of Book V (Pneumatica')
exists in a Latin translation from the Arabic. Another extant treatise on
engines of war was written by one Athenaeus and is addressed
to Marcellus.
Apollonius of Perga, known
as ‘the great geometer’ on the strength of his Conics, was born at Perga in Pamphylia, but studied for a long time at
Alexandria with the successors of Euclid. He flourished in the reign of
Ptolemy Euergetes I (246-221 BC). He refers in one of his prefaces to a
visit to Pergamum, and he dedicated the fourth and following Books
of the Conics to King Attalus I (241-197 BC).
The Conics was originally in eight Books. Of
these only the first four survive in Greek, but Books 5-7 exist in Arabic;
the eighth is lost. Eutocius (flor A.D.
520) edited the first four Books and added a commentary; it is probably to
this fact that we owe the survival of both text and commentary. The
monumental edition of Halley (Oxford, 1710') includes, besides the Greek
text of Books 1—4, a Latin translation of Books 5—6 from the Arabic and an
attempted restoration of Book 8.
From the outset we are struck by the remarkable
generality of treatment. Before Apollonius’ time the three conics were
called ‘sections of a right-angled, acute-angled, and obtuse-angled
cone’ respectively, because they were originally produced as sections
of right cones of these types by planes always perpendicular to
a generator (edge). Apollonius produces all three curves
(including the double-branch hyperbola) in the most general way from
any one cone (in general oblique). The diameter emerging from the construction
is in general not an axis; but the fundamental property is in each case
stated with reference to this diameter and the corresponding ‘parameter’
or ‘latus rectum.’ Though expressed in the language of ‘application of areas,’
the property is actually equivalent to the Cartesian equation of the curve
referred to any diameter and the tangent at its extremity as axes; the
square on the ‘ordinate’ is stated to be equal to a rectangle of width
equal to the ‘abscissa’ applied to the parameter, but in the case of
the parabola the rectangle is applied exactly (parabole simply) to the parameter, whereas in the case of the ellipse or
hyperbola it falls short (elleipsis) or
exceeds (hyperbole) respectively by a rectangle of given shape.
Hence the names parabola, ellipse, hyperbola applied to the conics
for the first time by Apollonius. There is something impressive in a
treatise which only condescends to notice the principal axes of a conic,
as a particular case, so late as 1. 52, after it has been shown that the
conic has the same property with regard to any diameter and parameter as
it has with reference to the pair arising out of the original
construction.
Books 1—4 are described by Apollonius himself as
forming an elementary introduction. They evidently cover much the
same ground as earlier treatises (notably Euclid’s); only certain
propositions in Books 3, 4 are new, but Apollonius rightly claims to have
treated the whole subject more fully and generally than his predecessors.
The four Books contain all the well-known ordinate-, chord-, and
tangent-properties of the three conics (including the double-branch
hyperbola treated as one curve), as well as, in the case of the central
conics, the focal properties other than those which bring in the
directrix; the directrix does not appear at all; nor does Apollonius
mention the focus of a parabola.
Books 5-7 are, in Apollonius’ own words, more ‘by way
of surplusage’; they contain in fact highly specialized, and in part very
difficult, investigations. Book v is the most remarkable. It deals with normals to the conics regarded as maximum and minimum
straight lines drawn to a conic from any point, internal or external, and apart
from their relation to tangents. The Book is a tour de force which
must be read to be believed. Not only does it investigate the number of normals that can be drawn from any point, but
there are propositions in it (all established by pure geometry) which lead
immediately to the determination of the evolutes of a parabola and a
central conic.
Six other works by Apollonius are included by Pappus
in the ‘Treasury of Analysis.’ Only one has survived, and that in Arabic, On
the Cutting-off of a Ratio, The others are On the Cutting-off of
an Area, On Determinate Section, On Tangencies (in this the
historic problem of drawing a circle to touch each of three given
circles was solved by elementary methods), Plane Loci, and Inclinationes. Apollonius also wrote on (1) the Comparison of the Dodecahedron with the Icosahedron, (2) the Cochlias (or cylindrical helix), (3) Unordered
Irrationals. In a General Treatise he apparently
discussed fundamental principles, while in a book called Ocytocion (‘quick delivery’) he is said to have
found a closer approximation than that of Archimedes to the value of
. He also had a system
for expressing large numbers based on powers of the myriad; this may
have been a counterblast to Archimedes’s system
Apollonius was no less distinguished in astronomy. He
was master of (if he did not invent) the theories of epicycles
and eccentric circles as a means of accounting for the movements
of the planets; he may therefore have been the ancient Tycho Brahe following
the ancient Copernicus (Aristarchus).
The classical tradition was kept up for a time by
geometers who are mostly known for one particular achievement.
Thus Nicomedes was the discoverer of the conchoid or cochloid, a mechanically-constructed curve useful
for solving higher problems. Perseus investigated the spiric curves, namely certain sections of the speira,
one variety of which is the anchor-ring or tore. Zenodorus wrote on Isometric
figures, the problem being to compare the content of different figures,
plane or solid, having equal contours or surfaces respectively. Diodes,
the discoverer of the curve called the cissoid, also wrote on Burning
Mirrors.
But with Apollonius pure geometry had practically
reached the limit of what it could accomplish unaided. Advance only
became possible after many centuries, when men like Fermat and Descartes
showed that the symbolism and methods of algebra could be used as aids to
geometry, and when again in course of time the calculus increased the
resources of the new analytical geometry. The wonder is, not that Greek
geometry could go no farther, but that it could accomplish so much. And it
was, after all, the Greeks and notably Apollonius who pointed the way to
co-ordinate geometry, while Archimedes actually anticipated the calculus.
VI.
HIPPARCHUS
One branch of geometry remained to be developed,
namely trigonometry. So far as we know, the first person to make systematic use
of trigonometry was Hipparchus, with whom this chapter may suitably
conclude.
Hipparchus, the greatest astronomer of antiquity, was
born at Nicaea in Bithynia. His date can only be inferred from
Ptolemy’s references to observations made by him, the limits of which
are 161 and 126 BC. His characteristics were extraordinary
modesty and passionate love of truth. The only book of his which
has survived is the Commentary on the Phaenomena of Eudoxus
and Aratus (an early work); what we know of his work apart
from this is contained in Ptolemy (who based himself mainly
on Hipparchus) and in scattered references by other authors. Hipparchus,
we are told, wrote a treatise in twelve Books on straight lines (i.e. chords) in a circle, which no doubt included a table of chords (equivalent
to a table of trigonometrical sines) something like that given by
Ptolemy. Ptolemy sets out, in an admirably clear and succinct form, the
minimum number of geometrical propositions required for preparing his own
table, and these (including the well-known ‘Ptolemy’s theorem’) were
doubtless known to Hipparchus.
It remains to indicate briefly the main contributions
of Hipparchus to astronomy.
1. His greatest discovery was, no doubt, that of the precession of
the equinoxes. He found that the bright star Spica was, at the time of
his observation of it (129-128 BC), 6° distant from the autumnal
equinoctial point, whereas he deduced from observations recorded by one
Timocharis that Timocharis had made the distance 8°. Now Timocharis made
observations of fixed stars (among the first recorded) in 295 and 283-282 BC;
the interval was thus 166 or 154 years, and the difference of 2° gives for
the motion an average of 43-4" or 46-8" a year as compared
with the true figure of 50-3757".
2. In a work On the length of the Year Hipparchus compared observations of the summer solstice made by
Aristarchus in 281-280 BC and by himself in 136-135 BC and found that,
after the 145 years, the summer solstice occurred half a day and night
earlier than it should have done on the assumption of 365I days to the
year; hence he inferred that the tropical year contained about a day and
night less than 365
days. This agrees very nearly with
Censorinus’ statement that Hipparchus made his cycle 304 years or four
times that of Callippus (76 years), but with 111,035 days in it instead of
111,036 (= 27,759 x 4). Counting in the 304 years 3760 months in all (12 x
304+ 112 intercalary), Hipparchus made the mean lunar month 29 days,
12 hours, 44 minutes, 2
seconds, which is less than a second out
in comparison with the present accepted figure of 29•53059 days!
3. Hipparchus improved on Aristarchus’ figures for the
sizes and distances of the sun and moon, determining the
apparent diameters more exactly and noting the changes in them.
4. He accounted for the motions of the sun and moon by
the simple epicycle and eccentric hypotheses, but found it
necessary, in the case of the planets, to combine the two, i.e. to
superadd epicycles to motion in eccentric circles.
5. He made a catalogue of fixed stars to the number of
850 or more, distinguishing degrees of brightness and (apparently for
the first time) stating the position of each star in terms of coordinates
(latitude and longitude) referred to the ecliptic.
6. He constructed a globe showing the positions of the
fixed stars as determined by him, and made great improvements in
the instruments used for observations.
7. He wrote on geography, mainly by way of criticism
of Eratosthenes, and emphasizing the necessity of applying astronomy to
geography.
The debt of humanity to Hipparchus, and the greatness
of soul of the man whose system was practically not improved upon
until the days of Copernicus, Galilei and Kepler, cannot be
better illustrated than by his motive in compiling the catalogue.
He discovered a new star which first appeared in his time, and
this set him thinking whether the same phenomenon might not occur at
any time, and whether there might not be movement on the part even of the
stars which we deem ‘fixed’; in making his catalogue, therefore, he ‘left
the heavens as a sort of heritage to all and sundry’ (as Pliny says), in
order that it might be easy for posterity to judge, not only whether stars
vanish and new ones appear, but whether changes occur in their relative
positions, and whether they wax or wane.
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