their general theories, laid the foundation of science. Pythagoras
taught the obliquity of the ecliptic, probably learned in Egypt, and the
identity of the morning and evening stars. It is supposed that he
maintained that the sun was the centre of the universe, and that the
earth revolved around it; but this he did not demonstrate, and his whole
system was unscientific, assuming certain arbitrary principles, from
which he reasoned deductively. "He assumed that fire is more worthy than
earth; that the more worthy place must be given to the more worthy; that
the extremity is more worthy than the intermediate parts,--and hence,
as the centre is an extremity, the place of fire is at the centre of the
universe, and that therefore the earth and other heavenly bodies move
round the fiery centre." But this was no heliocentric system, since the
sun moved, like the earth, in a circle around the central fire. This was
merely the work of the imagination, utterly unscientific, though bold
and original. Nor did this hypothesis gain credit, since it was the
fixed opinion of philosophers that the earth was the centre of the
universe, around which the sun, moon, and planets revolved. But the
Pythagoreans were the first to teach that the motions of the sun, moon,
and planets are circular and equable. Their idea that the celestial
bodies emitted a sound, and were combined into a harmonious symphony,
was exceedingly crude, however beautiful "The music of the spheres"
belongs to poetry, as well as to the speculations of Plato.
Eudoxus, in the fifth century before Christ, contributed to science by
making a descriptive map of the heavens, which was used as a manual of
sidereal astronomy to the sixth century of our era.
The error of only one hundred and ninety days in the periodic time of
Saturn shows that there had been for a long time close observations.
Aristotle--whose comprehensive intellect, like that of Bacon, took in
all forms of knowledge--condensed all that was known in his day into a
treatise concerning the heavens. He regarded astronomy as more
intimately connected with mathematics than any other branch of science.
But even he did not soar far beyond the philosophers of his day, since
he held to the immobility of the earth,--the grand error of the
ancients. Some few speculators in science (like Heraclitus of Pontus,
and Hicetas) conceived a motion of the earth itself upon its axis, so as
to account for the apparent motion of the sun; but they also thought it
was in the centre of the universe.
The introduction of the gnomon (time-pillar) and dial into Greece
advanced astronomical knowledge, since they were used to determine the
equinoxes and solstices, as well as parts of the day. Meton set up a
sun-dial at Athens in the year 433 B.C., but the length of the hour
varied with the time of the year, since the Greeks divided the day into
twelve equal parts. Dials were common at Rome in the time of Plautus,
224 B.C.; but there was a difficulty in using them, since they failed at
night and in cloudy weather, and could not be relied on. Hence the
introduction of water-clocks instead.
Aristarchus is said to have combated (280 B.C.) the geocentric theory so
generally received by philosophers, and to have promulgated the
hypothesis "that the fixed stars and the sun are immovable; that the
earth is carried round the sun in the circumference of a circle of
which the sun is the centre; and that the sphere of the fixed stars,
having the same centre as the sun, is of such magnitude that the orbit
of the earth is to the distance of the fixed stars as the centre of the
sphere of the fixed stars is to its surface." Aristarchus also,
according to Plutarch, explained the apparent annual motion of the sun
in the ecliptic by supposing the orbit of the earth to be inclined to
its axis. There is no evidence that this great astronomer supported his
heliocentric theory with any geometrical proof, although Plutarch
maintains that he demonstrated it. This theory gave great offence,
especially to the Stoics; and Cleanthes, the head of the school at that
time, maintained that the author of such an impious doctrine should be
punished. Aristarchus left a treatise "On the Magnitudes and Distances
of the Sun and Moon;" and his methods to measure the apparent diameters
of the sun and moon are considered theoretically sound by modern
astronomers, but practically inexact owing to defective instruments. He
estimated the diameter of the sun at the seven hundred and twentieth
part of the circumference of the circle which it describes in its
diurnal revolution, which is not far from the truth; but in this
treatise he does not allude to his heliocentric theory.
Archimedes of Syracuse, born 287 B.C., is stated to have measured the
distance of the sun, moon, and planets, and he constructed an orrery in
which he exhibited their motions. But it was not in the Grecian colony
of Syracuse, but of Alexandria, that the greatest light was shed on
astronomical science. Here Aristarchus resided, and also Eratosthenes,
who lived between the years 276 and 196 B.C. The latter was a native of
Athens, but was invited by Ptolemy Euergetes to Alexandria, and placed
at the head of the library. His great achievement was the determination
of the circumference of the earth. This was done by measuring on the
ground the distance between Syene, a city exactly under the tropic, and
Alexandria, situated on the same meridian. The distance was found to be
five thousand stadia. The meridional distance of the sun from the zenith
of Alexandria he estimated to be 7° 12', or a fiftieth part of the
circumference of the meridian. Hence the circumference of the earth was
fixed at two hundred and fifty thousand stadia,--which is not very
different from our modern computation. The circumference being known,
the diameter of the earth was easily determined. The moderns have added
nothing to this method. He also calculated the diameter of the sun to be
twenty-seven times greater than that of the earth, and the distance of
the sun from the earth to be eight hundred and four million stadia, and
that of the moon seven hundred and eighty thousand stadia,--a close
approximation to the truth.
Astronomical science received a great impulse from the school of
Alexandria, the greatest light of which was Hipparchus, who flourished
early in the second century before Christ. He laid the foundation of
astronomy upon a scientific basis. "He determined," says Delambre, "the
position of the stars by right ascensions and declinations, and was
acquainted with the obliquity of the ecliptic. He determined the
inequality of the sun and the place of its apogee, as well as its mean
motion; the mean motion of the moon, of its nodes and apogee; the
equation of the moon's centre, and the inclination of its orbit. He
calculated eclipses of the moon, and used them for the correction of his
lunar tables, and he had an approximate knowledge of parallax." His
determination of the motions of the sun and moon, and his method of
predicting eclipses evince great mathematical genius. But he combined
with this determination a theory of epicycles and eccentrics which
modern astronomy discards. It was however a great thing to conceive of
the earth as a solid sphere, and to reduce the phenomena of the heavenly
bodies to uniform motions in circular orbits. "That Hipparchus should
have succeeded in the first great steps of the resolution of the
heavenly bodies into circular motions is a circumstance," says Whewell,
"which gives him one of the most distinguished places in the roll of
great astronomers." But he did even more than this: he discovered that
apparent motion of the fixed stars round the axis of the ecliptic, which
is called the Precession of the Equinoxes,--one of the greatest
discoveries in astronomy. He maintained that the precession was not
greater than fifty-nine seconds, and not less than thirty-six seconds.
Hipparchus also framed a catalogue of the stars, and determined their
places with reference to the ecliptic by their latitudes and longitudes.
Altogether he seems to have been one of the greatest geniuses of
antiquity, and his works imply a prodigious amount of calculation.
Astronomy made no progress for three hundred years, although it was
expounded by improved methods. Posidonius constructed an orrery, which
exhibited the diurnal motions of the sun, moon, and five planets.
Posidonius calculated the circumference of the earth to be two hundred
and forty thousand stadia, by a different method from Eratosthenes. The
barrenness of discovery from Hipparchus to Ptolemy,--the Alexandrian
mathematician, astronomer, and geographer in the second century of the
Christian era,--in spite of the patronage of the royal Ptolemies of
Egypt, was owing to the want of instruments for the accurate measure of
time (like our clocks), to the imperfection of astronomical tables, and
to the want of telescopes. Hence the great Greek astronomers were unable
to realize their theories. Their theories however were magnificent, and
evinced great power of mathematical combination; but what could they do
without that wondrous instrument by which the human eye indefinitely
multiplies its power? Moreover, the ancients had no accurate almanacs,
since the care of the calendar belonged not so much to the astronomers
as to the priests, who tampered with the computation of time for
sacerdotal objects. The calendars of different communities differed.
Hence Julius Caesar rendered a great service to science by the reform of
the Roman calendar, which was exclusively under the control of the
college of pontiffs, or general religious overseers. The Roman year
consisted of three hundred and fifty-five days; and in the time of
Caesar the calendar was in great confusion, being ninety days in
advance, so that January was an autumn month. He inserted the regular
intercalary month of twenty-three days, and two additional ones of
sixty-seven days. These, together with ninety days, were added to three
hundred and sixty-five days, making a year of transition of four hundred
and forty-five days, by which January was brought back to the first
month in the year after the winter solstice; and to prevent the
repetition of the error, he directed that in future the year should
consist of three hundred and sixty-five and one-quarter days, which he
effected by adding one day to the months of April, June, September, and
November, and two days to the months of January, Sextilis, and
December, making an addition of ten days to the old year of three
hundred and fifty-five. And he provided for a uniform intercalation of
one day in every fourth year, which accounted for the remaining
quarter of a day.
Caesar was a student of astronomy, and always found time for its
contemplation. He is said even to have written a treatise on the motion
of the stars. He was assisted in his reform of the calendar by
Sosigines, an Alexandrian astronomer. He took it out of the hands of the
priests, and made it a matter of pure civil regulation. The year was
defined by the sun, and not as before by the moon.
Thus the Romans were the first to bring the scientific knowledge of the
Greeks into practical use; but while they measured the year with a great
approximation to accuracy, they still used sun-dials and water-clocks to
measure diurnal time. Yet even these were not constructed as they should
have been. The hour-marks on the sun-dial were all made equal, instead
of varying with the periods of the day,--so that the length of the hour
varied with the length of the day. The illuminated interval was divided
into twelve equal parts; so that if the sun rose at five A.M., and set
at eight P.M., each hour was equal to eighty minutes. And this rude
method of measurement of diurnal time remained in use till the sixth
century. Clocks, with wheels and weights, were not invented till the
twelfth century.
The last great light among the ancients in astronomical science was
Ptolemy, who lived from 100 to 170 A.D., in Alexandria. He was
acquainted with the writings of all the previous astronomers, but
accepted Hipparchus as his guide. He held that the heaven is spherical
and revolves upon its axis; that the earth is a sphere, and is situated
within the celestial sphere, and nearly at its centre; that it is a mere
point in reference to the distance and magnitude of the fixed stars, and
that it has no motion. He adopted the views of the ancient astronomers,
who placed Saturn, Jupiter, and Mars next under the sphere of the fixed
stars, then the sun above Venus and Mercury, and lastly the moon next to
the earth. But he differed from Aristotle, who conceived that the earth
revolves in an orbit around the centre of the planetary system, and
turns upon its axis,--two ideas in common with the doctrines which
Copernicus afterward unfolded. But even Ptolemy did not conceive the
heliocentric theory,--the sun the centre of our system. Archimedes and
Hipparchus both rejected this theory.
In regard to the practical value of the speculations of the ancient
astronomers, it may be said that had they possessed clocks and
telescopes, their scientific methods would have sufficed for all
practical purposes. The greatness of modern discoveries lies in the
great stretch of the perceptive powers, and the magnificent field they
afford for sublime contemplation. "But," as Sir G. Cornewall Lewis
remarks, "modern astronomy is a science of pure curiosity, and is
directed exclusively to the extension of knowledge in a field which
human interests can never enter. The periodic time of Uranus, the nature
of Saturn's ring, and the occultation of Jupiter's satellites are as far
removed from the concerns of mankind as the heliacal rising of Sirius,
or the northern position of the Great Bear." This may seem to be a
utilitarian view, with which those philosophers who have cultivated
science for its own sake, finding in the same a sufficient reward, can
have no sympathy.
The upshot of the scientific attainments of the ancients, in the
magnificent realm of the heavenly bodies, would seem to be that they
laid the foundation of all the definite knowledge which is useful to
mankind; while in the field of abstract calculation they evinced
reasoning and mathematical powers that have never been surpassed.
Eratosthenes, Archimedes, and Hipparchus were geniuses worthy to be
placed by the side of Kepler, Newton, and La Place, and all ages will
reverence their efforts and their memory. It is truly surprising that
with their imperfect instruments, and the absence of definite data,
they reached a height so sublime and grand. They explained the doctrine
of the sphere and the apparent motions of the planets, but they had no
instruments capable of measuring angular distances. The ingenious
epicycles of Ptolemy prepared the way for the elliptic orbits and laws
of Kepler, which in turn conducted Newton to the discovery of the law of
gravitation,--the grandest scientific discovery in the annals of
our race.
Closely connected with astronomical science was geometry, which was
first taught in Egypt,--the nurse and cradle of ancient wisdom. It arose
from the necessity of adjusting the landmarks disturbed by the
inundations of the Nile. There is hardly any trace of geometry among the
Hebrews. Among the Hindus there are some works on this science, of great
antiquity. Their mathematicians knew the rule for finding the area of a
triangle from its sides, and also the celebrated proposition concerning
the squares on the sides of the right-angled triangle. The Chinese, it
is said, also knew this proposition before it was known to the Greeks,
among whom it was first propounded by Thales. He applied a circle to the
measurement of angles. Anaximander made geographical charts, which
required considerable geometrical knowledge. Anaxagoras employed
himself in prison in attempting to square the circle. Thales, as has
been said, discovered the important theorem that in a right-angled
triangle the squares on the sides containing the right angle are
together equal to the square on the opposite side of it. Pythagoras
discovered that of all figures having the same boundary, the circle
among plane figures and the sphere among solids are the most capacious.
Hippocrates treated of the duplication of the cube, and wrote elements
of geometry, and knew that the area of a circle was equal to a triangle
whose base is equal to its circumference and altitude equal to its
radius. The disciples of Plato invented conic sections, and discovered
the geometrical foci.
It was however reserved for Euclid to make his name almost synonymous
with geometry. He was born 323 B.C., and belonged to the Platonic sect,
which ever attached great importance to mathematics. His "Elements" are
still in use, as nearly perfect as any human production can be. They
consist of thirteen books. The first four are on plane geometry; the
fifth is on the theory of proportion, and applies to magnitude in
general; the seventh, eighth, and ninth are on arithmetic; the tenth on
the arithmetical characteristics of the division of a straight line; the
eleventh and twelfth on the elements of solid geometry; the thirteenth
on the regular solids. These "Elements" soon became the universal study
of geometers throughout the civilized world; they were translated into
the Arabic, and through the Arabians were made known to mediaeval
Europe. There can be no doubt that this work is one of the highest
triumphs of human genius, and it has been valued more than any single
monument of antiquity; it is still a text-book, in various English
translations, in all our schools. Euclid also wrote various other works,
showing great mathematical talent.
Perhaps a greater even than Euclid was Archimedes, born 287 B.C. He
wrote on the sphere and cylinder, terminating in the discovery that the
solidity and surface of a sphere are two thirds respectively of the
solidity and surface of the circumscribing cylinder. He also wrote on
conoids and spheroids. "The properties of the spiral and the quadrature
of the parabola were added to ancient geometry by Archimedes, the last
being a great step in the progress of the science, since it was the
first curvilineal space legitimately squared." Modern mathematicians may
not have the patience to go through his investigations, since the
conclusions he arrived at may now be reached by shorter methods; but the
great conclusions of the old geometers were reached by only prodigious
mathematical power. Archimedes is popularly better known as the inventor
of engines of war and of various ingenious machines than as a
mathematician, great as were his attainments in this direction. His
theory of the lever was the foundation of statics till the discovery of
the composition of forces in the time of Newton, and no essential
addition was made to the principles of the equilibrium of fluids and
floating bodies till the time of Stevin, in 1608. Archimedes detected
the mixture of silver in a crown of gold which his patron, Hiero of
Syracuse, ordered to be made; and he invented a water-screw for pumping
water out of the hold of a great ship which he had built. He contrived
also the combination of pulleys, and he constructed an orrery to
represent the movement of the heavenly bodies. He had an extraordinary
inventive genius for discovering new provinces of inquiry and new points
of view for old and familiar objects. Like Newton, he had a habit of
abstraction from outward things, and would forget to take his meals. He
was killed by Roman soldiers when Syracuse was taken; and the Sicilians
so soon forgot his greatness that in the time of Cicero they did not
know where his tomb was.
Eratosthenes was another of the famous geometers of antiquity, and did
much to improve geometrical analysis. He was also a philosopher and
geographer. He gave a solution of the problem of the duplication of the
cube, and applied his geometrical knowledge to the measurement of the
magnitude of the earth,--being one of the first who brought
mathematical methods to the aid of astronomy, which in our day is almost
exclusively the province of the mathematician.
Apollonius of Perga, probably about forty years younger than Archimedes,
and his equal in mathematical genius, was the most fertile and profound
writer among the ancients who treated of geometry. He was called the
Great Geometer. His most important work is a treatise on conic sections,
which was regarded with unbounded admiration by contemporaries, and in
some respects is unsurpassed by any thing produced by modern
mathematicians. He however made use of the labors of his predecessors,
so that it is difficult to tell how far he is original. But all men of
science must necessarily be indebted to those who have preceded them.
Even Homer, in the field of poetry, made use of the bards who had sung
for a thousand years before him; and in the realms of philosophy the
great men of all ages have built up new systems on the foundations which
others have established. If Plato or Aristotle had been contemporaries
with Thales, would they have matured so wonderful a system of
dialectics? Yet if Thales had been contemporaneous with Plato, he might
have added to the great Athenian's sublime science even more than did
Aristotle. So of the great mathematicians of antiquity; they were all
wonderful men, and worthy to be classed with the Newtons and Keplers of
our times. Considering their means and the state of science, they made
as _great_ though not as _fortunate_ discoveries,--discoveries which
show patience, genius, and power of calculation. Apollonius was one of
these,--one of the master intellects of antiquity, like Euclid and
Archimedes; one of the master intellects of all ages, like Newton
himself. I might mention the subjects of his various works, but they
would not be understood except by those familiar with mathematics.
Other famous geometers could also be named, but such men as Euclid,
Archimedes, and Apollonius are enough to show that geometry was
cultivated to a great extent by the philosophers of antiquity. It
progressively advanced, like philosophy itself, from the time of Thales
until it had reached the perfection of which it was capable, when it
became merged into astronomical science. It was cultivated more
particularly by the disciples of Plato, who placed over his school this
inscription: "Let no one ignorant of geometry enter here." He believed
that the laws by which the universe is governed are in accordance with
the doctrines of mathematics. The same opinion was shared by Pythagoras,
the great founder of the science, whose main formula was that _number_
is the essence or first principle of all things. No thinkers ever
surpassed the Greeks in originality and profundity; and mathematics,
being highly prized by them, were carried to the greatest perfection
their method would allow. They did not understand algebra, by the
application of which to geometry modern mathematicians have climbed to
greater heights than the ancients; but then it is all the more
remarkable that without the aid of algebraic analysis they were able to
solve such difficult problems as occupied the minds of Archimedes and
Apollonius. No positive science can boast of such rapid development as
geometry for two or three hundred years before Christ, and never was the
intellect of man more severely tasked than by the ancient
mathematicians.
No empirical science can be carried to perfection by any one nation or
in any particular epoch; it can only expand with the progressive
developments of the human race itself. Nevertheless, in that science
which for three thousand years has been held in the greatest honor, and
which is one of the three great liberal professions of our modern times,
the ancients, especially the Greeks, made considerable advance. The
science of medicine, having in view the amelioration of human misery and
the prolongation of life itself, was very early cultivated. It was,
indeed, in old times another word for _physics_,--the science of
Nature,--and the _physician_ was the observer and expounder of physics.
The physician was supposed to be acquainted with the secrets of
Nature,--that is, the knowledge of drugs, of poisons, of antidotes to
them, and the way to administer them. He was also supposed to know the
process of preserving the body after death. Thus Joseph, seventeen
hundred years before the birth of Christ, commanded his physician to
embalm the body of his father; and the process of embalming was probably
known to the Egyptians before the period when history begins. Helen, of
Trojan fame, put into wine a drug that "frees man from grief and anger,
and causes oblivion of all ills." Solomon was a great botanist,--a realm
with which the science of medicine is indissolubly connected. The origin
of Hindu medicine is lost in remote antiquity. The Ayur Veda, written
nine hundred years before Hippocrates was born, sums up the knowledge of
previous periods relating to obstetric surgery, to general pathology, to
the treatment of insanity, to infantile diseases, to toxicology, to
personal hygiene, and to diseases of the generative functions.