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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

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.

Thus Hippocrates, the father of European medicine, must have derived his
knowledge not merely from his own observations, but from the writings of
men unknown to us and from systems practised for an indefinite period.
The real founders of Greek medicine are fabled characters, like Hercules
and Aesculapius, - that is, benefactors whose fictitious names alone
have descended to us. They are mythical personages, like Hermes and
Chiron. Twelve hundred years before Christ temples were erected to
Aesculapius in Greece, the priests of which were really physicians, and
the temples themselves hospitals. In them were practised rites
apparently mysterious, but which modern science calls by the names of
mesmerism, hydropathy, the use of mineral springs, and other essential
elements of empirical science. And these temples were also medical
schools. That of Cos gave birth to Hippocrates, and it was there that
his writings were begun. Pythagoras - for those old Grecian philosophers
were the fathers of all wisdom and knowledge, in mathematics and
empirical sciences as well as philosophy itself - studied medicine in the
schools of Egypt, Phoenicia, Chaldaea, and India, and came in conflict
with sacerdotal power, which has ever been antagonistic to new ideas in
science. He travelled from town to town as a teacher or lecturer,
establishing communities in which _medicine_ as well as _numbers_
was taught.

The greatest name in medical science in ancient or in modern times, the
man who did the most to advance it, the greatest medical genius of whom
we have any early record, was Hippocrates, born on the island of Cos,
460 B.C., of the great Aesculapian family. He received his instruction
from his father. We know scarcely more of his life than we do of Homer
himself, although he lived in the period of the highest splendor of
Athens. Even his writings, like those of Homer, are thought by some to
be the work of different men. They were translated into Arabic, and were
no slight means of giving an impulse to the Saracenic schools of the
Middle Ages in that science in which the Saracens especially excelled.
The Hippocratic collection consists of more than sixty works, which were
held in the highest estimation by the ancient physicians. Hippocrates
introduced a new era in medicine, which before his time had been
monopolized by the priests. He carried out a system of severe induction
from the observation of facts, and is as truly the creator of the
inductive method as Bacon himself. He abhorred theories which could not
be established by facts; he was always open to conviction, and candidly
confessed his mistakes; he was conscientious in the practice of his
profession, and valued the success of his art more than silver and gold.
The Athenians revered Hippocrates for his benevolence as well as genius.
The great principle of his practice was _trust in Nature_; hence he was
accused of allowing his patients to die. But this principle has many
advocates among scientific men in our day; and some suppose that the
whole successful practice of Homoeopathy rests on the primal principle
which Hippocrates advanced, although the philosophy of it claims a
distinctly scientific basis in the principle _similia similibus
curantur_. Hippocrates had great skill in diagnosis, by which medical
genius is most severely tested; his practice was cautious and timid in
contrast with that of his contemporaries. He is the author of the
celebrated maxim, "Life is short and art is long." He divides the causes
of disease into two principal classes, - the one comprehending the
influence of seasons, climates, and other external forces; the other
including the effects of food and exercise. To the influence of climate
he attributes the conformation of the body and the disposition of the
mind; to a vicious system of diet he attributes innumerable forms of
disease. For more than twenty centuries his pathology was the foundation
of all the medical sects. He was well acquainted with the medicinal
properties of drugs, and was the first to assign three periods to the
course of a malady. He knew but little of surgery, although he was in
the habit of bleeding, and often employed the knife; he was also
acquainted with cupping, and used violent purgatives. He was not aware
of the importance of the pulse, and confounded the veins with the
arteries. Hippocrates wrote in the Ionic dialect, and some of his works
have gone through three hundred editions, so highly have they been
valued. His authority passed away, like that of Aristotle, on the
revival of science in Europe. Yet who have been greater ornaments and
lights than these two distinguished Greeks?

The school of Alexandria produced eminent physicians, as well as
mathematicians, after the glory of Greece had departed. So highly was it
esteemed that Galen in the second century, - born in Greece, but famous
in the service of Rome, - went there to study, five hundred years after
its foundation. It was distinguished for inquiries into scientific
anatomy and physiology, for which Aristotle had prepared the way. Galen
was the Humboldt of his day, and gave great attention to physics. In
eight books he developed the general principles of natural science known
to the Greeks. On the basis of the Aristotelian researches, the
Alexandrian physicians carried out extensive inquiries in physiology.
Herophilus discovered the fundamental principles of neurology, and
advanced the anatomy of the brain and spinal cord.

Although the Romans had but little sympathy with science or philosophy,

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