Books: The Old Roman World

J >> John Lord >> The Old Roman World

Pages:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47

[Sidenote: Socrates.]

[Sidenote: Pythagoras.]

Socrates, who belonged to another school, avoided all barren
speculations concerning the universe, and confined himself to human
actions and interests. He looked even upon geometry in a very practical
way, so far as it could be made serviceable to land measuring. As for
the stars and planets, he supposed it was impossible to arrive at a true
knowledge of them, and regarded speculations upon them as useless. The
Greek astronomers, however barren were their general theories, still
laid the foundation of science. Pythagoras, born 580 B.C., 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 and 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 they 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 the speculations of Plato.

[Sidenote: Eudoxus.]

Eudoxus, who was born 406 B.C., may be considered the founder of
scientific astronomical knowledge among the Greeks. He is reputed to
have visited Egypt with Plato, and to have resided thirteen years in
Heliopolis, in constant study of the stars, communing with the Egyptian
priests. His contribution to the science was a descriptive map of the
heavens, which was used as a manual of sidereal astronomy to the sixth
century of our era. He distributed the stars into constellations, with
recognized names, and gave a sort of geographical description of their
position and limits, although the constellations had been named before
his time. He stated the periodic times of the five planets visible to
the naked eye, but only approximated to the true periods.

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 in a
treatise concerning the heavens. [Footnote: Delambre, _Hist. de
l'Astron. Anc._, tom. i. p. 301.] He regarded astronomy as more
intimately connected with mathematical science than any other branch of
philosophy. 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.

[Sidenote: Meton.]

The introduction of the gnomon 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.;
[Footnote: Ap. Gell., _N. A._, iii. 3.] but there was a difficulty
of using them, since they failed at night and in cloudy weather, and
could not be relied on. Hence the introduction of water-clocks instead.

[Sidenote: Aristarchus.]

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." [Footnote: Lewis, p. 190.] This
speculation, resting on the authority of Archimedes, was ridiculed by
him; but if it were advanced, it shows a great advance in astronomical
science, and considering the age, was one of the boldest speculations of
antiquity. Aristarchus also, according to Plutarch, [Footnote: Plut.,
_Plac. Phil._, ii. 24.] 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. [Footnote: _Quaest. Plat._, viii.
1.] This theory gave great offense, 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 has
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 sound by modern astronomers, [Footnote: Lewis, p. 193.]
but 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.

[Sidenote: Archimedes.]

[Sidenote: Eratosthenes.]

Archimedes, 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. He 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 degrees 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, not far from the truth.
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
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 very close approximation to the truth.

[Sidenote: Hipparchus.]

[Sidenote: Greatness of Hipparchus.]

Astronomical science received a great impulse from the school of
Alexandria, and Eratosthenes had worthy successors in Aristarchus,
Aristyllus, Apollonius. But the great light of this school was
Hipparchus, whose lifetime extended from 190 to 120 years B.C. He laid
the foundation of astronomy upon a scientific basis. "He determined,"
says Delambre, "the position of the stars by right ascensions and
declinations; he 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 likewise detected a second inequality, of which he could not,
for want of proper observations, discover the period and the law. His
commentary on Aratus shows that he had expounded, and given a
geometrical demonstration of, the methods necessary to find out the
right and oblique ascensions of the points of the ecliptic and of the
stars, the east point and the culminating point of the ecliptic, and the
angle of the east, which is now called the nonagesimal degree. He could
calculate eclipses of the moon, and use them for the correction of his
lunar tables, and he had an approximate knowledge of parallax."
[Footnote: Delambre, _Hist. de l'Astron. Anc._, tom. i. p. 184.]
His determination of the motions of the sun and moon, and 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 reduce the phenomena of the heavenly
bodies to uniform motions in of 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." [Footnote: _Hist. Ind. Science_, vol. i. p.
181.] But he even did 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 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.

[Sidenote: Posidonius.]

[Sidenote: The Roman Calendar.]

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, in spite of the
patronage of the Ptolemies, 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
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?--by which objects are
distinctly seen, which, without it, would be invisible? Moreover, the
ancients had no accurate almanacs, since the care of the calendar
belonged to the priests rather than to the astronomers, who tampered
with the computation of time for temporary and personal 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. 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 of 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.
[Footnote: Suet., _Caesar_, 49; Plut., _Caesar_, 59.]

"Ille moras solis, quibus in sua signa rediret,
Traditur exactis disposuisse notis.
Is decies senos tercentum et quinque diebus
Junxit; et pleno tempora quarta die.
Hic anni modus est. In lustrum accedere debet
Quae consummatur partibus, una dies."

[Footnote: Ovid, _Fast._, iii.]

[Sidenote: Caesar's labors.]

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. And even these were not constructed as they should
have been. The hours on the sun-dial were all made equal, instead of
varying with the length of the day, so that 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. But
clocks, with wheels and weights, were not invented till the twelfth
century.

The earlier Greek astronomers did not attempt to fix the order of the
planets; but when geometry was applied to celestial movements, the
difference between the three superior planets and the two inferior was
perceived, and the sun was placed in the midst between them, so that the
seven movable heavenly bodies were made to succeed one another in the
following order: 1. Saturn; 2. Jupiter; 3. Mars; 4. The Sun; 5. Venus;
6. Mercury; 7. The Moon. Archimedes adopted this order, which was
followed by the leading philosophers. [Footnote: Lewis, p. 247.]

[Sidenote: Ptolemy and his system.]

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 round the centre of the planetary system, and turns
upon its axis--two ideas in common with the doctrines which Copernicus
afterward unfolded. But even he did not conceive the heliocentric theory
that the sun is the centre of the universe. 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 reasoning powers, and the magnificent field they
afford for sublime contemplation. "But," as Sir G. Cornwall 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 occupation 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, as in
truth and virtue, can have no sympathy.

[Sidenote: Result of ancient investigations.]

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 which 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 laws
of gravitation--the grandest scientific discovery in the annals of our
race.

[Sidenote: Geometry.]

[Sidenote: Ancient Greek geometers.]

[Sidenote: Euclid.]

[Sidenote: Archimedes.]

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. Thales introduced the science to the Greeks. He
applied a circle to the measurement of angles. Anaximander invented the
sphere, the gnomon, and geographical charts, which required considerable
geometrical knowledge. Anaxagoras employed himself in prison in
attempting to square the circle. Pythagoras 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. He also discovered that of all figures having the
same boundary, the circle among plane figures and the sphere among
solids, are the most capacious. The theory of the regular solids was
taught in his school, and his disciple, Archytas, was the author of a
solution of the problem of two mean proportionals. Democritus of Abdera
treated of the contact of circles and spheres, and of irrational lines
and solids. 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 loci. They also attempted to resolve the
problems of the trisection of an angle and the duplication of a cube. To
Leon is ascribed that part of the solution of a problem, called its
_determination_, which treats of the cases in which the problem is
possible, and of those in which it cannot be resolved. Euclid has almost
given his name to the science of geometry. He was born B.C. 323, 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
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 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. But, perhaps, a
greater even than Euclid was Archimedes, born 287 B.C., who wrote on the
sphere and cylinder, which terminate in the discovery that the solidity
and surface of a sphere are respectively two thirds 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 only reached by prodigious
mathematical power. Archimedes is popularly better known as the inventor
of engines of war, and various ingenious machines, than as a
mathematician, great as were his attainments. 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. He 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
he built. He used also a 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. [Footnote: See article in Smith's
_Dictionary_, by Prof. Darkin, of Oxford.]

[Sidenote: Eratosthenes.]

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--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.

[Sidenote: Apollonius of Perga.]

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,
regarded with unbounded admiration by contemporaries, and, in some
respects, 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. 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? and if
Thales had been contemporaneous with Plato, he might have added to his
sublime science even more than 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. [Footnote: See Bayle's _Dict_.;
Bossuet, _Essai sur L'Hist. Gen. des Math_.; Simson's _Sectiones
Conicae_.]