BOOK I. THE GEOMETRICAL PERIOD

BOOK I. THE GEOMETRICAL PERIOD

1. PRIMITIVE ASTRONOMY AND ASTROLOGY.

The growth of intelligence in the humanrace has its counterpart in

that of the individual, especially in theearliest stages.

Intellectual activity and the developmentof reasoning powers are in

both cases based upon the accumulation ofexperiences, and on the

comparison, classification, arrangement,and nomenclature of these

experiences. During the infancy of each thesuccession of events can

be watched, but there can be no _à priori_anticipations.

Experience alone, in both cases, leads tothe idea of cause and effect

as a principle that seems to dominate ourpresent universe, as a rule

for predicting the course of events, and asa guide to the choice of a

course of action. This idea of cause andeffect is the most potent

factor in developing the history of thehuman race, as of the

individual.

In no realm of nature is the principle ofcause and effect more

conspicuous than in astronomy; and we fallinto the habit of thinking

of its laws as not only being unchangeablein our universe, but

necessary to the conception of any universethat might have been

substituted in its place. The firstinhabitants of the world were

compelled to accommodate their acts to thedaily and annual

alternations of light and darkness and ofheat and cold, as much as to

the irregular changes of weather, attacksof disease, and the fortune

of war. They soon came to regard the influence of the sun, in

connection with light and heat, as a cause.This led to a search for

other signs in the heavens. If theappearance of a comet was sometimes

noted simultaneously with the death of agreat ruler, or an eclipse

with a scourge of plague, these might wellbe looked upon as causes in

the same sense that the veering or backingof the wind is regarded as

a cause of fine or foul weather.

For these reasons we find that the earnestmen of all ages have

recorded the occurrence of comets,eclipses, new stars, meteor

showers, and remarkable conjunctions of theplanets, as well as

plagues and famines, floods and droughts,wars and the deaths of great

rulers. Sometimes they thought they couldtrace connections which

might lead them to say that a cometpresaged famine, or an eclipse

war.

Even if these men were sometimes led toevolve laws of cause and

effect which now seem to us absurd, let usbe tolerant, and gratefully

acknowledge that these astrologers, whenthey suggested such "working

hypotheses," were laying thefoundations of observation and deduction.

If the ancient Chaldæans gave to theplanetary conjunctions an

influence over terrestrial events, let usremember that in our own

time people have searched for connectionbetween terrestrial

conditions and periods of unusualprevalence of sun spots; while De la

Rue, Loewy, and Balfour Stewart[1] thoughtthey found a connection

between sun-spot displays and the planetarypositions. Thus we find

scientific men, even in our own time,responsible for the belief that

storms in the Indian Ocean, the fertilityof German vines, famines in

India, and high or low Nile-floods in Egyptfollow the planetary

positions.

And, again, the desire to foretell theweather is so laudable that we

cannot blame the ancient Greeks forannouncing the influence of the

moon with as much confidence as it isaffirmed in Lord Wolseley's

_Soldier's Pocket Book_.

Even if the scientific spirit ofobservation and deduction (astronomy)

has sometimes led to erroneous systems forpredicting terrestrial

events (astrology), we owe to the oldastronomer and astrologer alike

the deepest gratitude for their diligencein recording astronomical

events. For, out of the scanty recordswhich have survived the

destructive acts of fire and flood, ofmonarchs and mobs, we have

found much that has helped to a fullerknowledge of the heavenly

motions than was possible without theserecords.

So Hipparchus, about 150 B.C., and Ptolemya little later, were able

to use the observations of Chaldæanastrologers, as well as those of

Alexandrian astronomers, and to make somediscoveries which have

helped the progress of astronomy in allages.  So, also, Mr. Cowell[2]

has examined the marks made on the bakedbricks used by the Chaldæans

for recording the eclipses of 1062 B.C. and762 B.C.; and has thereby

been enabled, in the last few years, tocorrect the lunar tables of

Hansen, and to find a more accurate valuefor the secular acceleration

of the moon's longitude and the node of herorbit than any that could

be obtained from modern observations madewith instruments of the

highest precision.

So again, Mr. Hind [3] was enabled to traceback the period during

which Halley's comet has been a member ofthe solar system, and to

identify it in the Chinese observations ofcomets as far back as 12

B.C. Cowell and Cromellin extended the dateto 240 B.C. In the same

way the comet 1861.i. has been traced backin the Chinese records to

617 A.D. [4]

The theoretical views founded on Newton'sgreat law of universal

gravitation led to the conclusion that theinclination of the earth's

equator to the plane of her orbit (theobliquity of the ecliptic) has

been diminishing slowly since prehistorictimes; and this fact has

been confirmed by Egyptian and Chineseobservations on the length of

the shadow of a vertical pillar, madethousands of years before the

Christian era, in summer and winter.

There are other reasons why we must betolerant of the crude notions

of the ancients. The historian, wishing togive credit wherever it may

be due, is met by two difficulties.Firstly, only a few records of

very ancient astronomy are extant, and theauthenticity of many of

these is open to doubt. Secondly, it isvery difficult to divest

ourselves of present knowledge, and toappreciate the originality of

thought required to make the firstbeginnings.

With regard to the first point, we aregenerally dependent upon

histories written long after theevents.  The astronomy of Egyptians,

Babylonians, and Assyrians is known to usmainly through the Greek

historians, and for information about theChinese we rely upon the

researches of travellers and missionariesin comparatively recent

times. The testimony of the Greek writershas fortunately been

confirmed, and we now have in addition amass of facts translated from

the original sculptures, papyri, andinscribed bricks, dating back

thousands of years.

In attempting to appraise the efforts ofthe beginners we must

remember that it was natural to look uponthe earth (as all the first

astronomers did) as a circular plane,surrounded and bounded by the

heaven, which was a solid vault, orhemisphere, with its concavity

turned downwards. The stars seemed to befixed on this vault; the

moon, and later the planets, were seen tocrawl over it. It was a

great step to look on the vault as a hollowsphere carrying the sun

too. It must have been difficult to believethat at midday the stars

are shining as brightly in the blue sky asthey do at night. It must

have been difficult to explain how the sun,having set in the west,

could get back to rise in the east withoutbeing seen _if_ it was

always the same sun. It was a great step tosuppose the earth to be

spherical, and to ascribe the diurnalmotions to its rotation.

Probably the greatest step ever made inastronomical theory was the

placing of the sun, moon, and planets atdifferent distances from the

earth instead of having them stuck on thevault of heaven. It was a

transition from "flatland" to aspace of three dimensions.

Great progress was made when systematicobservations began, such as

following the motion of the moon andplanets among the stars, and the

inferred motion of the sun among the stars,by observing their

_heliacal risings_--i.e., the times of yearwhen a star

would first be seen to rise at sunrise, andwhen it could last be seen

to rise at sunset. The grouping of thestars into constellations and

recording their places was a usefulobservation. The theoretical

prediction of eclipses of the sun and moon,and of the motions of the

planets among the stars, became later thehighest goal in astronomy.

To not one of the above important steps inthe progress of astronomy

can we assign the author with certainty.Probably many of them were

independently taken by Chinese, Indian,Persian, Tartar, Egyptian,

Babylonian, Assyrian, Phoenician, and Greekastronomers.  And we have

not a particle of information about thediscoveries, which may have

been great, by other peoples--by theDruids, the Mexicans, and the

Peruvians, for example.

We do know this, that all nations requiredto have a calendar. The

solar year, the lunar month, and the daywere the units, and it is

owing to their incommensurability that wefind so many calendars

proposed and in use at different times. Theonly object to be attained

by comparing the chronologies of ancientraces is to fix the actual

dates of observations recorded, and this isnot a part of a history of

astronomy.

In conclusion, let us bear in mind thelimited point of view of the

ancients when we try to estimate theirmerit. Let us remember that the

first astronomy was of two dimensions; thesecond astronomy was of

three dimensions, but still purelygeometrical. Since Kepler's day we

have had a dynamical astronomy.

FOOTNOTES:

[1] Trans. R. S. E., xxiii. 1864, p. 499,_On Sun Spots_, etc., by

B. Stewart. Also Trans. R. S. 1860-70. AlsoProf. Ernest Brown, in

_R. A. S. Monthly Notices_, 1900.

[2] _R. A. S. Monthly Notices_, Sup.; 1905.

[Illustration: CHALDÆAN BAKED BRICK ORTABLET, _Obverse and reverse

sides_, Containing record of solar eclipse,1062 B.C., used lately by

Cowell for rendering the lunar theory moreaccurate than was possible

by finest modern observations. (BritishMuseum collection,

No. 35908.)]

[3] _R. A. S. Monthly Notices_, vol. x., p.65.

[4] R. S. E. Proc.,vol. x., 1880.

2. ANCIENT ASTRONOMY--THE CHINESE ANDCHALDÆANS.

The last section must have made clear thedifficulties the way of

assigning to the ancient nations theirproper place in the development

of primitive notions about astronomy. Thefact that some alleged

observations date back to a period beforethe Chinese had invented the

art of writing leads immediately to thequestion how far tradition can

be trusted.

Our first detailed knowledge was gatheredin the far East by

travellers, and by the Jesuit priests, andwas published in the

eighteenth century. The Asiatic Society ofBengal contributed

translations of Brahmin literature. The twoprincipal sources of

knowledge about Chinese astronomy weresupplied, first by Father

Souciet, who in 1729 published_Observations Astronomical,

Geographical, Chronological, and Physical_,drawn from ancient

Chinese books; and later by FatherMoyriac-de-Mailla, who in 1777-1785

published _Annals of the Chinese Empire,translated from

Tong-Kien-Kang-Mou_.

Bailly, in his _Astronomie Ancienne_(1781), drew, from these and

other sources, the conclusion that all weknow of the astronomical

learning of the Chinese, Indians,Chaldæans, Assyrians, and Egyptians

is but the remnant of a far more completeastronomy of which no trace

can be found.

Delambre, in his _Histoire de l'AstronomieAncienne_ (1817),

ridicules the opinion of Bailly, andconsiders that the progress made

by all of these nations is insignificant.

It will be well now to give an idea of someof the astronomy of the

ancients not yet entirely discredited.  China and Babylon may be taken

as typical examples.

_China_.--It would appear that Fohi, thefirst emperor, reigned

about 2952 B.C., and shortly afterwards Yu-Chimade a sphere to

represent the motions of the celestialbodies. It is also mentioned,

in the book called Chu-King, supposed tohave been written in 2205

B.C., that a similar sphere was made in thetime of Yao (2357

B.C.).[1] It is said that the EmperorChueni (2513 B.C.) saw five

planets in conjunction the same day thatthe sun and moon were in

conjunction. This is discussed by FatherMartin (MSS. of De Lisle);

also by M. Desvignolles (Mem. Acad. Berlin,vol. iii., p. 193), and by

M. Kirsch (ditto, vol. v., p. 19), who bothfound that Mars, Jupiter,

Saturn, and Mercury were all between theeleventh and eighteenth

degrees of Pisces, all visible together inthe evening on February

28th 2446 B.C., while on the same day thesun and moon were in

conjunction at 9 a.m., and that on March1st the moon was in

conjunction with the other four planets.But this needs confirmation.

Yao, referred to above, gave instructionsto his astronomers to

determine the positions of the solsticesand equinoxes, and they

reported the names of the stars in theplaces occupied by the sun at

these seasons, and in 2285 B.C. he gavethem further orders. If this

account be true, it shows a knowledge thatthe vault of heaven is a

complete sphere, and that stars are shiningat mid-day, although

eclipsed by the sun's brightness.

It is also asserted, in the book called_Chu-King_, that in the

time of Yao the year was known to have 365¼days, and that he

adopted 365 days and added an intercalaryday every four years (as in

the Julian Calendar). This may be true ornot, but the ancient Chinese

certainly seem to have divided the circleinto 365 degrees. To learn

the length of the year needed only patientobservation--a

characteristic of the Chinese; but manyyounger nations got into a

terrible mess with their calendar fromignorance of the year's length.

It is stated that in 2159 B.C. the royalastronomers Hi and Ho failed

to predict an eclipse. It probably createdgreat terror, for they were

executed in punishment for their neglect.If this account be true, it

means that in the twenty-second centuryB.C. some rule for calculating

eclipses was in use. Here, again, patientobservation would easily

lead to the detection of the eighteen-yearcycle known to the

Chaldeans as the _Saros_. It consists of235 lunations, and in

that time the pole of the moon's orbitrevolves just once round the

pole of the ecliptic, and for this reasonthe eclipses in one cycle

are repeated with very slight modificationin the next cycle, and so

on for many centuries.

It may be that the neglect of their dutiesby Hi and Ho, and their

punishment, influenced Chinese astronomy;or that the succeeding

records have not been available to laterscholars; but the fact

remains that--although at long intervalsobservations were made of

eclipses, comets, and falling stars, and ofthe position of the

solstices, and of the obliquity of theecliptic--records become rare,

until 776 B.C., when eclipses began to berecorded once more with some

approach to continuity. Shortly afterwardsnotices of comets were

added. Biot gave a list of these, and Mr.John Williams, in 1871,

published _Observations of Comets from 611B.C. to 1640 A.D.,

Extracted from the Chinese Annals_.

With regard to those centuries concerningwhich we have no

astronomical Chinese records, it is fair tostate that it is recorded

that some centuries before the Christianera, in the reign of

Tsin-Chi-Hoang, all the classical andscientific books that could be

found were ordered to be destroyed. Iftrue, our loss therefrom is as

great as from the burning of theAlexandrian library by the Caliph

Omar. He burnt all the books because heheld that they must be either

consistent or inconsistent with the Koran,and in the one case they

were superfluous, in the other caseobjectionable.

_Chaldæans_.--Until the last half centuryhistorians were

accustomed to look back upon the Greeks,who led the world from the

fifth to the third century B.C., as thepioneers of art, literature,

and science. But the excavations andresearches of later years make us

more ready to grant that in science as inart the Greeks only

developed what they derived from theEgyptians, Babylonians, and

Assyrians. The Greek historians said as much,in fact; and modern

commentators used to attribute theassertion to undue modesty. Since,

however, the records of the libraries havebeen unearthed it has been

recognised that the Babylonians were in noway inferior in the matter

of original scientific investigation toother races of the same era.

The Chaldæans, being the most ancientBabylonians, held the same

station and dignity in the State as did thepriests in Egypt, and

spent all their time in the study ofphilosophy and astronomy, and the

arts of divination and astrology. They heldthat the world of which we

have a conception is an eternal worldwithout any beginning or ending,

in which all things are ordered by rulessupported by a divine

providence, and that the heavenly bodies donot move by chance, nor by

their own will, but by the determinate willand appointment of the

gods. They recorded these movements, butmainly in the hope of tracing

the will of the gods in mundane affairs.Ptolemy (about 130 A.D.)

made use of Babylonian eclipses in theeighth century B.C. for

improving his solar and lunar tables.

Fragments of a library at Agade have beenpreserved at Nineveh, from

which we learn that the star-charts wereeven then divided into

constellations, which were known by thenames which they bear to this

day, and that the signs of the zodiac wereused for determining the

courses of the sun, moon, and of the fiveplanets Mercury, Venus,

Mars, Jupiter, and Saturn.

We have records of observations carried onunder Asshurbanapal, who

sent astronomers to different parts tostudy celestial phenomena. Here

is one:--

To the Director of Observations,--My Lord,his humble servant

Nabushum-iddin, Great Astronomer ofNineveh, writes thus: "May Nabu

and Marduk be propitious to the Director ofthese Observations, my

Lord. The fifteenth day we observed theNode of the moon, and the moon

was eclipsed."

The Phoenicians are supposed to have usedthe stars for navigation,

but there are no records. The Egyptianpriests tried to keep such

astronomical knowledge as they possessed tothemselves. It is probable

that they had arbitrary rules forpredicting eclipses. All that was

known to the Greeks about Egyptian scienceis to be found in the

writings of Diodorus Siculus. But confirmatoryand more authentic

facts have been derived from lateexplorations. Thus we learn from

E. B. Knobel[2] about the Jewish calendardates, on records of land

sales in Aramaic papyri at Assuan,translated by Professor A. H. Sayce

and A. E. Cowley, (1) that the lunar cycleof nineteen years was used

by the Jews in the fifth century B.C. [thepresent reformed Jewish

calendar dating from the fourth centuryA.D.], a date a "little more

than a century after the grandfathers andgreat-grandfathers of those

whose business is recorded had fled intoEgypt with Jeremiah" (Sayce);

and (2) that the order of intercalation atthat time was not

dissimilar to that in use at the presentday.

Then again, Knobel reminds us of "themost interesting discovery a few

years ago by Father Strassmeier of aBabylonian tablet recording a

partial lunar eclipse at Babylon in theseventh year of Cambyses, on

the fourteenth day of the Jewish monthTammuz."  Ptolemy, in the

Almagest (Suntaxis), says it occurred inthe seventh year of Cambyses,

on the night of the seventeenth andeighteenth of the Egyptian month

Phamenoth. Pingré and Oppolzer fix the date July 16th, 533 B.C. Thus

are the relations of the chronologies ofJews and Egyptians

established by these explorations.

FOOTNOTES:

[1] These ancient dates are uncertain.

[2] _R. A. S. Monthly Notices_, vol.lxviii., No. 5, March, 1908.

3. ANCIENT GREEK ASTRONOMY.

We have our information about the earliestGreek astronomy from

Herodotus (born 480 B.C.). He put thetraditions into writing. Thales

(639-546 B.C.) is said to have predicted aneclipse, which caused much

alarm, and ended the battle between theMedes and Lydians. Airy fixed

the date May 28th, 585 B.C. But othermodern astronomers give

different dates. Thales went to Egypt tostudy science, and learnt

from its priests the length of the year(which was kept a profound

secret!), and the signs of the zodiac, andthe positions of the

solstices. He held that the sun, moon, andstars are not mere spots on

the heavenly vault, but solids; that themoon derives her light from

the sun, and that this fact explains herphases; that an eclipse of

the moon happens when the earth cuts offthe sun's light from her. He

supposed the earth to be flat, and to floatupon water. He determined

the ratio of the sun's diameter to itsorbit, and apparently made out

the diameter correctly as half a degree. Heleft nothing in writing.

His successors, Anaximander (610-547 B.C.)and Anaximenes (550-475

B.C.), held absurd notions about the sun,moon, and stars, while

Heraclitus (540-500 B.C.)  supposed that the stars were lighted each

night like lamps, and the sun each morning.Parmenides supposed the

earth to be a sphere.

Pythagoras (569-470 B.C.) visited Egypt tostudy science. He deduced

his system, in which the earth revolves inan orbit, from fantastic

first principles, of which the followingare examples: "The circular

motion is the most perfect motion,""Fire is more worthy than earth,"

"Ten is the perfect number." Hewrote nothing, but is supposed to have

said that the earth, moon, five planets,and fixed stars all revolve

round the sun, which itself revolves roundan imaginary central fire

called the Antichthon. Copernicus in thesixteenth century claimed

Pythagoras as the founder of the systemwhich he, Copernicus, revived.

Anaxagoras (born 499 B.C.) studiedastronomy in Egypt. He explained

the return of the sun to the east eachmorning by its going under the

flat earth in the night. He held that in asolar eclipse the moon

hides the sun, and in a lunar eclipse themoon enters the earth's

shadow--both excellent opinions. But heentertained absurd ideas of

the vortical motion of the heavens whiskingstones into the sky, there

to be ignited by the fiery firmament toform stars. He was prosecuted

for this unsettling opinion, and formaintaining that the moon is an

inhabited earth. He was defended byPericles (432 B.C.).

Solon dabbled, like many others, in reformsof the calendar. The

common year of the Greeks originally had360 days--twelve months of

thirty days. Solon's year was 354 days. Itis obvious that these

erroneous years would, before long, removethe summer to January and

the winter to July. To prevent this it wascustomary at regular

intervals to intercalate days or months.Meton (432 B.C.) introduced a

reform based on the nineteen-year cycle.This is not the same as the

Egyptian and Chaldean eclipse cycle called_Saros_ of 223

lunations, or a little over eighteenyears.  The Metonic cycle is 235

lunations or nineteen years, after whichperiod the sun and moon

occupy the same position relative to thestars. It is still used for

fixing the date of Easter, the number ofthe year in Melon's cycle

being the golden number of ourprayer-books.  Melon's system divided

the 235 lunations into months of thirtydays and omitted every

sixty-third day. Of the nineteen years,twelve had twelve months and

seven had thirteen months.

Callippus (330 B.C.) used a cycle fourtimes as long, 940 lunations,

but one day short of Melon's seventy-sixyears. This was more correct.

Eudoxus (406-350 B.C.) is said to havetravelled with Plato in

Egypt. He made astronomical observations inAsia Minor, Sicily, and

Italy, and described the starry heavensdivided into constellations.

His name is connected with a planetarytheory which as generally

stated sounds most fanciful. He imaginedthe fixed stars to be on a

vault of heaven; and the sun, moon, andplanets to be upon similar

vaults or spheres, twenty-six revolvingspheres in all, the motion of

each planet being resolved into itscomponents, and a separate sphere

being assigned for each component motion.Callippus (330 B.C.)

increased the number to thirty-three. It isnow generally accepted

that the real existence of these sphereswas not suggested, but the

idea was only a mathematical conception tofacilitate the construction

of tables for predicting the places of theheavenly bodies.

Aristotle (384-322 B.C.) summed up thestate of astronomical knowledge

in his time, and held the earth to be fixedin the centre of the

world.

Nicetas, Heraclides, and Ecphantes supposedthe earth to revolve on

its axis, but to have no orbital motion.

The short epitome so far given illustratesthe extraordinary deductive

methods adopted by the ancient Greeks. Butthey went much farther in

the same direction. They seem to have beenin great difficulty to

explain how the earth is supported, just aswere those who invented

the myth of Atlas, or the Indians with thetortoise. Thales thought

that the flat earth floated on water.Anaxagoras thought that, being

flat, it would be buoyed up and supportedon the air like a kite.

Democritus thought it remained fixed, likethe donkey between two

bundles of hay, because it was equidistantfrom all parts of the

containing sphere, and there was no reasonwhy it should incline one

way rather than another. Empedoclesattributed its state of rest to

centrifugal force by the rapid circularmovement of the heavens, as

water is stationary in a pail when whirledround by a string.

Democritus further supposed that theinclination of the flat earth to

the ecliptic was due to the greater weightof the southern parts owing

to the exuberant vegetation.

For further references to similar effortsof imagination the reader is

referred to Sir George Cornwall Lewis's_Historical Survey of the

Astronomy of the Ancients_; London, 1862.His list of authorities

is very complete, but some of his conclusionsare doubtful.  At p. 113

of that work he records the real opinionsof Socrates as set forth by

Xenophon; and the reader will, perhaps,sympathise with Socrates in

his views on contemporary astronomy:--

With regard to astronomy he [Socrates]considered a knowledge of it

desirable to the extent of determining theday of the year or month,

and the hour of the night, ... but as tolearning the courses of the

stars, to be occupied with the planets, andto inquire about their

distances from the earth, and their orbits,and the causes of their

motions, he strongly objected to such awaste of valuable time. He

dwelt on the contradictions and conflictingopinions of the physical

philosophers, ... and, in fine, he heldthat the speculators on the

universe and on the laws of the heavenlybodies were no better than

madmen (_Xen_.  _Mem_, i. 1, 11-15).

Plato (born 429 B.C.), the pupil ofSocrates, the fellow-student of

Euclid, and a follower of Pythagoras,studied science in his travels

in Egypt and elsewhere.  He was held in so great reverence by all

learned men that a problem which he set tothe astronomers was the

keynote to all astronomical investigationfrom this date till the time

of Kepler in the sixteenth century. Heproposed to astronomers _the

problem of representing the courses of theplanets by circular and

uniform motions_.

Systematic observation among the Greeksbegan with the rise of the

Alexandrian school. Aristillus andTimocharis set up instruments and

fixed the positions of the zodiacal stars,near to which all the

planets in their orbits pass, thusfacilitating the determination of

planetary motions. Aristarchus (320-250B.C.) showed that the sun must

be at least nineteen times as far off asthe moon, which is far short

of the mark. He also found the sun'sdiameter, correctly, to be half a

degree. Eratosthenes (276-196 B.C.) measured the inclination to the

equator of the sun's apparent path in theheavens--i.e., he

measured the obliquity of the ecliptic, makingit 23° 51', confirming

our knowledge of its continuous diminutionduring historical times. He

measured an arc of meridian, fromAlexandria to Syene (Assuan), and

found the difference of latitude by thelength of a shadow at noon,

summer solstice. He deduced the diameter ofthe earth, 250,000

stadia. Unfortunately, we do not know thelength of the stadium he

used.

Hipparchus (190-120 B.C.) may be regardedas the founder of

observational astronomy. He measured theobliquity of the ecliptic,

and agreed with Eratosthenes.  He altered the length of the tropical

year from 365 days, 6 hours to 365 days, 5hours, 53 minutes--still

four minutes too much. He measured theequation of time and the

irregular motion of the sun; and allowedfor this in his calculations

by supposing that the centre, about whichthe sun moves uniformly, is

situated a little distance from the fixedearth. He called this point

the _excentric_. The line from the earth tothe "excentric" was

called the _line of apses_. A circle havingthis centre was

called the _equant_, and he supposed that aradius drawn to the

sun from the excentric passes over equalarcs on the equant in equal

times. He then computed tables forpredicting the place of the sun.

He proceeded in the same way to computeLunar tables. Making use of

Chaldæan eclipses, he was able to get anaccurate value of the moon's

mean motion.  [Halley, in 1693, compared this value withhis own

measurements, and so discovered theacceleration of the moon's mean

motion. This was conclusively established,but could not be explained

by the Newtonian theory for quite a longtime.] He determined the

plane of the moon's orbit and itsinclination to the ecliptic. The

motion of this plane round the pole of theecliptic once in eighteen

years complicated the problem. He locatedthe moon's excentric as he

had done the sun's.  He also discovered some of the minor

irregularities of the moon's motion, due,as Newton's theory proves,

to the disturbing action of the sun's attraction.

In the year 134 B.C. Hipparchus observed anew star.  This upset every

notion about the permanence of the fixedstars. He then set to work to

catalogue all the principal stars so as toknow if any others appeared

or disappeared. Here his experiencesresembled those of several later

astronomers, who, when in search of somespecial object, have been

rewarded by a discovery in a totallydifferent direction. On comparing

his star positions with those of Timocharisand Aristillus he found no

stars that had appeared or disappeared inthe interval of 150 years;

but he found that all the stars seemed tohave changed their places

with reference to that point in the heavenswhere the ecliptic is 90°

from the poles of the earth--i.e., theequinox. He found that this

could be explained by a motion of theequinox in the direction of the

apparent diurnal motion of the stars. Thisdiscovery of _precession of

the equinoxes_, which takes place at therate of 52".1 every year, was

necessary for the progress of accurateastronomical observations. It

is due to a steady revolution of theearth's pole round the pole of

the ecliptic once in 26,000 years in theopposite direction to the

planetary revolutions.

Hipparchus was also the inventor of trigonometry,both plane and

spherical. He explained the method of usingeclipses for determining

the longitude.

In connection with Hipparchus' greatdiscovery it may be mentioned

that modern astronomers have oftenattempted to fix dates in history

by the effects of precession of theequinoxes. (1) At about the date

when the Great Pyramid may have been builtγDraconis was near to the

pole, and must have been used as thepole-star. In the north face of

the Great Pyramid is the entrance to aninclined passage, and six of

the nine pyramids at Gizeh possess the samefeature; all the passages

being inclined at an angle between 26° and27° to the horizon and in

the plane of the meridian. It also appearsthat 4,000 years

ago--i.e., about 2100 B.C.--an observer atthe lower end of the

passage would be able to see γDraconis, thethen pole-star, at its

lower culmination.[1] It has been suggestedthat the passage was made

for this purpose. On other grounds the dateassigned to the Great

Pyramid is 2123 B.C.

(2) The Chaldæans gave names toconstellations now invisible from

Babylon which would have been visible in2000 B.C., at which date it

is claimed that these people were studyingastronomy.

(3) In the Odyssey, Calypso directsOdysseus, in accordance with

Phoenician rules for navigating theMediterranean, to keep the Great

Bear "ever on the left as he traversedthe deep" when sailing from the

pillars of Hercules (Gibraltar) to Corfu.Yet such a course taken now

would land the traveller in Africa.  Odysseus is said in his voyage in

springtime to have seen the Pleiades andArcturus setting late, which

seemed to early commentators a proof ofHomer's inaccuracy.  Likewise

Homer, both in the _Odyssey_ [2] (v. 272-5)and in the _Iliad_

(xviii. 489), asserts that the Great Bearnever set in those

latitudes. Now it has been found that theprecession of the equinoxes

explains all these puzzles; shows that inspringtime on the

Mediterranean the Bear was just above thehorizon, near the sea but

not touching it, between 750 B.C. and 1000B.C.; and fixes the date of

the poems, thus confirming other evidence,and establishing Homer's

character for accuracy. [3]

(4) The orientation of Egyptian temples andDruidical stones is such

that possibly they were so placed as toassist in the observation of

the heliacal risings [4] of certain stars.If the star were known,

this would give an approximate date. Up tothe present the results of

these investigations are far from beingconclusive.

Ptolemy (130 A.D.) wrote the Suntaxis, orAlmagest, which includes a

cyclopedia of astronomy, containing asummary of knowledge at that

date. We have no evidence beyond his ownstatement that he was a

practical observer. He theorised on theplanetary motions, and held

that the earth is fixed in the centre ofthe universe. He adopted the

excentric and equant of Hipparchus toexplain the unequal motions of

the sun and moon. He adopted the epicyclesand deferents which had

been used by Apollonius and others to explainthe retrograde motions

of the planets. We, who know that the earthrevolves round the sun

once in a year, can understand that theapparent motion of a planet is

only its motion relative to the earth. If,then, we suppose the earth

fixed and the sun to revolve round it oncea year, and the planets

each in its own period, it is onlynecessary to impose upon each of

these an additional _annual_ motion toenable us to represent truly

the apparent motions. This way of lookingat the apparent motions

shows why each planet, when nearest to theearth, seems to move for a

time in a retrograde direction. Theattempts of Ptolemy and others of

his time to explain the retrograde motionin this way were only

approximate. Let us suppose each planet tohave a bar with one end

centred at the earth.  If at the other end of the bar one end of a

shorter bar is pivotted, having the planetat its other end, then the

planet is given an annual motion in thesecondary circle (the

epicycle), whose centre revolves round theearth on the primary circle

(the _deferent_), at a uniform rate roundthe excentric. Ptolemy

supposed the centres of the epicycles ofMercury and Venus to be on a

bar passing through the sun, and to bebetween the earth and the

sun. The centres of the epicycles of Mars,Jupiter, and Saturn were

supposed to be further away than thesun.  Mercury and Venus were

supposed to revolve in their epicycles intheir own periodic times and

in the deferent round the earth in a year.The major planets were

supposed to revolve in the deferent roundthe earth in their own

periodic times, and in their epicycles oncein a year.

It did not occur to Ptolemy to place thecentres of the epicycles of

Mercury and Venus at the sun, and to extendthe same system to the

major planets. Something of this sort hadbeen proposed by the

Egyptians (we are told by Cicero andothers), and was accepted by

Tycho Brahe; and was as true arepresentation of the relative motions

in the solar system as when we suppose thesun to be fixed and the

earth to revolve.

The cumbrous system advocated by Ptolemyanswered its purpose,

enabling him to predict astronomical eventsapproximately. He improved

the lunar theory considerably, anddiscovered minor inequalities which

could be allowed for by the addition of newepicycles.  We may look

upon these epicycles of Apollonius, and theexcentric of Hipparchus,

as the responses of these astronomers tothe demand of Plato for

uniform circular motions. Their use becamemore and more confirmed,

until the seventeenth century, when theaccurate observations of Tycho

Brahe enabled Kepler to abolish thesepurely geometrical makeshifts,

and to substitute a system in which the sunbecame physically its

controller.

FOOTNOTES:

[1] _Phil. Mag_., vol. xxiv., pp. 481-4.

[2]

Πληιάδας τʽ ἐσορω̑ντε καὶ ὀψὲ δύοντα βοώτην

ʼΆρκτον θ̕, ἣν καὶ ἅμαξαν ἐπίκλησιν καλέουσιν,

ʽΉ τ̕ αὐτου̑ στρέφεταικαὶ τ̕ ʼΩρίωνα δοκεύει,

Οἴη δ̕ἄμμορος ἐστι λοετρων ʽΩκεανοι̑ο.

"The Pleiades and Boötes that settethlate, and the Bear,

which they likewise call the Wain, whichturneth ever in one

place, and keepeth watch upon Orion, andalone hath no part in

the baths of the ocean."

[3] See Pearson in the Camb. Phil. Soc.Proc., vol. iv., pt. ii., p.

93, on whose authority the above statementsare made.

[4] See p. 6 for definition.

4. THE REIGN OF EPICYCLES--FROM PTOLEMY TOCOPERNICUS.

After Ptolemy had published his book thereseemed to be nothing more

to do for the solar system except to go onobserving and finding more

and more accurate values for the constantsinvolved--viz., the periods

of revolution, the diameter of thedeferent,[1] and its ratio to that

of the epicycle,[2] the distance of theexcentric[3] from the centre

of the deferent, and the position of theline of apses,[4] besides the

inclination and position of the plane ofthe planet's orbit. The only

object ever aimed at in those days was toprepare tables for

predicting the places of the planets. Itwas not a mechanical problem;

there was no notion of a governing law offorces.

From this time onwards all interest inastronomy seemed, in Europe at

least, to sink to a low ebb.  When the Caliph Omar, in the middle of

the seventh century, burnt the library ofAlexandria, which had been

the centre of intellectual progress, thatcentre migrated to Baghdad,

and the Arabs became the leaders of scienceand philosophy. In

astronomy they made careful observations.In the middle of the ninth

century Albategnius, a Syrian prince,improved the value of

excentricity of the sun's orbit, observedthe motion of the moon's

apse, and thought he detected a smallerprogression of the sun's

apse. His tables were much more accuratethan Ptolemy's. Abul Wefa, in

the tenth century, seems to have discoveredthe moon's "variation."

Meanwhile the Moors were leaders of sciencein the west, and Arzachel

of Toledo improved the solar tables verymuch. Ulugh Begh, grandson of

the great Tamerlane the Tartar, built a fineobservatory at Samarcand

in the fifteenth century, and made a greatcatalogue of stars, the

first since the time of Hipparchus.

At the close of the fifteenth century KingAlphonso of Spain employed

computers to produce the Alphonsine Tables(1488 A.D.), Purbach

translated Ptolemy's book, and observationswere carried out in

Germany by Müller, known as Regiomontanus,and Waltherus.

Nicolai Copernicus, a Sclav, was born in1473 at Thorn, in Polish

Prussia. He studied at Cracow and in Italy.He was a priest, and

settled at Frauenberg.  He did not undertake continuous observations,

but devoted himself to simplifying theplanetary systems and devising

means for more accurately predicting thepositions of the sun, moon,

and planets. He had no idea of framing asolar system on a dynamical

basis. His great object was to increase the accuracy of the

calculations and the tables. The results ofhis cogitations were

printed just before his death in aninteresting book, _De

Revolutionibus Orbium Celestium_. It isonly by careful reading of

this book that the true position ofCopernicus can be realised. He

noticed that Nicetas and others hadascribed the apparent diurnal

rotation of the heavens to a real dailyrotation of the earth about

its axis, in the opposite direction to theapparent motion of the

stars. Also in the writings of MartianusCapella he learnt that the

Egyptians had supposed Mercury and Venus torevolve round the sun, and

to be carried with him in his annual motionround the earth. He

noticed that the same supposition, ifextended to Mars, Jupiter, and

Saturn, would explain easily why they, andespecially Mars, seem so

much brighter in opposition.  For Mars would then be a great deal

nearer to the earth than at other times. Itwould also explain the

retrograde motion of planets when inopposition.

We must here notice that at this stageCopernicus was actually

confronted with the system accepted laterby Tycho Brahe, with the

earth fixed. But he now recalled andaccepted the views of Pythagoras

and others, according to which the sun isfixed and the earth

revolves; and it must be noted that,geometrically, there is no

difference of any sort between the Egyptianor Tychonic system and

that of Pythagoras as revived by Copernicus,except that on the latter

theory the stars ought to seem to move whenthe earth changes its

position--a test which failed completelywith the rough means of

observation then available. The radicaldefect of all solar systems

previous to the time of Kepler (1609 A.D.)was the slavish yielding to

Plato's dictum demanding uniform circularmotion for the planets, and

the consequent evolution of the epicycle,which was fatal to any

conception of a dynamical theory.

Copernicus could not sever himself fromthis obnoxious tradition.[5]

It is true that neither the Pythagorean northe Egypto-Tychonic system

required epicycles for explainingretrograde motion, as the Ptolemaic

theory did. Furthermore, either systemcould use the excentric of

Hipparchus to explain the irregular motionknown as the equation of

the centre. But Copernicus remarked that he could also use an

epicycle for this purpose, or that he coulduse both an excentric and

an epicycle for each planet, and so bringtheory still closer into

accord with observation. And this heproceeded to do.[6] Moreover,

observers had found irregularities in themoon's motion, due, as we

now know, to the disturbing attraction ofthe sun.  To correct for

these irregularities Copernicus introducedepicycle on epicycle in the

lunar orbit.

This is in its main features the systempropounded by Copernicus. But

attention must, to state the case fully, bedrawn to two points to be

found in his first and sixth booksrespectively. The first point

relates to the seasons, and it shows astrange ignorance of the laws

of rotating bodies. To use the words ofDelambre,[7] in drawing

attention to the strange conception,

  heimagined that the earth, revolving round the sun, ought always to

  showto it the same face; the contrary phenomena surprised him: to

 explain them he invented a third motion, and added it to the two

 real motions (rotation and orbital revolution). By this third motion

  theearth, he held, made a revolution on itself and on the poles of

  theecliptic once a year ... Copernicus did not know that motion in

  astraight line is the natural motion, and that motion in a curve is

  theresultant of several movements. He believed, with Aristotle,

 that circular motion was the natural one.

Copernicus made this rotation of theearth's axis about the pole of

the ecliptic retrograde (i.e., opposite tothe orbital revolution),

and by making it perform more than onecomplete revolution in a year,

the added part being 1/26000 of the whole,he was able to include the

precession of the equinoxes in hisexplanation of the seasons. His

explanation of the seasons is given on leaf10 of his book (the pages

of this book are not all numbered, onlyalternate pages, or leaves).

In his sixth book he discusses theinclination of the planetary orbits

to the ecliptic. In regard to this thetheory of Copernicus is unique;

and it will be best to explain this in thewords of Grant in his great

work.[8] He says:--

 Copernicus, as we have already remarked, did not attack the

 principle of the epicyclical theory: he merely sought to make it

 more simple by placing the centre of the earth's orbit in the centre

  ofthe universe. This was the point to which the motions of the

  planetswere referred, for the planes of their orbits were made to

 pass through it, and their points of least and greatest velocities

 were also determined with reference to it.  By this arrangement the

  sunwas situate mathematically near the centre of the planetary

 system, but he did not appear to have any physical connexion with

  theplanets as the centre of their motions.

According to Copernicus' sixth book, theplanes of the planetary

orbits do not pass through the sun, and thelines of apses do not pass

through to the sun.

Such was the theory advanced by Copernicus:The earth moves in an

epicycle, on a deferent whose centre is alittle distance from the

sun. The planets move in a similar way onepicycles, but their

deferents have no geometrical or physicalrelation to the sun. The

moon moves on an epicycle centred on asecond epicycle, itself centred

on a deferent, excentric to the earth.  The earth's axis rotates about

the pole of the ecliptic, making onerevolution and a twenty-six

thousandth part of a revolution in thesidereal year, in the opposite

direction to its orbital motion.

In view of this fanciful structure it mustbe noted, in fairness to

Copernicus, that he repeatedly states thatthe reader is not obliged

to accept his system as showing the realmotions; that it does not

matter whether they be true, evenapproximately, or not, so long as

they enable us to compute tables from whichthe places of the planets

among the stars can be predicted.[9] Hesays that whoever is not

satisfied with this explanation must becontented by being told that

"mathematics are formathematicians" (Mathematicis mathematica

scribuntur).

At the same time he expresses hisconviction over and over again that

the earth is in motion. It is with him apious belief, just as it was

with Pythagoras and his school and withAristarchus. "But" (as Dreyer

says in his most interesting book, _TychoBrahe_) "proofs of the

physical truth of his system Copernicus hadgiven none, and could give

none," any more than Pythagoras orAristarchus.

There was nothing so startlingly simple inhis system as to lead the

cautious astronomer to accept it, as therewas in the later Keplerian

system; and the absence of parallax in thestars seemed to condemn his

system, which had no physical basis torecommend it, and no

simplification at all over theEgypto-Tychonic system, to which

Copernicus himself drew attention. It hasbeen necessary to devote

perhaps undue space to the interesting workof Copernicus, because by

a curious chance his name has become sowidely known. He has been

spoken of very generally as the founder ofthe solar system that is

now accepted. This seems unfair, and onreading over what has been

written about him at different times itwill be noticed that the

astronomers--those who have evidently readhis great book--are very

cautious in the words with which theyeulogise him, and refrain from

attributing to him the foundation of oursolar system, which is

entirely due to Kepler.  It is only the more popular writers who give

the idea that a revolution had beeneffected when Pythagoras' system

was revived, and when Copernicus supportedhis view that the earth

moves and is not fixed.

It may be easy to explain the associationof the name of Copernicus

with the Keplerian system. But the time haslong passed when the

historian can support in any way thispopular error, which was started

not by astronomers acquainted with Kepler'swork, but by those who

desired to put the Church in the wrong byextolling Copernicus.

Copernicus dreaded much the abuse heexpected to receive from

philosophers for opposing the authority ofAristotle, who had declared

that the earth was fixed.  So he sought and obtained the support of

the Church, dedicating his great work toPope Paul III. in a lengthy

explanatory epistle. The Bishop of Cracowset up a memorial tablet in

his honour.

Copernicus was the most refined exponent,and almost the last

representative, of the Epicyclical School.  As has been already

stated, his successor, Tycho Brahe,supported the same use of

epicycles and excentrics as Copernicus,though he held the earth to be

fixed. But Tycho Brahe was eminently apractical observer, and took

little part in theory; and his observationsformed so essential a

portion of the system of Kepler that it isonly fair to include his

name among these who laid the foundationsof the solar system which we

accept to-day.

In now taking leave of the system ofepicycles let it be remarked that

it has been held up to ridicule more thanit deserves. On reading

Airy's account of epicycles, in thebeautifully clear language of his

_Six Lectures on Astronomy_, the impressionis made that the

jointed bars there spoken of for describingthe circles were supposed

to be real. This is no more the case thanthat the spheres of Eudoxus

and Callippus were supposed to be real.Both were introduced only to

illustrate the mathematical conception uponwhich the solar,

planetary, and lunar tables wereconstructed.  The epicycles

represented nothing more nor less than thefirst terms in the Fourier

series, which in the last century hasbecome a basis of such

calculations, both in astronomy and physicsgenerally.

[Illustration: "QUADRANS MURALIS SIVETICHONICUS."  With portrait of

Tycho Brahe, instruments, etc., painted onthe wall; showing

assistants using the sight, watching theclock, and recording.  (From

the author's copy of the _AstronomiæInstauratæ Mechanica._)]

FOOTNOTES:

[1] For definition see p. 22.

[2] _Ibid_.

[3] For definition see p. 18.

[4] For definition see p. 18.

[5] In his great book Copernicus says:"The movement of the heavenly

bodies is uniform, circular, perpetual, orelse composed of circular

movements." In this he proclaimedhimself a follower of Pythagoras

(see p. 14), as also when he says:"The world is spherical because the

sphere is, of all figures, the mostperfect" (Delambre,

_Ast. Mod. Hist_., pp. 86, 87).

[6] Kepler tells us that Tycho Brahe waspleased with this

device, and adapted it to his own system.

[7] _Hist. Ast._, vol. i., p. 354.

[8] _Hist. of Phys. Ast._, p. vii.

[9] "Est enimAstronomi proprium, historiam motuum coelestium

diligenti etartificiosa observatione colligere. Deinde causas

earundem, seuhypotheses, cum veras assequi nulla ratione possit

... Neque enim necesseest, eas hypotheses esse veras, imo ne

verisimiles quidem, sed sufficit hoc usum,si calculum observationibus

congruentem exhibeant."