Full text of "Problems in time and space; a collection of essays relating to the earth, physically and astronomically, and cognate matters"

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PEOBLEMS IN TIME AND SPACE.

' Te many twinkling stars, who yet do hold

Your brilliant places in the sable vault
Of night's dominions ! — Planets, and central orbs
Of other systems; big as the burning Sun
Which lights this nether Globe — yet to our eye
Small as the glow-worm's lamp ! To you I raise
My lowly orisons, while all bewildered,
My vision strays o'er your ethereal hosts;
Too vast, too boundless for our narrow mind.
Warped with low prejudices, to unfold,
And sagely comprehend. Thence higher soaring.
Through ye I raise my solemn thoughts to Him,
The mighty Founder of this wondrous maze.
The Great Creator ! "

H. Kirkc-White ;
On a Swrvey of the Heavens.

"Clerk-Maxwell, the great physicist of Cambridge,
was wont to say that progress was symbolized in the
clock, the balance, and the foot-rule ... It is because
this later day has contrived subtle ways of counting and
measuring that we have come to know something of
the way our familiar world is constructed."

Carl Snyder ;
New Conceptions in Science.

PROBLEMS

Time and Space

A COLLECTION OF ESSAYS

RELATING TO

THE EARTH, PHYSICALL\ AND
ASTRONOMICALLY

AND COGNATE MATTERS

F. A. BLACK, F.R.S.E.

AUTHOR OP

"Terrestrial Magnetism and its Cavaes"

and

"Natural Phenomena,"

?• f

T^/

/^'
J

GALL & INGLIS, 31 Henribtta St, CovEirr Gabden

AND EDINBUROB

Works by F. A. BLACK, F.RS.E.

Terrestrial Magnetism. — Suggesting theories explaining the
daily and secular movements of the magnetic needle, the
position of the magnetic poles, &c. 238 pp., with ma})s,
illustrations, &c., 6/- net.

Natural IPhenomena.— Descriptive and speculative Essays in
By-Pa ths of Nature : — " Does the Weather move in Cycles?"
"The Zodiacal Light," &c., Ac. ^384 pp., illustrated, 6/- net.

GALL & INGLIS: London and Edinbubgh.

PREFACE.

The following Essays form a collection rather than a series —
that is to say they are mutually independent of each other
and may be read in any order. This necessarily results in
some repetition when the subject-matter of one Essay trenches
upon that of another. I have thought it preferable, however,
that such repetition should occur, to such an extent as is
requisite to avoid obscurity in relation to the subject specially
dealt with, rather than that certain Essays should be
mutually dependent.

Although the Essays are thus in a measure disconnected,
they are allied in topic. Each relates to, or has some con-
nection with, the Earth in one or other of its diverse aspects.
They might all in fact be fairly comprised within the subject
** Universal Geography " — being the science of the Earth both
as a distinct body in the Universe and as connected with the
Systems of Heavenly Bodies.

More particularly, the Essays may be divided into three
classes — astronomical, chronological (or arithmetical), and
geographical. In the first class I would place the following
Essays, "How the Distance and Size of the Sun were
measured;" "Solar and Sidereal Time;" "The Movements of
the Sun and of the Earth in Space;" and "Twilight and
Dawn." The second class comprises three Essays, "A Simple

vi PREFACE.

Means of Ascertaining the Day of the Week of any Given
Date in the Christian Era;" "The Reform of the Calendar;"
and "Logarithms and their Inventor." The Essays in the
third class — geographical — all relate to the Earth physically,
viz. : — " Measuring the Earth ; " " The Magnetism of the
Earth;" and "Gravitation the Chief Cause of ijae General
Oceanic Circulation."

The sequence adopted gives an Essay of each class in turn.
Thus the subject of the first Eassy is astronomical, that of the
second, chronological ; while the third Essay is comprised in
the class which I have described as geographical.

I have invariably endeavoured to be as clear as possible,
so that anyone, though having little or no knowledge of the
subjects treated, may have no avoidable difficulty in grasping
the meaning. My purpose in this respect will, I trust, be
furthered by the various diagrams.

I have to express my thanks to Dr. C. G. Knott, of
Edinburgh University, for most kindly going over several of
the Essays in manuscript, and to Mr. Norman D. Mackintosh,
C.E., Inverness, for much assistance in connection with the
diagrams.

I am indebted to the Rt. Hon. Lord Napier and Ettrick
for the portrait of his illustrious ancestor, the Inventor of
Logarithms; and to Mr. Andrew Davidson, Sculptor, of
Rome and Inverness, for the portrait of Pope Gregory XIII.,
the Reformer of the Calendar.

I have also to acknowledge my obligations to the
Publishers for many valuable suggestions.

FREDK. A. BLACK.

78 Academy Street, Inverness,
llth Dec. 1909.

CONTENTS.

/. HOW THB DISTANCE AND SIZE OF THE SUN
WERE MEASURED.

Early aspirations to determine Sun's distance —
Relative apparent velocity accepted from earliest
times as proof of relative distance — Estimates
of Sun's distance by Aristarchus, Hipparchus,
Ptolemy, Copernicus, and Kepler — The first fairly
accurate estimate — The growth in our conceptions
of the size of the solar system — Early measure-
ments vitiated by lack of appliances — Method
suggested by Aristarchus still impracticable —
Eclipse method of Hipparchus only recently
rendered capable of testing — The method described
— The size of the Earth as anciently estimated
and as now known — Distance of Moon — Apparent
size of Earth's shadow on lunar disc — Practical
test of the plan of Hipparchus — Terrestrial
diameter — Apparent solar diameter — Apparent
diameter of Earth's shadow on lunar disc —
Moon's mean distance — Meaning of term "ap-
parent diameter " — The lines from centre of Earth
to extremities of solar diameter — The lines bound-
ing the shadow -cone prolonged to Sun — The
junction of these pairs of lines — How the relations

viii CONTENTS.

PAOC

of the respective pairs of lines are affected — The
distance from the Earth at which the junction
occurs — The separation of the lines at the solar
diameter — The distance from centre of Earth to
centre of Sun — Comparison of results with the
accepted figures ... ... ... ... , ... 1-28

//. A SIMPLE MEANS OF ASCERTAINING
THE DA Y OF THE WEEK OF ANY GIVEN
DATE IN THE CHRISTIAN ERA.

Irregularities of the calendar — Rule as to
variation in length of month — Mnemonic rhymes
for determining relatively the week-day opening
the respective months in the same year — Fixing
day of week of given date in another year, past
or future — Hypothetical question in general know-
ledge — " Perpetual Calendars " — Rules regulating
the calendar — Adoption of New Style in British
Dominions — Facts simplifying ascertainment of
day of week of any given date — Determination of
day of the week of given date in current month or
current year — Application of the method to other
periods — Fractional portion of leap-year period —
Practical examples — Allowance for change of
Style — Application of system to foreign countries —
Diversities in regard to adoption of New Style —
Date-difference between Old and New Styles and
its cause — J ulian chronology — Changes in date of
commencement of year — Additional examples of
calculations illustrating the system ... ... 29-55

///. MEASURING THE EARTH

The first scientific attempt to ascertain the size
of the Earth — Method adopted by Eratosthenes —

CONTENTS. ix

rtam

Causes of error in result arrived at — Recent re-
discovery of the Well utilized by Eratosthenes —
Theoretical simplicity of the measurement of
terrestrial circumference or of a parallel of
latitude — Determination of latitude — Determina-
tion of terrestrial circumference from measurement
of parallel — Divergence of Earth's figure from
perfect sphericity — Measurement of ^n arc of
meridian — Measurement of an arc both of latitude
and longitude at about latitude 45° — How the
length of an arc at the equator can be got from
the length of an arc in any other latitude and vice-
versa — Difficulties of actual measurement of a
terrestrial arc — Means adopted — Essential part —
Possibility of measuring the Earth without
dependence on celestial observations — Method
described — Curvature as an indication of the size
of a sphere — Application to the Earth — Ratio
of diameter to curvature — Distinction between
"level" and "horizontal" — Importance of extreme
precision in determination of size of Earth . . . 57-90

ir. SOLAR AND SIDEREAL TIME.

CONTENTS.

PAGE

day distinguished from the sidereal day — Efifects
of precession in changing the apparent positions
of the Stars — The first point of Aries — Celestial
latitude and longitude — Right ascension and
declination — Terrestrial analogies — Use of term
"sidereal day" for time of Earth's rotation —
Difference between sidereal day and mean star
day — Variability of star day — Mean sidereal day
or time of Earth's rotation — Nutation — Aber-
ration — Proper motion — Similarities between solar
and sidereal day — Change in period of Earth's
rotation — Infinitesimal character of the change —
Contrast between the evanescence of humanity and
the constancy of the Earth's rotational movement. 9 1-1 27

THE REFORM OF THE CALENDAR.

Civil calendar based on that of the Romans —
Ancient confusion in civil year — Julian reform of
calendar — Why August has 31 days — "Why the
year begins on 1st January — The introduction of
the Julian system — The use of the system in
ancient chronology — Defects of the system — Pope
Gregory's reform — Incompleteness of the altera-
tion — The date of the vernal equinox — The real
author of the New Style — Adoption of New Style
in Britain and America — Change of commence-
ment of year in British Dominions — Where " Old
Style" still retained and present difference in
Styles — Day of week unaffected by change of
Style — The excellence of the Gregorian reform —
Proposed slight amendment — Practical disadvan-
tages of calendar — Want of harmony between week
and year — Proposed remedy by "non-counting"
days — The re-arrangement of the length of the

CONTENTS. xi

PAGS

montlis — The desirability of year being an exact
multiple of week — How this might be attained —
Application to existing circumstances — Rules
securing harmony between week and year —
Precessional correction — Desirability of changing
commencement of year to Christmas Day — Applic-
ability of new system to months and quarters —
Its applicability to the moveable Feasts — Import-
ance of international agreement in any future
reform of calendar ... ... ... ... 129-171

VI. THE MAGNETISM OF THE EARTH.

3di CONTENTS.

man
hemispheres — The causes of the needle's move-
ments — Early investigations — Gilbert's conjecture
— Barlow's hypothesis — Discoveries by Arago and
Ampere — Application of these discoveries to
terrestrial magnetism — Sun-spots and aurorae—
The Sun as a distributor of electricity — Superficial
character of the Earth's magnetization — Increase
of temperature with descent — Magnetization lost
by heating — Conclusions suggested by the argu-
ment — Magnetism and gravity ... ... 173-201

ril. THE MOVEMENTS OF THE SUN AND OF
THE EARTH IN SPACE.

Progress of knowledge in regard to the move-
ments of the Sun and the Earth — Terrestrial
orbit generally described as an ellipse of slight
eccentricity — The amount of the eccentricity —
The description true if orbit considered merely in
relation to Sun — Otherwise it conveys a false
impression — How this arises — Early speculations
as to the Sun's movement in space — Sir Wm.
Herschel's investigations and his conclusion —
Position of the solar apex — Discovery of two
stellar streams in opposite directions — Their ap-
proximate courses — Their relation to the Sun's
movement — Analogy of stellar drifts to planetary
orbital revolutions — Is there a central source of
stellar attraction 1 — Proper motion of Solar System
— Its velocity — True character of Earth's orbital
revolution — Earth's seasonal change of position —
Plane of the ecliptic — Inequality of Earth's orbital
progress in corresponding periods — Diversity in
estimated velocity of Sun's movement — Analogy
between Earth's revolution around a moving Sun,

CONTENTS. xiii

PAOK

and a satellite's revolution around a moving
planet — Earth's orbit as a movement in space
not an ellipse 203-231

YIIL LOGARITHMS AND THEIR INVENTOR.

Purpose of the invention — Subject treated
chiefly from popular standpoint — First method of
calculating Logarithms purely arithmetical — Early
connection between arithmetic and Logarithms —
John Napier's birth and ancestry — Religious pub-
lication — Anecdote of boyhood — Personal history
— Why he is sometimes called " Lord Napier " —
Son raised to peerage — First literary venture —
Various inventions and practical suggestions —
Publication of volume on Logarithms — Its recep-
tion by scientific world — Surprise of Professor
Briggs — Briggs visits Napier — Napier's system of
Logarithms not that now in general use — Decimal
base of Logarithms first suggested by Briggs —
Practical explanation of Logarithms — Variation
in base illustrated — Extraction of square root —
Advantages of base 10 — Decimal enumeration —
Logarithms of fractions — Calculation of Loga-
rithms — Convenient rules — How the multiplica-
tion or division of a number by its diverse roots
affects the relative Logarithms — Application to
calculation of Logarithms — Tables of Logarithms
— Table abbreviating calculation of Logarithms
and Anti-logarithms — Napier's death and burial —
The decimal point — "Napier's Bones" — Posthu-
mous volume — Purpose and value of Logarithms —
National distinction derived through Napier. 233-273

xiv CONTENTS.

PACK

IX. GRAVITATION THE CHIEF CAUSE OF THE
GENERAL OCEANIC CIRCULATION.

Gravitation in relation to ocean movements
usually associated chiefly with those of a sul)-
ordinate character — More important connection
suggested — Figure of the Earth — Gravitation in
its terrestrial aspect — Terrestrial, solar, and lunar
gravity, and centrifugal force — Effects on the
waters of the ocean — Superficial poleward move-
ment from equatorial regions — Course of the flow
— The gradient from equator to poles — Eastward
drift of poleward movement — Other causes affect-
ing the flow of the water polewards — Temperature
of ocean in polar regions — ^ Counter-flow of water
from polar areas to equatorial regions — Solar and
lunar gravity the inducing causes — The influence
of the Moon — The gradient from the poles to the
equator in relation to Moon — Influence of Sun —
Gradient in relation to Sun — Solar and lunar
attraction on the waters the converse of terres-
trial attraction — Movement of water from the
depths in the polar oceans towards the surface in
tropics — Westward deflection, &c. — Balancing of
converse movements — Effects of Earth's rotation —
Ocean ridges — Temperature — Relative salinity —
Centrifugal force — Ocean temperature and salinity
in tropics — Relative density — Diurnal, periodic
and seasonal variations in the movements of the
waters — Ascensional and descensional circulation
— Prevailing winds in relation to ocean circulation
— Relative temperature and salinity contributing
causes to the general movements primarily in-
duced by gravitation ... ... ... ... 275-301

CONTENTS. XV

PAGE

X. TWILIGHT AND DA WN.

The term "twilight" and its diverse applica-
tions — The term "dawn" — Original use of the
word "twilight" — Its derivation and literal mean-
ing — Distinction between morning and evening
twilight — The direction of the chief effect of
twilight — Cause of twilight — The atmosphere and
its relations to twilight — Duration of twilight —
The apparent daily solar movement in its geograph-
ical relations — Effect of refraction — The Sun's
apparent annual movement — Geographical section
illuminated merely through the size of the solar
disc — Section illuminated through refraction —
The refraction belt — Requisites for determination
of length of day — Geographical separation from
the Sun during exposure to the refraction and
twilight belts respectfully — Obliquity of the
ecliptic — Poleward recession of "the land of the
the midnight Sun " — Calculation of times of sun-
rise and sunset — Example of method adopted —
The equation of time — Explanation of almanac
variations in times of sunrise and sunset — Correc-
tion for longitude — Correction in respect of semi-
diurnal change in Sun's declination — Relation
between times of sunset and sunrise and length of
day and night respectively — Calculation of dura-
tion of twilight — Atmospheric diffusion of light —
The afterglow — Effects of the volcanic eruptions
inKrakatoa 303-343

APPENDIX 345-350

INDEX 351-362

LIST OF TABLES, &c.

FAOB

1. Apparent periods of revolution of Sun and Planets

as observed in ancient times .... 3

2. Estimates of Sun's distance, from early times to

present date ....... 8

3. Amount of change in the day of the week between

corresponding dates in different months . 36

4. The divergence between the proposed new system

of the Calendar and the present system, year by
year for a period of four hundred years (com-
mencing 1st January 1928 and ending 31st
December 2327, both dates inclusive) . 152-161

5. Mean annual ratio of change in declination of the

magnetic needle in London between 1550 and
1657 (in four periods) 181

6. Early calculations of position of solar apex . .211

7. More recent calculations of position of solar apex , 212

8. Estimated positions of apices of stellar drifts . .215

9. Tables for finding the Logarithms of Numbers and

the Numbers corresponding to Logarithms by
simple arithmetical methods . . . 268-269

10. The length of Twilight in different latitudes and

seasons 340

LIST OF ILLUSTRATIONS AND
DIAGRAMS.

1. The Well of Eratosthenes . G^ Frontispiece

2. Method by which Aristarchus endeavoured to

measure the distance of the Sun ... 4

3. The Universe according to Ptolemy . . ^ 5

4. Relative position of Sun, Earth, and Mars, with

Mars in " opposition " to the Sun ... 6

5. The Copernican conception of the Universe . 9 6

6. Method adopted by Cassini in measuring the

distance of the Sun ...... 7

7. Method suggested by Hipparchus for measuring

the distance of the Sun ..... 10

8. An annular eclipse of the Sun . . . .14

9. A partial eclipse of the Moon showing ill-defined

margin of shadow . . . . . S*) 14

10. Lines from Earth's centre to extremities of solar

diameter . . . . . . . .17

11. Enlargement of dark shadow-cone in passing from

the Moon to the Earth in a lunar eclipse . . 20

12. Diagram illustrating the calculation of the ratio in

length between lines from Earth's centre to ex-
tremities of solar diameter, and prolongations
towards the same point of lines bounding the
shadow-cone in a lunar eclipse . . . .23

xviii LIST OF ILLUSTRATIONS AND DIAGRAMS.

PAOB

13. Diagram illustrating (1) the mode of calculation of

the distance to which lines from the centre of
the Earth to the respective extremities of the
Sun's diameter must be prolonged, to unite with
corresponding prolongations of lines bounding
the shadow-cone in a lunar eclipse, and (2) the
mode of calculation of the separation of the
respective pairs of lines on reaching the Sun . 25

14. Diagram illustrating method of calculating the

distance separating the Earth and the Sun —
centre to centre ...... 26

15. Method adopted by Eratosthenes to determine the

size of the Earth ...... 60

16. Triangle illustrating error in measurement of

terrestrial arc by Eratosthenes . . .61

17. Sketch-Map showing relative positions of Alexandria

and Syene ........ 62

18. Sketch-Map illustrating measurement of terrestrial

arc between Quito and the Galapagos Islands . 67

19. Sketch-Map illustrating measurement of terrestrial

arc between Greenwich and Cardiff . . .69

20. Sketch-Map illustrating measurement of terrestrial

arc between The Balearic Islands and Castellon . 73

21. Sketch-Map illustrating measurement of terrestrial

arc across the Firth of Forth at Leven . . 76

22. Sketch-Map illustrating measurement of terrestrial

arc on the 45th parallel, at the Mouth of the
Danube 77

23. Diagram illustrating amount of curvature in a

distance of 1700 miles on the surface of the Sun,
Earth, and Moon, respectively .... 83

24. Earth's orbital movement in relation to the Sun,

showing constancy of direction of terrestrial axis. ^' 96

25. Movement of the Earth in an arc of its orbit, illus-

trating difference between period of rotation and
length of solar day 97

LIST OF ILLUSTRATIONS AND DIAGRAMS, xix

PAOR

26. The relation between the plane of the Earth's

equator and the plane of the Earth's orbit . 98

27. Diagram illustrating how there is a difference of

exactly one day between the solar year and the
sidereal year . . . . . . .103

28. The path traced out by the northern extremity of

the Earth's axis owing to the precession of the
equinoxes ....... ® 104

29. Conical circuit described by the Earth's axis in

about 25,868 years through the precession of the
equinoxes ........ 108

30. Celestial latitude and longitude and, right ascension

and declination . . . . . . .110

31. Diagram illustrating the variation in length of star

day in relation to stars situated within the pro-
cessional circuit described by the Earth's axis . 120

32. The effect of nutation on the processional circuit

described by the Earth's axis (p. 122) . ^ 104

33. Sketch illustrating effect of aberration of light . 123

34. Diagram illustrating effect smalogous to aberration

of light . ® 123

35. Effect of proper motion on stellar appearance . V^ 124

36. Julius Caesar, the Inaugurator of the Julian system

of Chronology ® 132

37. Pope Gregory XIIL, Reformer of the Calendar ^^^ 168

38. Direction indicated by the compass needle in London

at various dates . . . . . ^182

39. Lines of magnetic declination in 1907 . . H^ 184

40. Similarity of action of the magnetic needle as

regards dip on the surface of the Globe and on a
magnetic bar . . . . . . .186

41. Dip of magnetic needle in London at various dates . 187

42. Lines of equal magnetic dip in 1907 . . Hr* 188

43. Vibrations of magnetic needle during a magnetic

storm <S 191

XX LIST OF ILLUSTRATIONS AND DIAGRAMS.

I'AOK

44. Magnetization of a steel needle by right-handed

and left-handed spiral coils . , . .193

45 "Star Drift" in the constellations of Cancer and

Gemini ^217

46. The character of the Earth's orbital revolution as

a movement in space . . . . . *^ 222

47. The seasonal variation of the Earth's orbital position

and of the plane of the terrestrial equator in
relation to the Sun's movement in space . ^ 224

48. The path of the Moon around the Earth . ^^I^ 228

49. John Napier of Merchiston, Inventor of Logarithms '' 240

50. Memorial Tablet to John Napier of Merchiston, in

St. Cuthbert's Church, Edinburgh . . ^ 270

51. Diagram illustrating method of finding the differ-

ence between the Moon's mean distance from (1)
the geographical poles, and (2) the geographical
position at the equator turned towards the Moon
for the time being 290

52. Diagram showing the diversity according to

geographical situation in the relation of the
waters of the Ocean to the Sun or the Moon . 292

53. Sketch-Map illustrating Daylight, Twilight and

Night ^313

54. Sunshine, Twilight, and Darkness, at (1) The

Equinoxes, (2) The Summer Solstice and (3)
The Winter Solstice ^316

55. Diagram illustrating method of calculating the

length of the Day and the duration of Twilight ^i^ 325

56. Diagram illustrating method of calculating the

length of the Day and the duration of Twilight .'''* 336

HOW THE DISTANCE AND SIZE OF
THE SUN WERE MEASURED.

SYIf^OPSIS.

Early aspirations to determine Sun's distance —
Relative apparent velocity accepted from earliest
times as proof of relative distance — Estimates
of Sun's distance by Aristarchus, Hipparchus,
Ptolemy, Copernicus, and Kepler — The first fairly
accurate estimate — The growth in our conceptions
of the size of the solar system — Early measure-
ments vitiated by lack of appliances — Method
suggested by Aristarchus still impracticable —
Eclipse method of Hipparchus only recently
rendered capable of testing — The method described
— ^The size of the Earth as anciently estimated
and as now known — Distance of Moon — Apparent
size of Earth's shadow on lunar disc — Practical
test of the plan of Hipparchus — Terrestrial
diameter — Apparent solar diameter — Apparent
diameter of Earth's shadow on lunar disc —
Moon's mean distance — Meaning of term "ap-
parent diameter " — The lines from centre of Earth
to extremities of solar diameter — The lines bound-
ing the shadow - cone prolonged to Sun — The
junction of these pairs of lines — How the relations
of the respective pairs of lines are affected — The
distance from the Earth at which the junction
occurs — The separation of the lines at the solar
diameter — The distance from centre of Earth to
centre of Sun — Comparison of results with the
accepted figures.

HOW THE DISTANCE AND SIZE OF
THE SUN WERE MEASURED.

Since men first began, many thousands of years ago,
to gaze with wonder and awe on the glories of the
heavens, not a few, doubtless, have aspired — vaguely
and gropingly it may be — to determine the distance of
the great centre of our system. Some of the results
arrived at in the early efforts to solve this grand pro-
blem, absurdly inaccurate though we now know them to
be, were not without a certain amount of reason.

Thus from very early times relative rapidity of
apparent motion in the heavens was accepted as proof
of relative distance. The argument certainly appeals to
the intelligence. If one heavenly body appears to us to
be moving more rapidly than another, we have in this,
evidently, some indication of its greater proximity. It
was found that the Sun and the different planets made
the circuit of the heavens in approximately the follow-
ing periods: —

Saturn ... ... ... 29| years

Jupiter

Mars ..,
The Sun
Venus
Mercury
The Moon

225

88
27i

days

4 HOW THE DISTANCE AND SIZE OF

Here, then, was evidence of relative distance, the
Earth being accepted as the centre of the system. Thus
Ptolemy, who lived in the second century of our era, in
following the ^'ancient mathematicians" represents the
universe in this manner, the Moon being nearest to the
Earth, then, in their order. Mercury, Venus, the Sun^
Mars, Jupiter, and Saturn, and then "the firmament of
stars."

The most ancient estimates of the distance of the
Sun were purely relative. Aristarchus of Samos, who
lived in the third century B.C., and who is believed to
have been the first astronomer to maintain that the
Earth rotates on its axis and revolves around the Sun —
a belief which later astronomers abandoned — calculated,
by an ingenious method, that the distance of the Sun
was from 18 to 20 times that of the Moon. This result
was arrived at by forming a triangle connecting the
Earth, the Sun, and the Moon, when the Moon was
exactly half-illuminated, the triangle thus having a right
angle at the Moon, and the angle to be determined being

Method by which Aristarchus endeavoured to measure the
distance of the Sun. By determining the angle at a, the dis-
tance of the Sun might be fixed.

that between the two sides of the triangle meeting at
the Earth. His method was geometrically sound, but it
was impracticable; as, even with modern appliances, it
is impossible, owing to the irregularities of the lunar

THE SUN WERE MEASURED. 5

surface, to determine with accuracy the moment when
the Moon is exactly half -illuminated. As the angle at
the Sun is exceedingly small, a very slight error in the
determination of the time of the semi-illumination of
the Moon completely vitiates the final results. In reality
the Sun's distance is nearly 400 times that of the Moon,
instead of between 18 and 20 times as calculated by
Aristarchus. As the mean distance of the Moon is about
238,840 miles, the result arrived at by Aristarchus, if we
accept the mean of his estimate — being 19 times the
distance of the Moon — would make the distance of the
Sun about 4,537,960 miles, say 4| millions of miles.
This, we may accept, as the earliest estimate of the
Sun's distance, having any scientific value.

Hipparchus, who lived about a century after Aris-
tarchus (from, it is supposed, 190 to 120 B.C.), and who
made great advances in astronomy, seems to have deter-
mined that the distance of the Sun was about 1200
times the radius of the Earth. Taking the terrestrial
radius at 4000 miles, this would make the distance of
the Sun about 4,800,000 miles. This calculation Ptolemy,
about two hundred and fifty years later, increased to
1210 terrestrial radii, being, we may take it, about
4,840,000 miles.

These estimates of the Sun's distance remained prac-
tically unaltered for the next fourteen centuries. Coper-
nicus, in the sixteenth century, revolutionized the con-
ception of the universe by placing the Sun at the centre
of our system, and making the Earth revolve around it,
as had been maintained to be the case by Aristarchus
about eighteen hundred years previously. Copernicus
estimated the Sun's distance from the Earth to be 1500

6 HOW THE DISTANCE AND SIZE OF

times the terrestrial radius, thus making it about
6,000,000 miles.

Kepler, early in the seventeenth century, again in-
creased the estimated distance of the Sun, although for
little other than imaginative reasons. His conclusion
was that the distance of the Sun must be dbout three
times as great as had previously been estimated. As
this change was made on the calculation of Hipparchus,
the estimated distance of the Sun was thus increased

Relative position of Sun, Earth, and Mars, with Mars in
" opposition " to the Sun.

to 3600 terrestrial radii, being equivalent to about
14,400,000 miles.

It was about the year 1673, being forty-three year*
after the death of Kepler, that the first fairly accurate
estimate of the distance of the Sun was arrived at.
The basis of the new calculation was the observation,
from two geographical positions, of Mars, when at its
nearest distance to the Earth. As Mars moves around
the Sun in an orbit outside that of the Earth, the planet,
when at its nearest distance, is on the meridian at mid-
night, and the Earth is then situated between the Sun
and Mars, on a straight line joining the three bodies.
The places from which the planet was at the same time

THE SUN WERE MEASURED. r

observed were Cayenne, in French Guiana, and Paris, the
places being separated by about 4,500 miles. It was
found under these circumstances that there was distinct
parallax in relation to the planet, that is to say that its
celestial position as seen from the widely separated
stations, differed quite appreciably in relation to the
neighbouring stars. Thus the two angles at the base of
a triangle joining the two stations to each other and

I^AR*

Method adopted by Cassini in measuring the distance of the Sun.*

each of them to Mars were obtained, and the base of the
triangle — being the terrestrial chord joining the two
stations — was also easily obtainable, so that the triangle
could readily be solved, and the distance of Mars when
at its nearest to the Earth determined. By then making
use of the ascertained distance of Mars, it was possible,
from the planet's known orbit around the Sun, and the
fact, discovered by Kepler, that the squares of the
periodic times of the planets are proportional to the
cubes of their mean distance from the Sun, to calculate

* A similar method, applied to the minor planet Eros at its next
favourable opposition, is expected to furnish a more accurate determina-
tion of the Sun's distance than any yet made.

S HOW THE DISTANCE AND SIZE OF

the distance of the Sun from the Earth. These investi-
gations were made by Caasini, who at the time was the
French astronomer-royal. He determined the distance
of the Sun to be about 87,000,000 miles, being about 5|
millions of miles le.ss than the true distance.

Since then, the efforts of astronomers have been
directed to the more precise solution of this important
problem with ever-increasing refinement of detail, and
it is now definitely known that the mean distance of the
Sun is about 92,897,000 miles, subject to a plus or minus
correction probably not exceeding 200,000 miles.

Thus the estimates of the Sun's distance have
gradually increased in the course of the ages, a great
bound upward taking place when Cassini faced the
problem by purely scientific methods. Thus also our
conceptions of the bounds of the solar system have been
correspondingly enlarged. The following statement
shows summarily the estimates to which we have
referred : —

Date. Name. Estimate of Stm's distance.

3rd Century B.C. Aristarchus.

(19 times the distance of the Moon) 4,500,000 miles.
2nd Century B.C. Hipparchus.

(1200 radii of the Earth) 4,800,000 „
2nd Century a.d. Ptolemy

(1210 radii of the Earth) 4,840,000 „
About 1620 Kepler

(About 3600 radii of the Earth) 14,400,000 „

„ 1673 Cassini 87,000,000 „

Present Day Various Astronomers 92,897,000 „

It might very naturally be argued that the slow
progress, from what we now know to be ridiculously
inadequate estimates, towards the measure of precision
which characterizes the estimate of the present day, is

THE SUN WERE MEASURED. 9

symbolical of a similar gradual advance in the mental
capacity of the race. There is, however, no justification
for any such contention. Strange as it may appear,
certain of the methods suggested in the most ancient
times for the determination of the distance of the Sun
were scientifically just as correct as the methods made
use of in our own day. The advance in knowledge
which has come with the passing centuries does not
seem to afford any argument in favour of a corresponding
advance in mental power. The "grey matter" is to-day
essentially as it was in the time of Aristarchus, although
knowledge is now more general and the average capacity
is consequently raised.

How then does it come about that the early estimates
of the Sun's distance were so utterly erroneous ? The
explanation seems to be simply that the early astro-
nomers were without the requisite appliances to cany
out their designs with the necessary precision. They
were capable of grasping the methods which had to be
adopted, but the tools for carrying out their mental
conceptions were awanting. Science had to wait for
art. And thus it has been through all the ages. Science
and art have had to go hand in hand in the gradual
advance of knowledge.

Although many centuries have elapsed since Aris-
tarchus put forward his ingenious and geometrically
accurate method of solving the problem of the Sun's
distance on the basis of triangulating the Moon when
exactly half-illuminated, we are still unable to utilize
his method practically, as even yet we cannot determine
the exact moment of semi-illumination in view of the
irregularities of the lunar surface. Art in this particular

10 HOW THE DISTANCE AND SIZE OF

lias lagged far behind science, and has rendered the pro-
posed method quite valueless.

Following on Aristarchus, Hipparchus, about twa
thousand years ago, put forward a method of determin-
ing the Sun's distance. During nearly two milleniums
this method, like that of his great predecessor, proved
impracticable. It was not until the nineteenth century
that art had advanced sufficiently to render it possible
to put to the test the suggestions made by this great
astronomer in the second century before our era. It is
therefore interesting in this late age, with our present

Method suggested by Hipparchus for measuring the distance of the Sun.

day knowledge, to examine the plan put forward by
one who has been well named "the father of modern
astronomy."

The method proposed by Hipparchus for determin-
ing the distance of the Sun is characterized by a de-
lightful simplicity which appeals to persons without
any astronomical knowledge. It is indeed based on
matters of common life. Supposing, reasoned Hippar-
chus, that we have three things, (1) a light of any
description; (2) an opaque body intercepting the light;
and (3) a third body, situated in such a direction as to
receive the shadow of the opaque body produced by the
interception of the light; then, provided we know (1)
the actual dimensions of the opaque body; (2) the

THE SUN WERE MEASURED. 11

apparent, or angular, dimensions of both the light and
the shadow; and (3) the distance between the shadow
and the opaque body, we can determine both the distance
and the dimensions of the light.

If, for instance, a jet of gas is burning at one side of
a room and the shadow of an intercepting object is
thereby cast upon the opposite wall, we can, if we know
(1) the width of the intercepting object at any particular
part; (2) the corresponding apparent or angular width
of the light and of the shadow in relation to the inter-
cepting object; and (3) the actual distance between the
object and its shadow, determine both the distance of
the light from the intercepting object and the actual
size of the jet. We have, as it were, a cone connecting
the light and the shadow, the taper of which is decided
by the relative dimensions of the light and of the object
casting the shadow. If the light is smaller than the
object the cone as it departs from the borders of the
light is an enlarging one, and the shadow is corres-
pondingly larger than the object casting it. If, on the
other hand, the light is larger than the object the cone
narrows from the light to the shadow, and the latter is
therefore correspondingly smaller^than the object. The
argument appeals to reason.

Now it is manifest that during an eclipse of the
Moon the relation of three such bodies — a light-giving
body, an intervening object, and a body in the resulting
shadow — is brought about in the case of the Sun, the
Earth, and the Moon. We have, therefore, the requisites
called for by Hipparchus to allow of a determination of
the distance and the size of the Sun, provided only that
we can ascertain (1) the size of the Earth; (2) the ap-

12 HOW THE DISTANCE AND SIZE OF

parent, or angular, size of both the Sun and the shadow ;
and (3) the distance of the Moon from the Earth.

As regards the first of these requisites — the size of
tlie Earth — it is the case that shortly before the time of
Hipparchus, say about the year 200 B.C., or rather
earlier, the first rude but ingenious attempt had been
made to estimate the size of the Earth.* This was
made by Eratosthenes of Alexandria, who, by measur-
ing the angular distance of the Sun from the zenith in
Alexandria when it was known to be in the zenith at
Syene about seven degrees south of Alexandria, com-
puted, from the known length of these seven degrees,
and on the confident assumption that the Earth was
round, that the circumference of the Earth was 250,000
stadia. The stadium is believed to have been equiva-
lent to about 606 feet 9 inches according to the British
standard. This would make the Earth's circumference,
in accordance with the measurement of Eratosthenes,
about 28,700 miles. This, as we now know, is nearly
four thousand miles greater than the mean circumfer-
ence of the Earth — (24,857 miles). Still, the measure-
ment cannot be said to be grossly inaccurate, and there
is no reason to doubt that it must have been known to
Hipparchus.

In regard to the third requisite — the distance of the
Moon from the Earth — we know that Hipparchus him-
self estimated the Moon's distance to be about fifty-nine
times the radius of the Earth, which is, in fact, very
nearly the Moon's mean distance from the Earth. If we
accept, however, as Hipparchus probably did, the result
arrived at by Eratosthenes for the length of the Earth's
* See Essay No. III.

THE SUN WERE MEASURED. 13

circumference, the length of the terrestrial radius would
be about 4570 miles. This would make the distance of
the Moon as estimated by Hipparchus about 269,630
miles. The mean distance of the Moon is now known
to be about 238,840 miles, so that, like the estimated
size of the Earth, this ancient estimate is too great.
Still, however, like the other, it is not excessively-
erroneous, the difference of 30,790 miles being less than
thirteen per cent, of the Moon's mean distance. It is
noticeable also that both in the measurement of the
Earth and in the measurement of the Moon's distance,
the error is on the side of excess, the distance in
both cases being considerably over-estimated.

Yet, as we have noticed, Hipparchus, proceeding by
a sound method, made a grossly inadequate estimate of
the Sun's distance notwithstanding these excess errors
in his premises. How, then did this come about ?

It is very evident that the fatal stumbling block has
to be sought in the second requisite — and that it lies in
the determination of the apparent or angular size of
the Earth's shadow on the lunar disc. The first difficulty
in this connection arises from the fact that the Earth's
shadow at the Moon's distance is greater than the size
of the Moon. Thus, although the variation in the
Moon's distance is extreme — the distance ranging from
about 252,948 miles as a maximum, to about 221,593
miles as a minimum, a difference of over 31,000 miles —
we do not have in the case of the Moon, as in the case
of the Sun, what are known as " annular " eclipses, that
is to say eclipses in which the shadow is insufficient to
cover the disc, the border of which is consequently
visible as a surrounding ring. Had this been the case

14 HOW THE DISTANCE AND SIZE OF

the measurement of the shadow would during an annular
lunar eclipse be somewhat simplified. As it is, however,
the measurement of the apparent diameter of the Earth's
shadow on the Moon is possible only during partial
eclipse, and as the shadow is in rapid motion across the
disc the difficulties of measurement are extreme. Not
only is this so but the difficulties are intensified by the

An annular eclipse of the Sun.
The dark portion represents the Moon, which appears smaller
than the Sun, so that the Sun shines around it like a bright ring.

great irregularity of the surface of the Moon and the
consequent ill-defined margin of the moving shadow.
As if this were not enough, the difficulties in the deter-
mination of the absolute edge of the shadow are further
increased by the fact that the terrestrial atmosphere
through the partial and irregular interception of the
light of the Sun contributes to the formation of a still
more indefinite margin to the shadow.

The intricacies arising from these complex causes
were insuperable at the time of Hipparchus, particularly
in view of the extreme nicety of the problem involved,
and, consequently, the estimate which this brilliant as-
tronomer of ancient times made of the Sun's distance
was devoid of value.

In this we have but another example of science
having to wait for the development of art. If Hip-
parchus had had at his disposal appliances of the re-

THE SUN WERE MEASURED. 15

quisite precision to enable him with reasonable accuracy
to have faced the various difficulties of the problem, we
can scarcely doubt that some approach to a reasonable
conception of the vast distance of the Sun from the
Earth, and consequently of the stupendous size of the
centre of our system would have been acquired even
before the dawn of the Christian era.

In truth, however, the determination with accuracy
of the dimensions of the Earth's shadow on the Moon's
disc during an eclipse is a problem which, even in our
own day, is involved in extreme difficulty, notwith-
standing the degree of perfection which has now been
reached in scientific appliances. The various intricacies
to which we have referred make this fact abundantly
evident. Thus, even to this day, the determination
of the distance and size of the Sun by means of the
method suggested by Hipparchus has never been
undertaken.

While this is so, it is interesting to ascertain, now
that the requisite data have become available, how these
fit in with the plan of Hipparchus; and to determine the
size and distance of the Sun as Hipparchus, by the
method we have described, would have done, had the
data been at his disposal. In this enquiry it is desir-
able in view of the various measurements required as
data — such, for instance, as the diameter of the Earth
(the exact length of which depends on whether the
diameter is equatorial, or meridional, or otherwise) and
the distance of the Moon — being themselves indefinite,
to accept the mean as the required amount in each case,
with the purpose of thereby obtaining the mean distance
of the Sun.

16 HOW THE DISTANCE AND SIZE OF

The three requisites then, to start with, are (1) the
size of the Earth; (2) the angular extent or apparent
diameter of the Sun and of the shadow; and (3) the
distance of the Moon.

By the size of the Earth we, of course, in this
connection, mean its diameter, and as this is, per-
haps, the most important factor in the problem it is
necessary to exercise great care in its determination,
as any error must necessarily be greatly magnified
in the final result. Bessel, a Prussian astronomer,
who occupied a position of eminence in the scientific
world in the first half of the nineteenth century,
concluded from exhaustive measurement and calcula-
tion, that the equatorial diameter of the Earth was
about 41,847,192 feet in length, and the polar
diameter about 41,707,324 feet. These figures have
been accepted as fairly reliable by Sir John Herschel,
and more recently by Sir Archibald Geikie. On the
basis of these figures, the length of the mean diameter
of the Earth is about 41,777,258 feet, which is equivalent
to 7912-35947 miles.

We come now to the second requisite — the apparent
diameter of both the Sun and the shadow. It has been
determined, as the result of observations made at the
Royal Observatory, Greenwich, during thirty-three
years, that the mean apparent diameter of the Sun
is about 32' 2*36". We may, therefore, accept that
extent without question. As regards the apparent
diameter of the dark shadow-cone at the Moon's mean
distance, when an eclipse occurs with the Earth at its
mean distance from the Sun, there is more uncertainty,
and there is some discrepancy between authorities.

THE SUN WERE MEASURED. 17

The weight of evidence, however, is in favour of the
apparent diameter of the cone in the given circum-
stances, calculated as from the centre of the Earth,
measuring 1* 22' 8"57" and we shall, accordingly, accept
this as correct.

The third requisite is the mean distance of the
Moon from the Earth, and it is agreed by recent writers
that this may be accepted as being 238,840 miles.

Let us now, on these particulars, apply ourselves to
the solution of the problem of determining the mean
distance separating the Earth from the Sun, and also
the size of the Sun.

When we say that the mean apparent diameter of
the Sun is 32' 2*36" what is really meant is that if the
Sun were observed from a distance corresponding to
that occupied by the Earth's centre (when the Earth

■» tiv. /hum m» Sm-fh,

Linea from Earth's centre to extremities of solar diameter.*

is at its average distance from the Sun) it would occupy
32' 2'36'' on the circumference of a circle having as its
centre the point of observation and as its radius the
mean distance separating the Earth's centre from the
extremities of the solar diameter. Thus, with the Elarth
at its mean position, two radii extending from the

•Here and in the following diagrams we assume, for the sake of
simplicity, that linea drawn from the Earth to opposite sides of the Sun
would touch the Sun at the respective extremities of a solar diameter.
Owing to the sphericity of the Sun, contact would really ocour hefort the
lines would reach these extremities. But the difference between the actual
and the assumed points of contact does not affect the argument.
C

18 HOW THE DISTANCE AND SIZE OF

centre of the Earth to the opposite extremities of the
Sun's diameter, would form an angle with each other at
the centre of the Earth of 32' 2-36". It follows that
these radii, on reaching the surface of the Earth, would
enclose an arc of the terrestrial surface of the same
angular extent. Consequently, the distance separating
the radii at the surface of the Earth would be the
chord of a terrestrial arc of 32' 2*36". What then, is
the length of such a chord ?

Assuming, as is of course necessary, that the Earth
is a perfect sphere — which, although not exactly the
case, we are, in the present circumstances, justified in
doing, from the fact that we are proceeding on the
Tnean measurements — we can readily determine the
length of the given chord by the aid of trigonometrical
ratios. The chord of any arc is twice the "natural sine"
of half the arc. By consulting a Table of these sines,
as given in most Mathematical Tables, we can note the
sine corresponding to one half of 32' 236" — being 16'
I'lS" — and by doubling this we shall get the required
chord, on the ratio of radius representing 1. The sine
of 16' 1-18" is -0046599, so that the chord is 0093198.
As the radius in the present case is not 1 but 3956"179735
— being one-half of the terrestrial diameter — we have
now to multiply the chord by this amount. This makes
the length of the chord 368708039 miles.

We find in this way that the two radii, in their
course from the centre of the Earth to the opposite
extremities respectively of the solar diameter, must
diverge, by the time they reach the Earth's surface, to
the extent of 36*8708039 miles. A divergence of this
amount in a distance of 3956'179735 miles will be found.

THE SUN WERE MEASURED. 19

by simple proportion, to represent a divergence of
•93198000 of a mile— being 1640-2848000 yards— in one
hundred miles. It is evident that this divergence must
continue at exactly the same rate until at last the radii
in their prolongation touch the respective extremities
of the diameter of the solar orb.

Let us now turn to the consideration of the dark
shadow-cone in a lunar eclipse with Sun and Moon at
mean distance from the Earth. We have seen that at
the distance of the Moon the apparent diameter of the
shadow-cone is 1° 22' 8*57". Let us ascertain what this
represents in miles. Supposing a circle to be formed
having as its centre the centre of the Earth, and as its
radius the distance from the Earth's centre of the re-
spective extremities of the diameter of the shadow as
appearing on the Moon in the specified circumstances,
then the apparent diameter of the shadow-cone
would occupy an arc of 1* 22' 8*57" on the circum-
ference of the circle. Evidently, therefore, the real
diameter of the cone is the chord corresponding to an
arc of this extent. We can determine the length of this
chord in the same way as we determined the length of
the terrestrial chord of the arc representing the Sun's
apparent diameter. Thus, we have to ascertain the
sine of one-half of 1° 22' 8-57"— being 41' 4'285"— and
double it, and then multiply by the Moon's mean
distance. The sine of 41' 4285" is '01194687. Doubling
tliis — and dropping the last figure as uncertain — we get
•0238937. Multiplying this by 238,840 we get 5706-77
as the diameter in miles of the shadow-cone at the
Moon's mean distance during a lunar eclipse with the
Sun at its mean distance from the Earth.

20 HOW THE DISTANCE AND SIZE OF

Supposing now that from each extremity of the
diameter of the cone at the Moon's mean distance a line
passes to the corresponding extremity of the diameter
of the Earth, it is evident that these two lines must
mark the margin of the cone. In the given circum-

EM=Mean distance between Earth and Moon, centre to centre=238,840
miles.

ABD is apparent or angular diameter of dark shadow-oone at Moon'a
mean distance =1* 22' 8'57".

A MD= Actual diameter of dark shadow-cone at Moon's mean distances
570677 miles.

BC= Actual diameter of Earth=7912-35947 miles.

The diameter of the shadow-cone thus increases in length by »bout (say)
2205-58947 mUes in 238,840 miles.

stances the distance separating the lines at the place
of the Moon must represent the diameter of the shadow
in that position, and the distance separating the linea
on their arrival at the Earth must be the mean terres-
trial diameter. As the latter is 7912-35947 miles in
length and as the diameter of the cone at the Moon's
mean distance is 570677 miles in length we find that
in their progress from the Moon to the Earth the
divergence of the lines increases by, say, 2205 58947
miles. As this increase in the separation of the two
lines occurs in a distance of 238,840 miles, we can
easily determine the ratio of increase. It will be found

THE SUN WERE MEASURED. 21

by proportion that a divergence of 2205'58947 miles
in 238,840 miles is equivalent to -92345900 of a mile
— ^being 1625*2878400 yards — in one hundred miles.

Although the shadow which causes the lunar eclipse,
if followed from the Moon earthwards, has its termina-
tion at the Earth, we may imagine the prolongation of
the lines bounding the shadow-cone from the Earth to
the Sun. As the Sun is the source of the shadow it is
clear that these lines, if prolonged to the Sun, would
respectively touch the opposite extremities of the solar
diameter just as they touch the opposite extremities of
the terrestrial diameter. As the shadow, however,
necessarily, when regarded in this way, ends at the
Earth, the lines in their prolongation would, of course,
not mark the margin of a shadow-cone but rather of a
cone of light — the light whose interception by the Earth
results in the formation of the shadow. Evidently,
however, the bounding lines would continue to diverge
at the same rate in their prolongation from the Earth
onwards as in their passage from the Moon to the
Elarth.

Thus we have two pairs of lines passing from the
Earth to the Sun, one pair passing from the centre of
the Earth and the other from the extremities of the
terrestrial diameter. Both pairs are directed to the
respective extremities of the same solar diameter, each
radius in its prolongation being directed towards the
same extremity of the Sun's diameter as the line bound-
ing the cone which, on reaching the Earth's surface, the
radius adjoins. We may describe the extended radii as
the inner pair of lines, and the prolongations of the linea
bounding the shadow-cone as the outer pair.

22 HOW THE DISTANCE AND SIZE OF

We have seen that the inner lines diverge at the rate
of -93198000 of a mile, or 16402848000 yards, in every
hundred miles of their progress sunwards, while we see
that the outer lines diverge at the slightly less rapid
rate of 92345900 of a mile, or 16252878400 yards in
the same distance. Thus the respective lines of the
inner pair, through their more rapid divergence, ap-
parently approach the corresponding lines of the outer
pair by the difference between the two rates of diver-
gence, that is to say they apparently approach them by
•00852100 of a mile, being 149969600 yards in every
hundred miles. As the destination of the adjoining
lines of each pair is identical, being the same extremity
of the Sun's diameter, the position of the Sun is ap-
parently fixed as the place at which each inner line
shall, through its excess divergence, touch the corres-
ponding outer line. It would seem, therefore, that all
we have to do to fix the place of the Sun is to find what
distance is required to make up 7912'35947 miles — that
being the separation of the outer lines through their
position at the respective extremities of the terrestrial
diameter when the inner lines are still unseparated at
the centre of the Earth — at the rate of 00852100 of a
mile in a hundred miles. This is practically the case
although it is not precisely so, and it is desirable that
the slight complication which arises in this respect
should not be overlooked.

It is evident that, as the rate of divergence of the
lines of each pair differs, the adjoining lines of the inner
and outer pairs respectively are not exactly parallel.
No doubt the deviation from the parallel is very small,
but still it must follow from the want of parallelism

THE SUN WERE MEASURED.

that if, say, at a distance of 10,000 miles from the
Earth's centre, measuring along one of the extended
radii, a line parallel to the terrestrial diameter is made
to intersect this extended radius, and also the adjoining
line of the other pair, the length of the outer line lying
between the extremity of the terrestrial diameter and
the point of intersection will not be exactly 10,000
miles. Let us find what is actually the length of the
outer line thus cut off. We can do this by calculation
based on the angles which the respective adjoining lines
^/o ,_.._„ ,., _ ..

AC =10,000 milcB.
BC=Terre8trial Radius.
DA is parallel to Terrestrial Baclius.
Length of DB is 9999 9976989 miles.

The dotted line shows direction of a straight line towards centre of Sun.

Diagram illustrating the calculation of the ratio in length between
lines from Earth's centre to extremities of solar diameter, and
prolongations towards the same point of lines bounding the shadow-
oone in a lunar eclipse.

make with the terrestrial diameter and with the inter-
secting line parallel thereto. We find that the length
of the outer line thus cut off is 99999976989* miles in
the distance from the terrestrial diameter corresponding
to 10,000 miles when measured on the inner line. Pro-
portionally, 100 miles measured on the inner line would
represent a separation from the terrestrial diameter, or
from any line parallel thereto in the course of the
respective lines sunwards, of 99999976989 miles.

* This calculation ia detailed in the Appendix.

^ HOW THE DISTANCE AND SIZE OF

If, now, we proportionally modify the divergence of
the outer lines to this distance instead of to exactly one
handred miles, we shall find that the divergence of these
lines is at the rate of -92345879 of a mile — being
1625*2874704 yards — in the distance corresponding to
100 miles measured on either of the inner lines.

We see, therefore, that, in every hundred miles of
progress sunwards, the inner lines diverge to the extent
of -93198000 of a mile, or 1640-2848000 yards, while, in
the distance corresponding thereto, the outer lines
diverge '92345879 of a mile or 16252874704 yards.
The inner lines thus approach the outer lines respec-
tively by -00852121 of a mile, or 149973296 yards,
in every hundred miles; each inner line approaching
the corresponding outer line by half this amount.

It is clear that at the position at which the excess
divergence of the inner lines is sufficient to counteract
the preliminary advantage pertaining to the outer lines
through their terrestrial place being at the extremities
of the Earth's diameter, while that of the inner lines is
at the Earth's centre, the respective lines of each pair
must unite, and that this must occur at the extremities
of the solar diameter. As the preliminary advantage
pertaining to the outer lines is 7912*35947 miles, and as
it is counteracted at the rate of '00852121 of a mile in
every hundred miles, all we have to do to ascertain the
place of union is to divide the former figures by the
latter, and multiply the quotient by 100. We find in
this way that the inner lines touch the respective outer
lines at a distance of 92,854,882 miles from the Earth's
centre, as measured on the inner lines. This, therefore,
according to the data and our calculations, is the dis-

THE SUN WERE MEASURED. 25

tance from the Earth's centre of the respective ex-
tremities of the diameter of the solar orb.

Let us now see what is the distance separating the
respective lines of each pair at the points of union. It
will be found that if the inner lines diverge at the
rate of 'OSIOS of a mile in 100 miles, their separation
in 92,854,882 miles must be 865,389 miles. As 100
is to 92,854,882 so is -93198 to 865,389. Similarly, in
the case of the outer lines — if they diverge at the rate
of '923459 of a mile in a hundred miles their separa-
tion in 92,854,882 miles must be 857,476-76 miles, and if

Ectr-Vb^

AB being Terrestrial diameter is 7912-35947 miles.
3> CD „ Solar „ is 865,389 „

Diagram ilhistrating (1) the mode of calculation of the distance
to which lines from the centre of the Earth to the respective ex-
tremities of the Sun's diameter must be prolonged, to unite with
corresponding prolongations of lines bounding the shadow-cone in
a lunar eclipse ; and (2) the mode of calculation of the separation
of the respective pairs of lines on reaching the Sun.

we add thereto the amount of their separation at the
Earth (7912-36 miles) we find the amount of separation
at the given distance to be, as in the case of the inner
lines, 865,389 miles. This, then, according to the specified
data, is the length of the solar diameter, or, as we have
called it, " the size of the Sun."

We have found that the distance from the centre of
the Earth of the extremities of a diameter of the Sun
is 92,854,882 miles. This, however, is not quite the
same thing as the distance of the Sun — that is to say,
the distance from the centre of the Earth to the centre

26 HOW THE DISTANCE AND SIZE OF

of the Sun. The lines which we have made use of
in our calculations diverge, as we have seen, more and
more as they are prolonged from the Earth towards the
Sun, whereas the true distance from the Sun's centre
would be represented by a direct line from the centre of
the Earth to the Sun, midway between these diverging
lines. It is evident that such a line would be somewhat
shorter than any sloping line can be. We shall now,
therefore, by making use of the foregoing calculations,
ascertain the length, on the given data, of a direct line
from the centre of the Earth to the centre of the Sun.

Let us form a triangle having as two of its sides the
extended radii from the centre of the Earth to the
extremities respectively of the Sun's diameter and
having the Sun's diameter as its third side. The first

CA=CB=92,854,882 miles.

AB=865,389 miles. C=32' 2-36".

Diagram illustrating method of calculating the distance separating

the Earth and the Sun — centre to centre.

two sides are each 92,854,882 miles in length and the
length of the third side is 865,389 miles. The angle at
the Elarth's centre, being the apparent mean diameter of
the Sun, is 32' 2-36". Let us now bisect the solar
diameter and draw a straight line from the point of
bisection to the centre of the Earth. This line, evi-
dently, represents the distance from the centre of the
Sun to the centre of the Earth, and it has the effect of

. THE SUN WERE MEASURED. 27

giving us two right-angled triangles, each having one
side measuring 92,854,882 miles, and a second side
measuring 432,694'5' miles, the latter being the semi-
diameter of the Sun. The angle at the Earth's centre,
being halved by the line of bisection, is now 16' US" in
each triangle. We have to find the length of the third
side, being the side which is common to both triangles.
This, according to the rules of elementary trigonometry,
can be obtained by squaring the length of each of the
two known sides, subtracting the less from the greater,
and then extracting the square root.

We find, in this way, that the length of the third
side is 92,853.874 miles— say, 92,854,000 miles— which,
therefore, is the distance indicated by the data as form-
ing the mean separation of the Sun from the Earth.

The accepted mean distance of the Sun from the
Earth is, as we have noticed, 92,897,000 miles. The
distance found is, therefore, about 43,000 miles less than
the accepted distance, but as the margin of possible
error in the latter may be anything less than 200,000
miles, the distance found is far within the permissible
limits. A similar discrepancy arises in regard to the
length of the solar diameter, the accepted length being
about 866,000 miles and the length we have found being
865,389 miles. Of course the distance and the size of
the Sun are mutually dependent. The difference be-
tween the accepted figures and the amounts found is
practically about '05 per cent., the difi*erence in both
cases being on the side of deficiency in our results.

After all, however, it should be noticed that in at
least one particular, it may fairly be contended that we
«re arguing in a circle. Evidently the measurement of

28 DISTANCE AND SIZE OF SUN.

the apparent diameter of the dark shadow-cone at the
Moon's mean distance during a lunar eclipse, with the
Earth at its mean distance from the Sun, is based to
some extent on the reliability of the accepted measure-
ment of the Sun's mean distance. It is doubtful
whether the measurement of the apparent diameter of
the cone could be obtained directly with the requisite
precision. The other measurements on which we have
based our calculations, do not appear to be subject to
this objection, being, evidently, made in complete in-
dependence of the actual distance or size of the Sun,

Owing to the difficulty in determining by direct
observation the length of the diameter of the shadow-
cone with precision to the extent of, say, two decimal
figures in seconds of arc, it is improbable that this
method could independently be made the means of
measuring the distance and size of the Sun with the
accuracy obtainable by other methods in modem use, al-
though it could, with the necessary care in detail, furnish
a very fair approximation to the correct measurements.

It seems strange that this ingenious method of ascer-
taining approximately the distance and size of the Sun
should have been thought of about a century and a half
before the dawn of our era, while, yet, it is only in our
own time it has, through the gradual accumulation of the
requisite particulars, become capable of being practically
tested. It is notewoi-thy also that by the method now
described the details requiring greatest accuracy to enable
us to solve the grand problem of the distance and size of
the centre of our system are the measurements of the
Globe which we inhabit, and measurements of, and in rela-
tion to, our own Satellite — our nearest neighbour in space.

MEASURING THE EARTH.

SYJsropsis.

The first scientific attempt to ascertain the size
of the Earth — Method adopted by Eratosthenes —
Causes of error in result arrived at — Recent re-
discovery of the Well utilized by Eratosthenes —
Theoretical simplicity of the measurement of
terrestrial circumference or of a parallel of
latitude — Determination of latitude — Determina-
tion of terrestrial circumference from measurement
of parallel — Divergence of Earth's figure from
perfect sphericity — Measurement of an arc of
meridian — Measurement of an arc both of latitude
and longitude at about latitude 45° — How the
length of an arc at the equator can be got from
the length of an arc in any other latitude and vice-
versa — Difficulties of actual measurement of a
terrestrial arc — Means adopted — Essential part —
Possibility of measuring the Earth without
dependence on celestial observations — Method
described — Curvature as an indication of the size
of a sphere — Application to the Earth — Ratio
of diameter to curvature — Distinction between
"level" and "horizontal" — Importance of extreme
precision in determination of size of Earth.

MEASURING THE EARTH.

To Eratosthenes, one of the most distinguished astro-
nomers of the Alexandrine School, belongs the honour
of having made the first scientific attempt, of which
any record remains, to ascertain the size of the Earth.
This celebrated scientist was born at Cyrene, in Barca,
Northern Africa, in the year 276 B.C., and he survived
until 196 or 195^B.c. He was appointed superintendent
of the great library in Alexandria by Ptolemy Euergetes.
It was in Alexandria, more than two hundred years
before the commencement of our era, that he made his
famous effort to measure the Earth.

Eratosthenes learned that at the city of Syene, in
Upper Egypt, at noon on the day of the summer solstice,
the Sun was exactly in the zenith, so that the dial cast
no shadow and the Sun shone to the very bottom of a
deep well, that, in fact, Syene was exactly on the tropic.
Eratosthenes believed that Syene lay due south of
Alexandria, these two cities being, he understood, on
the same meridian. As he did not doubt that the Earth
was practically spherical, and that the distance of the
Sun was almost infinitely great compared with the size
of the Earth, he perceived that by obtaining the angular
separation of the Sun from the zenith in Alexandria,
when it was actually in the zenith at Syene, due south

MEASURING THE EARTH.

<^T,

of Alexandria, he would secure the angular measurement
of the arc of the Earth's surface extending from Syene
to Alexandria. By then ascertaining the length of this
terrestrial arc in miles, he would be able, from the
known proportion of the arc to the complete circle, to
determine the length of the
circumference of the Earth.
The reasoning was absolutely
coiTect.

Eratosthenes found that
at noon at the summer sols-
tice the centre of the Sun
was 7° 12' to the south of
the zenith of Alexandria.
The surveyors of Ptolemy

gave the distance between

7° 12' of arc.

Method adopted by Eratosthenes
to determine the size of the Earth.

Alexandria and Syene as
5000 stadia, the length of
the stadium being, it is believed, about 606 feet 9 inches.
As 7° 12' is the fiftieth part of a circle, it followed that
5000 stadia formed the fiftieth part of the Earth's
circumference. This made the length of the circum-
ference 250,000 stadia, or, say, 28,729 miles.

It is now known that the length of the meridional
or polar circumference of the Earth is 24,816 miles, so
that the measurement made by Eratosthenes was about
3,913 miles — or nearly sixteen per cent. — too great.
Considering, however, the time at which it was made,
and the accompanying circumstances, the approximation
to accuracy is surprising, and the method adopted bj^ this
early scientist, who may be called the father of modern
geodesy, may well command the respect of these later ages.

MEASURING THE EARTH,

Seeing that the process by which Eratosthenes
hoped to determine the length of the Earth's circum-
ference is scientifically sound, it is interesting to notice
the reasons of the discrepancy between the result arrived
at by him and the true result.

The chief cause of the erroneous result is not, as
might be supposed, inaccurate measurement of the celes-
tial arc through defective instruments. There are really
two prime sources of error, and each is quite apart from
the observations made by Eratosthenes of the separation
of the Sun from the zenith as seen in Alexandria. The

first of these was the belief
of Eratosthenes that Syene
was due south of Alex-
andria, and the second was
his acceptance of the meas-
urement made by Ptolemy's
surveyors of the distance
between Alexandria and
Syene.

Though Eratosthenes
believed that Syene was on
the same meridian as Alex-
andria, it was in reality
rather more than three degrees to the east of that
meridian. Let us make a triangle connecting (1) Syene
and Alexandria, (2) Syene and the position in the same
latitude due south of Alexandria, and (3) the last men-
tioned position and Alexandria. We see by this means
that when Eratosthenes made his observations in Alex-
andria fixing the angular separation of the Sun from
the zenith he was really determining the angular

Triangle illustrating error in
measurement of terrestrial arc
by Eratosthenes.

MEASURING THE EARTH.

measurement of the arc separating Alexandria from the
position due south of it in the latitude of Syene — he
was, in fact, measuring the third side of our triangle.
Having determined the angular length of this side to be
7° 12', he then applied his measurement to the distance
between Alexandria and Syene, being the first side of
the triangle. It is evident at a glance that the first side
— which is the hypotenuse of the triangle — is consider-
ably larger than the third side. The former may indeed
be taken as being about 522 miles in length, while the

Sketch-map showing relative positions of Alexandria and Syene.

latter is about 488 miles. An angular measurement of
7° 12' applied to 522 miles would indicate that the
length of the degree was exactly 72^ miles, while, applied

MEASURING THE EARTH. 63

to 488 miles, it would make the length rather less than
68 miles, a very considerable difference when multiplied
by the number of degrees in a circle.

Eratosthenes accepted 5000 stadia — say 574*6 miles —
as the true distance between Alexandria and Syene,
being the distance as determined by an official survey.
While there is no reason to doubt the approximate
accuracy of the actual measurement, it is clear that such
a measurement could not possibly have been absolutely
direct or on one level. As we have noticed, the direct
and level distance between Alexandria and Syene may
be accepted as having been about 522 miles, being 52'6
miles less than the surveyed distance. Thus in accepting
574*6 miles instead of 522 miles as the length of T 12'
of the Earth's surface, Eratosthenes increased the length
of the degree from 72| miles to nearly 80 miles.

As might be expected, there were also minor errors in
connection with the measurement made by Eratosthenes,
but they were very insignificant in comparison with
those to which we have referred. He acted on the
belief that Syene was on the tropic, but this is not
exactly the case. In the time of Eratosthenes the
obliquity of the ecliptic — or the range of the Sun's
apparent northward and southward movement — was
rather greater than it now is. The extent of the Sun's
movement as found by Eratosthenes himself — which in
all likelihood was substantially correct — was 47" 42' 39",
the sun passing by one-half that extent to the south of
the equator, and by the remaining half to the north of
the equator. The northern limit of the Sun's movement
in its apparent annual journey was thus latitude 23* 51'
19 5" north. Consequently at noon on the day of the

64 MEASURING THE EARTH.

June solstice, the centre of the solar disc would be in
the zenith at a position in the last mentioned latitude.
The latitude of Syene, however, was 24° 5' 20" north,
so that the city was about 14' 0"5" — rather more than 16
miles — north of the tropic. Thus it was not the case
that the centre of the Sun was in the zenith at Syene at
noon at the solstice, although, as the mean diameter of
the Sun is about 32' and the semi-diameter consequently
slightly greater than the angular separation of Syene
from the tropic, it would have been the case that the
edge of the solar disc occupied the zenith as seen from
Syene. The error arising from this discrepancy is
trifling, and it is evident that its effect would have been
to lessen, not to increase, the length obtained for the
Earth's circumference. This would follow from the fact
that Eratosthenes, while he thought he was measuring the
arc separating Alexandria from the latitude of Syene,
was really measuring the arc separating Alexandria
from a point over 16 miles south of the latitude of Syene.
If, then, he applied his angular measurement to a less
distance in miles than that to which it really applied he
would be proportionately lessening the ultimate result.

Strangely enough, however, this error, trifling as it
is, was partially compensated by another small error.
The true distance in angular measurement between
Alexandria and the latitude of Syene is about 7° 6^',
not, as determined by Eratosthenes, 7° 12'. The differ-
ence of 5^' — which is equivalent to about 6*3 miles on
the Elarth's surface — compensated by this amount the
error arising from the assumption that Syene lay exactly
on the tropic, while in actual fact it lay over 14' to the
north of the tropic. This left the difference of rather more

MEASURING THE EARTH. 65

thau 9J miles — the angular measurement being really
to a position over 9^ miles south of the latitude of
Syene — as compensating to some extent the large errora
occasioned by the belief that Alexandria and Syene
were on the same meridian and 5000 stadia apart.

Of course, it is very evident also, that Eratosthenes
simply accepted round figures, and did not calculate
with a view to absolute precision. Had he done so, it is
scarcely conceivable that the length of his terrestrial
arc should have been exactly 5000 stadia and of his
celestial arc 7° 12', being exactly the fiftieth part of a circle.

This early attempt to measure the Earth, although it
took place more than two millenniums before our time,
is connected in an interesting manner with our own
day. Eratosthenes, as we have noticed, was satisfied
that at noon on the day of the summer solstice, the Sun
was exactly in the zenith at Syene, as it then shone " to
the bottom of a deep well" at that city. This well,
although covered up and lost for centuries, has recently
been re-discovered and is now restored. The ancient
city of Syene is now represented by the flourishing
community of Aswan, the head-quarters of the great
irrigation works on the Nile. Through these works the .
well of Eratosthenes has again been brought to light
and the Sun now again at the summer solstice shines
to the bottom, although, owing to the lessened obliquity
of the ecliptic, not exactly so completely as it did in the
time of the celebrated librarian of Alexandria. Thus in
Upper Egypt at the present day we have a remarkable
link with the great astronomer who first set himself in
a scientific manner to "put a girdle round about the
Earth."

66 MEASURING THE EARTH.

Although even before the time of Eratosthenes
vague and crude attempts were, no doubt, made to
estimate the size of the Earth, such attempts were little
better than mere guesses, and no measurement more
accurate than his would appear to have been made until
more than a thousand years after his time.

It may, indeed, be said that the method adopted by
Eratosthenes in his attempt to gauge the size of the
Earth comprised the essential features of geodetical
operations even as still carried out. Doubtless, these
operations are conducted with instruments of a precision,
and with an attention to detail, undreamt of in his day,
but still the fundamental bases of the operations are
really, as in the time of this ancient astronomer, the
measurement of a celestial arc, and the accurate survey
and determination of a corresponding arc on the terres-
trial surface.

Although in the practical work of measuring a
geographical arc extreme care is requisite, and instru-
ments of the utmost refinement have to be made use of
to secure reasonable accuracy, it is yet the case that
theoretically the measurement of the circumference of
the Earth with a fair approximation to correctness is
not a matter of great difficulty.

Let us suppose, for instance, that it is desired to
ascertain the length of the equatorial circumference of
the Earth. This might be done by simply observing
with care the exact time at which the Sun crosses the
meridian at two positions on the equator whose distance
apart is known. Let us suppose the two positions to
be the important city of Quito, in Ecuador in South
America, which lies near the coast of the Pacific Ocean,

MEASURING THE EARTH.

and the Galapagos Islands in the Pacific, which are
associated with Darwin's investigations. The distance
from Quito to a selected point in the Galapagos Group
■of Islands is, we may suppose, 830 miles. It will be
found to be the case that, on the mean, the Sun, in its
apparent daily westward journey, is on the meridian at

Sketch-map illustrating measurement of terrestrial arc between
Quito and the Galapagos Islands.

the selected position in the Galapagos Islands forty-
eight minutes later than the time at which it is on the
meridian at Quito. Thus the Sun takes forty-eight
minutes to pass westward in relation to the Earth's-
surface a distance of 830 miles measured on the equator.
The Sun, however, takes on the average twenty-four
hours to return to the meridian of Quito. If, then, the
Sun takes forty-eight minutes to travel a distance of
830 miles, what must be the length of the circuit which
it travels in the twenty-four hours which intervene
between its successive returns to the meridian of Quito ?
As forty-eight minutes is to twenty-four hours, so is
830 miles to the length of the circuit. The latter, which

68 MEASURING THE EARTH.

is the equatorial circumference of the Earth, is, there-
fore, 24,900 miles. This method of ascertaining the
equatorial circumference of tlie Earth involves, of courser
the measuring of the distance between the two places at
which the transit of the Sun is observed, as well as the
observation of the exact time at which the sun is on the
meridian at each of these places.

If this method were made use of at two places lying
either to the north or the south of the equator and
situated due east and west of each other — that is to say
having the same latitude — the amount obtained would
clearly not be the length of the equatorial circumference
of the Earth, but simply the length of the parallel of
latitude of the places of observation.

Thus, for instance, Cardiff lies as nearly as possible
due west of Greenwich, their direct and level distance
apart being practically 186*04 miles. The Sun is on
the meridian at Cardiff about twelve minutes forty
seconds later than it is on the meridian at Greenwich.
Therefore, in the latitude of these towns, the Sun takes,
on the mean, twelve minutes forty seconds to pass west-
ward in its apparent daily journey, a distance of 136 "04
miles measured on the Earth's surface. This being so,
what distance on the parallel must the Sun travel
during the twenty-four hours which elapse between its
successive returns to the meridian of Greenwich ? It is a
matter of simple proportion to find that the distance is
15,465"6 miles, which, consequently, is the length of the
parallel on which Greenwich and Cardiff are situated.
If, now, we divide the length of the parallel by 360, we
shall get the length of one degree of the parallel in the
latitude of Greenwich and Cardiff. We thus find that

MEASURING THE EARTH.

at the distance from the equator at which these towns
are situated, one degree of the parallel is about 42*96
miles in length.

It is quite possible from this measurement in relation
to Greenwich and Cardiff to find approximately the

Sketch-map illustrating measurement of terrestrial arc between
Greenwich and Cardiff.

length of the circumference of the Earth at the equator,
just as we have found it by the measurement at the
equator itself. It is necessary, however, to assume — in
view of our information being supposed to be restricted
to the operations in relation merely to the Greenwich
parallel — that the Earth is a perfect sphere, as the
measurement on the parallel furnishes no information
as to the deviations of the figure of the Earth in this
respect. As a matter of fact the deviation from perfect
sphericity is not great, the polar radius being only
about thirteen miles less than the equatorial radius.

70 MEASURING THE EARTH.

Working, then, from the basis of the measurement of
the distance between Greenwich and Cardiff, and the
interval which elapses between the Sun's appearance on
the meridian in these towns respectively, how are we to
find the length of the equator ?

We must first determine the angular distance from
the equator at which Greenwich and Cardiff lie, that is
to say their latitude. Restricting our attention still
simply to the Sun, we can ascertain the latitude. The
celestial equator, which may be considered as merely an
extension of the terrestrial equator, is a great circle of
the heavens, extending due east and west, situated
exactly midway between the most northerly and the
most southerly positions of the Sun in its annual course.
If, therefore, we determine the Sun's position in relation
to the celestial equator and the angular separation of
the Sun, when on the meridian, from the zenith of the
place of observation, we shall have the means of fixing
the latitude of the place. Thus at the June solstice the
Sun is 23° 27' north of the celestial equator. On the same
date it crosses the Greenwich meridian 28° 1' 38" to the
south of the zenith of Greenwich. Adding these two
angular distances, we find that Greenwich lies 51° 28' 38"
north of the equator.

A similar calculation of the local latitude can, of
course, be made whatever be the position of the Sun in
relation to the celestial equator, and whatever be the
situation of the place of observation. The angular
separation from the celestial equator of the Sun when
on the meridian of the place of observation, is added to
or subtracted from the angular separation of the Sun
from the zenith of the place of observation, according as

MEASURING THE EARTH. 71

the Sun is at the time to the north or to the south of
the celestial equator. If the Sun is to the north of the
celestial equator or, as it is usually expressed, is in
north declination, then, in the northern hemisphere, the
declination is added to the angular separation of the
mid-day Sun from the zenith, while in the southern
hemisphere it is subtracted therefrom. The converse is
the case if the Sun is in south declination. If the Sun
is actually on the celestial equator, then its angular
separation from the zenith of the place of observation is
itself the latitude. In every case the purpose is to
determine the angular separation of the zenith of the
place of observation from the celestial equator, as this
angular separation is the latitude. The observation of
Sun (or stars) is merely an aid to this purpose.

Having now fixed the latitude of the place of
observation — which in view of there being a slight
difference between the latitude of the two positions,
Greenwich and Cardiff, we shall consider not as
51" 28' 38" north but simply as 51° 28' north — we can
determine the length of the equatorial circumference of
the Earth, from the data found, by the use of trigono-
metrical ratios. These ratios have certain relations
to arcs of circles and to triangles. The ratios which suit
our present purpose — the determination of the length of
the equator from our knowledge of the length of a
parallel at a distance of 51° 28' from the equator — are
the values of the " natural cosines " of arcs of a circle in
proportion to the value of the radius of the same circle.
These values have been calculated numerically for all
arcs up to ninety degrees, and are expressed as decimal
fractions of the radius, the latter being taken as repre-

Y2 MEASURING THE EARTH.

sented by 1. Tables of " natural cosines " are included
in most Mathematical Tables.

The natural cosine for 51° 28', being the latitude of
Greenwich and Cardiff, is '6229698. We have already-
found that the length of the parallel at Greenwich is
about 15,4656 miles. As the natural cosine of 51° 28
(•6229698) is to 1 (being the value of the radius of a
circle in proportion to the natural cosine) so is 15,465'6
miles (the length of the parallel of Greenwich) to the
length of the equator. We find by this means that the
proportional length of the equator is about 24,825-6
miles. This is about 73 miles less than the correct
amount, the discrepancy being, as we have already
indicated, chiefly due to the fact that the Earth bulges
at the equator to a somewhat greater extent than would
be the case were it perfectly spherical. Naturally also
some discrepancy is accounted for by our bases of
calculation not being absolutely precise.

If, now, we should measure an arc of latitude at a
sufficient distance from the parallel of Greenwich as to
make evident the divergence from perfect sphericity in
the figure of the Earth, we might, by noting the differ-
ence arrived at in the final result and the angular
separation of the two parallels, determine, very roughly,
the amount of the divergence which exists between the
length of the equator as calculated from the length of
the Greenwich parallel and its true length.

For this purpose let us measure, in similar manner,
the arc separating Castellon, on the east coast of Spain,
from the northern extremity of the largest of the
Balearic Islands. The direct distance separating these
two positions, which lie due east and west of each other

MEASURING THE EARTH.

is slightly less than 172 miles — say 171'88 miles — and,
on the mean, the Sun is on the meridian at Castellon
about thirteen minutes later than it is on the meridian
at the northern extremity of the main Island of the
Balearic group. As twenty-four hours of solar move-
ment represent the circuit of 360°, four minutes of solar
movement represent 1°, and thirteen minutes, therefore,
represent a westward change of position of 3J*. Thus

Sketch-map illustrating meaaurement of terrestrial arc between
the Balearic Islands and Castellon.

in the latitude of Castellon, 3^° on the parallel repre-
sent about 171 "88 miles, and one degree, therefore,
represents 52*886 miles. This makes the total length
of the 360° forming the parallel about 19,038-96 milea
We may suppose that, by observation of the Sun, we
find that the latitude of Castellon is 40° north. The
natural cosine for 40° is -7660444. As this natural
cosine is to 1, so — supposing the Earth to be a perfect

74 MEASURING THE EARTH.

sphere in proportion to this parallel — is 19,038'96 miles
to the length of the equator. This makes the length of
the equator about 24.853*6 miles.

According, however, to our measurements in the
latitude of Greenwich — 51° 28' north — the length of the
equator should be about 24,825*6 miles, so that the
measurement, at a distance southward of 11° 28', reveals
an excess over true sphericity on the basis of the Green-
wich parallel of about 28 miles. If, then, a difference of
11° 28' indicates this excess, a difference of 51' 28' —
being the angular distance of Greenwich from the equator
— should indicate a proportionally greater excess. As
11° 28' is to 51° 28' so is the excess indicated by 11° 28'
to the excess required by the larger arc. This indicates
an excess in the length of the equator over its length in
proportion to the Greenwich parallel of about 1257
miles. As the length of the equator in proportion to
the length of the Greenwich parallel is, as we have
seen, about 24,825*6 miles this would indicate an actual
length of about 24,951 miles. This is 52 miles more than
the true length — which is 24,899 miles. It is evident
that to secure a closer approximation it would be
necessary to have much greater exactitude in all the
details than we have adopted in our illustration.

The method, however, sufficiently indicates how the
deviation from true sphericity in the figure of the Earth
can be made apparent by the exact measurement of
small arcs on parallels suflficiently separated. This can,
of course, be also done, and in practice is more commonly
done, by the measurement of widely-separated corres-
ponding arcs of the meridian, as, owing to the polar
flattening and equatorial bulging of the figure of the

MEASURING THE EARTH. 75

Earth the degree increases in length with separation
from the equator.

In order to determine the approximate length of
the polar circumference of the Earth an arc may (as
was attempted by Eratosthenes) be measured not on a
parallel of latitude but on a meridian, and, except in so
far as variation in the length of the arc results from the
figure of the Earth not being perfectly spherical, the
positions in latitude between which the measurement is
made are not of consequence. It is necessary only that
the terminal points of the arc should lie on the same
meridian, or that allowance should be made trigono-
metrically for their difference in longitude.

Thus we can obtain a fair approximation to the
polar circumference of the Earth by measuring, for
instance, the distance across the Firth of Forth between,
say, the town of Leven and a point on the opposite side
of the Forth due south of Leven. We may take the
distance separating these positions as being about 17*1
miles. We can determine the latitude of each position
by observation of the Sun or the stars. The latitude of
Leven is about 56" 12' north, while that of the opposite
point on the southern side of the Firth is about 55' 57|^'
north. Thus 14f minutes in angular measurement re-
present 171 miles on the meridian. Proportionally one
degree will represent 69*17 miles. Multiplying this by
360, as the number of degrees in a circle, we get the
polar circumference of the Earth, in proportion to the
length of the short arc between Leven and the opposite
side of the Firth of Forth, as 24,901 miles. The true
length is 24,816 miles, so that our measurement is 85
miles too great. The discrepancy arises chiefly from

76 MEASURING THE EARTH.

the fact that the length of the degree at the Firth of
Forth is, owing to the flattening which occurs with
approach to the poles, quite appreciably greater than
the 'mean length of a degree of the meridian.

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Sketch-map illustrating measurement of terrestrial arc across
the Firth of Forth at Leven.

As the parallel 45° north, if considered in its angular
«,spect, is exactly midway between the pole and the
equator, it is interesting to notice the results as regards
the length of the terrestrial circumference indicated by
the measurement of an arc at that parallel, whether
stretching east and west or north and south.

We may take a point at the mouth of the Danube
exactly in the latitude specified, and a point on the
same parallel on the coast of Crimea. The distance

MEASURING THE EARTH.

between these two points is 190'31 miles and the time
difference between them is about fifteen minutes thirty-
six seconds, which, at the rate of four minutes in time
for each degree of arc, represents a difference in longi-
tude of 3° 54'. This makes the length of one degree on
the parallel 48797 miles making the complete parallel
about 17,567 miles. The natural cosine for 45° i&
•7071068. Dividing this into the length of the parallel
we get the length of the equator in proportion to the
parallel as 24,843 miles. This is very nearly the nuan

Sketch-map illustrating measurement of terrestrial arc on the 45th
parallel at the Mouth of the Danube.

circumference of the Earth, the length found being only
about fourteen miles less than the mean circumference
— 24,857 miles. The difference in all probability chiefly
arises from the want of absolute precision which
necessarily characterizes our calculations.

Having thus measured an arc of the parallel let us
now measure, in a similar position, an arc of the
meridian. The accepted length of a degree of the
meridian at the equator is 362,746*4 feet, while at either

78 MEASURING THE EARTH.

pole it is 366,479'8 feet, from which it follows — the
flattening with separation from the equator and ap-
proach to the pole being of gradual increase — that the
length of one degree of the meridian having the parallel
45° north or south latitude as the intermediate position
is 364,613-1 feet. Multiplying this by 360, as the
number of degrees in the circle, we get 131,260,716 feet,
being almost exactly 24,860 miles. This is only three
miles more than the accepted length of the mean circum-
ference of the Earth. The trifling difference probably
arises either from an insignificant error in our figures
or from the mean length of the degree of the meridian
not occuring precisely at the forty-fifth parallel.

It would seem to follow from these two calculations
that one degree measured on the parallel intermediate
between the equator and the pole, or one degree measured
on the meridian with the same parallel as the intermediate
position, is practically the 360th part of the mean cir-
cumference of the Earth, that is to say is practically the
rnean length of one degree on the terrestrial surface.

It will be observed that by taking the length of one
degree, or any other angular distance, on any parallel
and dividing the amount by the natural cosine applic-
able to the latitude, we obtain the length of a corres-
ponding angular distance on the equator; and by taking
the length of one degree, or any other angular distance,
on the equator, and multiplying by the natural cosine
for any given latitude, we get the length of a corres-
ponding arc in the given latitude, the result arrived at
being, in both cases, on the basis of the Earth being a
perfect sphere. Thus, on account of the equatorial bulge
of the Earth, the result, in working from a higher

MEASURING THE EARTH. 79

latitude to the equator is rather less, and in working
from the equator to a higher latitude is rather more
than the true value. A correction may be applied in
proportion to the latitude, on the basis of 83 miles being
the excess of the length of the equator (24,899 miles)
over true sphericity in proportion to the polar circum-
ference (24,816 miles). Thus for every degree of separa-
tion from the equator any parallel is lessened by
practically one mile — being |f of a mile — from its
length if proportioned to the equator, and for every
degree of approach to the equator the parallel is in-
creased by practically one mile over its length if pro-
portioned to the polar circumference of the Earth.

We can, for example, find the length of one degree
on the parallel in, say, latitude 51° 30' with reasonable
accuracy if we know the length of one degree on the
equator and apply the stated correction. The length of
one degree on the equator being accepted as 69'17 miles,
we should proceed as follows: — Multiply 6917 miles by
natural cosine of 51* 30', being -6225146. The product
is 43*06 miles. Subtract from this the 360th part oi 51^
miles — being one mile deducted from the parallel for
every degree of separation from the equator. The 360th
part of 51 1 miles is '143 of a mile, which, if subtracted
from 4306 miles, leaves 42917 miles as the length of
one degree of the parallel in latitude 51° 30'.

Similarly, to get the length of one degree on the
equator supposing we know that the length of one
degree of the parallel in latitude 66' north is 2806
miles, we should proceed as follows: — Divide 2806
miles by the natural cosine of 66°, being -4067366. The
quotient is 68988 miles. Add to this the 360th part of

80 MEASURING THE EARTH.

66 miles, being one mile of increase in the parallel for
every degree of approach to the equator. The 360th
part of 66 miles is '183 of a mile. This makes the
length of one degree on the equator 69'171 miles.

It will be observed that a degree of the meridian is
really a degree of latitude, while a degree of the parallel
is really a degree of longitude, although at first sight
the converse may appear to be the case.

Comparatively simple as it is in theory to measure
a limited stretch of sea or land, and to determine the
angular extent of the celestial arc corresponding thereto,
it is, in actual practice, one of the most difficult problems
which can be presented to the engineer. It is obvious
that when the results of the measurement of a very
limited arc come to be applied to the solution of the
length of the circumference of a circle, an error which
in the first instance is insignificant is multiplied to such
an extent as to be of great consequence. Geodetical
measurements have to be carried out with the most
extreme care, with instruments of the utmost precision.

Col. A. R. Clarke, F.R.S., of the Ordnance Survey,
a distinguished authority on the practical work of earth-
measurement states that : —

"The basis of every extensive survey is an accurate
triangulation, and the operations of geodesy consist in
the measurement, by theodolites, of the angles of the
triangles, the measurement of one or more sides of those
triangles on the ground, the determination by astronomi-
cal observations of the azimuth of the whole network of
triangles, the determination of the actual position of the
same on the surface of the Earth by observations, first
for latitude at some of the stations, and secondly for
longitude."

MEASURING THE EARTH. 81

The actual measurement on the ground of one of
the sides of a triangle for use in the operation of tri-
angulation is in itself, simple as it appears, by no means
an easy process to effect with the requisite accuracy.
The corrections for refraction, instrumental errors, &c.,
have to be applied with extreme care. The measuring-
rods or wires have to be compensated for changes of
temperature and adjusted by micrometer screws, and
the lengths procured have to be reduced to sea-level.
From the triangles first determined new triangles are
obtained until a net-work of triangles is calculated.
From these is obtained finally the length of an arc of
the terrestrial surface, the exact positions of the ends of
which are precisely determined.

As to the practical work of earth-measurement, it is
unnecessary to say more. The essential part of the
work is, as we have seen, the determination of the
actual distance on the surface of the Earth, at sea-level,
corresponding to any given arc of angular measurement.
Whenever the relation between the angular measure-
ment and the mileage is established the work of the
geodetical engineer may be considered as accomplished.

It will be noticed that, in the measurement of the
Earth, the angular length of the given arc of the terres-
trial surface is determined from the observation of a
corresponding arc of the heavens. The sky is considered
as a hollow sphere surrounding the Earth, and our Globe
is regarded as occupying the centre of this apparent
sphere. As the distance of the Sun and stars — being
the objects which afford the means of identifying an arc
of the celestial sphere — may reasonably be regarded as
infinite in comparison with terrestrial distances, this

82 MEASURING THE EARTH.

attitude is justified. Thus the terrestrial sphere is en-
closed centrically in a great celestial sphere, and as a
circle, no matter what its size, necessarily consists of 360
degrees, the angular measurement of any portion of the
celestial circle exactly corresponds with the angular
measurement of the terrestrial arc which it covers. In
this way terrestrial measurements depend upon accurate
celestial observations.

Although this is so, it is — at any rate theoretically —
quite possible to measure the Earth without any de-
pendence on celestial observations. It is quite conceiv-
able that the size of the Earth might be fixed by direct
observation.

As the figure of the Earth is practically spherical, it
is evident that — if we disregard the merely superficial
elevations and depressions which form mountains and
hills, ocean-floors, and valleys, which, even at their
greatest extent, are quite insignificant in comparison
with the size of the Earth — it may truly be said that
from every point on the surface the surrounding parts
gradually slope away. It is evident that on the
surface of any sphere the amount of this sloping-
away, or, we may say, the curvature, in any given
distance, is exactly proportioned to the size of the
sphere. As the sphere increases in size, the amount of
curvature in a given distance lessens, and, on the other
hand, as the curvature in a given distance increases, the
size of the sphere lessens. The amount of curvature in
a distance of one yard on, for instance, the rim of a
cart-wheel is incomparably greater than the amount of
curvature in the same distance on the Earth's surface
while the length of the rim of the cart-wheel is incom-

MEASURING THE EARTH. 83

parably less than the length of the circumference of the
Earth. Again, as the Sun is vastly larger than the
Earth, it is evident that the amount of curvature in a
distance of, say, ten miles on the Sun's spherical surface
must be vastly less than it is in the same distance on the

Sun, 1700 miles, Curvature 0*22* Earth, 1700 miles. Moon, 1700 miles,

(almost imperceptible). Curvature 24 •62°. Curvature 90*.

Diagram illustrating amount of curvature in a distance of 1700
miles on the surface of the Sun, Earth, and Moon, respectively.

surface of the Earth; while the Moon, being considerably
smaller than the Earth, must have considerably more
curvature in a given distance on its spherical surface
than the Earth has. Evidently, then in the surface-
curvature we have an independent means of earth-
measurement.

Supposing now, that, by observation with instru-
ments of extreme precision, and after due allowance for
refraction and instrumental errors and reduction to sea-
level, we find that the curvature of the surface of the
Earth in a distance of, say, one mile is, on the mean
8*0076 inches — which is indeed almost exactly the case —
how can we, from that single circumstance — which we
may suppose to have been determined by observation —
fix the size of the Earth ?

We might proceed as follows: — Divide one mile —
63,360 inches — by the curvature in one mile — 80076
inches. The quotient is 791248. We have now to find
what arc of a circle has the same ratio of length to

84 MEASURING THE EARTH.

curvature. Such an arc will be the angular measure-
ment of one mile on the surface of the Earth. The very-
small proportion which the curvature bears to the dis-
tance — being as 1 to 7912"48 — and the fact that the
ratio which the surface-curvature of a sphere bears to
the distance increases rapidly as the arc lengthens, show
that in this case the arc must be a very small one. In
the absence of any Table of such ratios let us assume, to
commence with, that the arc is one minute. In order to
test the accuracy of this assumption, let us ascertain the
ratio subsisting between the length and the curvature of
1' of arc.

We may suppose the length of the circumference of a
circle to be represented by 1. The ratio of the circumfer-
ence of a circle to the diameter is as 3'14159,26535,8979
to 1, being the ratio which is universally distinguished by
the Greek letter tt (pi). If, then, the circumference is
represented by 1, the diameter will be represented by 1
divided by 3-14159,26535,8979. The diameter will, there-
fore, be represented by '31830,98861,8379, and the radius,
being one-half of the diameter, by 15915,49430,91895.

As a circle consists of 360 degrees, each of which
consists of 60 minutes, a circle consists of 21,600
minutes of arc. If, then, we divide 1, as representing
the circumference, by 21,600, we shall get the propor-
tionate value of 1' of arc. The latter is, therefore,
•00004629.

Our next step is to square the value of the
radius (-15915,49430,91895) and the value of 1' of arc
(-00004629), the squares being respectively —
•02533.02959,10584,33598,24146,910
and -00000,00021,43347,05046,29629,630.

MEASURING THE EARTH, 85

Subtracting the latter from the former we have
02533,02937,67237,28551,94517,280, and extracting the
square root thereof we have '15915,49303,5837. Subtract-
ing this square root from the value of the radius of the
circle (-15915,49430,9189) we get -00000,00067,3352 as the
curvature in 1' of arc of a circle, the circumference of
which is represented by 1. By dividing this curvature
into the value of 1' of arc (-00004629) we find that the
proportion which the curvature in 1' of arc bears to the
length of the arc is as 1 to 6875-50, the arc containing
the amount of its curvature 6875.50 times. *

As the ratio which the surface-curvature of a sphere
bears to the distance increases, as we have seen, as the
arc lengthens, and as the surface-curvature in one
minute of arc bears a larger ratio to one minute of arc
— being as 1 to 6875-50 — than the curvature in one
mile on the Earth's surface bears to one mile — being as
1 to 7912-48 — it is evident that one minute of terrestrial
arc must, in actual length be more than one mile. It is
also evident, from the fact that the difference in the
respective ratios is not extreme, that one minute of arc
is not very much more than one mile.

There are different courses now open to us in order
to ascertain the true angular extent of one mile on
the surface of the Earth. We may find, by proportion
between the ratios, a closer approximation to the re-
quired angular extent, and test this, as we have already
tested 1' of arc, repeating this operation as often as may
be requisite to secure either absolute con-espondence or
a mere trifling difference between the ratios; or we
may, if satisfied with the approach already found to
correspondence between the ratios, proceed by simple

* Difference in length between arc of 1' and its chord is considered negligible.

86 MEASURING THE EARTH.

proportion, and accept the fourth proportional as the
angular length of one mile on the surface of the Earth.
We seem to be justified by the present circumstances
in adopting the latter and simpler course. By propor-
tion, therefore, from the ratio which the curvature in
1' of arc bears to the length of 1' of arc, we shall fix
approximately the angular measurement of one mile.

As 7912*48 (being the number of times the curvature
in one mile is contained in one mile) is to 6875"50
(being the number of times the curvature in 1' of arc is
contained in 1' of arc) so is 1' of arc to the angular
extent of one mile on the Earth's surface. The fourth
proportional is 52*14". Accepting this as the angular
length of one mile, we find by another statement in
proportion that the mean circumference of the Earth is
24,856 miles. This is, in fact, just one mile less than the
accepted length of the mean circumference, the accuracy
of the final result being indeed greater than might be
looked for in view of the want of extreme precision in
our calculations. It is clear, however, that the accuracy
of such a method of measuring the Earth is limited only
by the accuracy attainable in observation of curvature.

There is another and still more simple means of
ascertaining the size of the Earth directly from tlie
curvature in a given distance.

It is one of the properties of a circle that in any
short arc — say anything less than a degree and a half —
the amount of the curvature bears practically the same
proportion to the arc to which it pertains as the latter
bears to the diameter of the circle. The shorter the
arc the more perfect is the correspondence. Up to the
extent of about a degree and a half, the variation from

MEASURING THE EARTH. 87

exactitude is comparatively trifling, although as the
arc is increased to a greater extent the variation very
rapidly rises.

Applying this fact to the Earth we can readily
measure the size of our Globe by the observation of the
curvature in one mile or any less distance, or in any
distance up to a limit of, say, one hundred miles. The
mean curvature in one mile is, as we have noticed,
8'0076 inches. If we divide one mile by 8'0076 inches
— first, of course, reducing the mile to inches — we get
7912*48, which is the mean length in miles of the
diameter of the Earth. This quotient is simply the
fourth proportional to the following statement in pro-
portion. As 80076 inches (being the mean curvature
in one mile) is to 63,360 inches (being one mile reduced
to inches) so is 1 mile to the mean diameter of the
Earth in miles.

Having found the diameter, the circumference is
obtained by multiplying the diameter by 3*1415927.
This makes the length of the mean circumference
of the Earth 24,8578 miles, which is the accepted
length.

We may test this method by applying it to two •
other distances, say, in the first place, three miles, and,
in the second place, 250 yards.

As the curvature for a limited distance may be
accepted as varying according to the square of the
distance — which is virtually correct for the Earth up to,
say, about one hundred miles — we can fix the curvature
for the distances specified from that which we have
accepted as suflficiently accurate in the case of one mile.

As the curvature in one mile is 80076 inches, the

88 MEASURING THE EAUTH.

curvature in three miles is nine times 80076 inches, that
is to say it is 720684 inches. It has to be noticed,
however, that an inappreciable error in the curvature in
one mile is, by this means, multiplied nine times in the
curvature in three miles. Setting aside this as of very
slight importance in the circumstances, let us measure
the Earth from the specified curvature in three miles.

As 720684 inches is to 190,080 inches (being three
miles reduced to inches) so is three miles to the diameter
of the Earth in miles. The fourth proportional, being
the length of the diameter of the Earth, is again found
to be 7912-48 miles.

We have now to apply the method to the curvature
in 250 yards. As this distance is contained in one mile
7*04 times, we obtain the curvature applicable to the
distance by dividing the curvature in one mile (8'0076
inches) by the square of 7*04. The square of 7 '04 is
49-5616, and by dividing 49-5616 into 8-0076 we get
•1615686 of an inch as the proportional mean curvature
in 250 yards. The corresponding statement in propor-
tion is, therefore, as follows: — As "1615686 (being the
curvature in 250 yards) is to 9000 (being the number
of inches in 250 yards) so is 250 yards to the diameter
of the Earth in yards. The fourth proportional is
13,925,973. Dividing this sum by 1760 as the number
of yards in one mile we get 7912-48 as the length of the
mean diameter of the Earth in miles, as in the two
preceding calculations.

Incidentally it may be noticed, although the matter
is outside our subject, that this relation between the
curvature of the Earth in a limited distance — being
applicable up to, say, one hundred miles — and the

MEASURING THE EARTH. 89

diameter of the Earth, affords, conversely to our present
purpose, a ready means of ascertaining the curvature in
a given distance from the known diameter. As the
diameter of the Earth — say 7912"5 miles — is to the
given distance so is the given distance to its curvature.*

In view of the relation between the curvature in any
given distance, and the size of the sphere to which the
curvature applies, it is possible to imagine the occurrence
of a time in the future when, through the perfection of
mechanical appliances, the circumference of the Earth,
on the assumption of perfect sphericity, may be deter-
mined in any desired direction from the position occupied
by an observer at the centre of a level surface specially
prepared for curvature observations.

In this connection it is well to keep in mind that a
distinction exists betw^een "level" and "horizontaL"
The word level really means parallel to the curvature of
the surface of the unruffled sea, while a horizontal line
is a tangent to the level. A level line, in this relation,
is not necessarily in all its parts at an equal distance
from the Earth's centre. It is evident, indeed, that,
owing to the irregularity in the figure of the Earth, the
surface of still water at the pole is about thirteen miles
nearer to the centre of the Earth than is the surface at
the equator. Yet a level line would follow the surface
from equator to pole. Observations for curvature would
have te be made on a surface absolutely level in this
sense of the word, though such a surface might be
either natural or artificial. The curvature would have
to be adjusted te sea-level, if determined at any other
level, as sea-level is practically the mean level of the
terrestrial surface. If the curvature could be ascertained

* The curratore is the versed sine of the angular extent of the distance.

90 MEASURING THE EARTH.

with precision, the extent of the area subjected to
observation would be unimportant, an area with a
radius of 250 yards being as satisfactory as one with
a radius of several miles.

As the diameter of the Earth is necessarily the unit
in celestial measurements, the determination of the size
of the Earth with exactness is a matter of supreme im-
portance. Although it is scarcely open to doubt that
this important and difficult problem has now, as regards
both the polar and the mean equatorial circumference,
been solved with an amount of precision which can
scarcely be surpassed, it is certainly the case that the
various irregularities of the terrestrial figure have not
even yet been fully ascertained. Geodetic observations
are, therefore, still being energetically pursued with the
view of testing and either confirming or correcting the
information already gained, and of further adding to our
knowledge of the deviations of the figure of the Earth
from that of a true sphere. It thus comes about that,
just as the blank spaces in the map of the Earth are
gradually and with increasing rapidity being filled up,
so every year as it passes witnesses some growth in the
great tree of geodetical knowledge which has sprung up
from the tiny seedling planted by Eratosthenes of
Alexandria, in the days of the Ptolemies, a tree which,
no doubt, will ere long spread its branches over the face
of the whole Earth.

SOLAR AND SIDEREAL TIME.

SYNOPSIS.

The solar day — Its mean length — The dates on
which the solar day is nearest the mean length —
Cause of the Sun's irregularity — Constancy in
the direction of inclination of the terrestrial axis
— The plane of the ecliptic — "Ascending" and
"Descending" movements of the Earth — Variation
in distance of Earth from Sun — The equation of
time — The sidereal day — Difference between solar
day and sidereal day — Relation of this difference
to the year — Solar or tropical year and sidereal
year — The precession of the equinoxes — The star
day distinguished from the sidereal day — Effects
of precession in changing the apparent positions
of the Stars — The first point of Aries — Celestial
latitude and longitude — Right ascension and
declination — Terrestrial analogies — Use of term
"sidereal day" for time of Earth's rotation —
Difference between sidereal day and mean star
day — Variability of star day — Mean sidereal day
or time of Earth's rotation — Nutation — Aber-
ration — Proper motion — Similarities between solar
and sidereal day — Change in period of Earth's
rotation — Infinitesimal character of the change —
Contrast between the evanescence of humanity and
the constancy of the Earth's rotational movement.

SOLAR AND SIDEREAL TIME.

The solar day, considered fundamentall}^ is the interval
between two consecutive appearances of the Sun on the
meridian. This is the basis of all our principal time-
units in civil matters — the day, hour, minute, and
second. But although this is so, it is impossible to
make any direct use as a time standard of the interval
between the successive returns of the Sun to the
meridian. It is clearly essential to have uniformity in
any standard of time-measurement, and this is exactly
what the Sun does not supply. Our prime time-keeper,
strange as it may appear, is regular only in its irregu-
larity.

Thus it comes about that, in order to get rid of the
want of punctuality on the part of the Sun, we have to
fix a medium on the constant errors and nullify them
by putting the occasions on which the Sun is "before"
time against the occasions on which the Sun is "behind"
time. We, so to speak, put the credit balances against
the debit balances, and by this method we get an
average, or mean, of the interval which occui-s between
the Sun's consecutive appearances. This average forms
the mean solar day, the length of which is exactly
twent^'-'four hours; or, rather, we should say, the mean
solar day being accepted as a time-basis in civil life
we call the period of which it consists twenty-four
hours.

94 SOLAR AND SIDEREAL TIME.

It is a somewhat remarkable fact that although the
length of the solar day on the mean is twenty-four
hours, there are, throughout the year, very few, if any,
days which are precisely of this length. This is most
nearly the case on or about 11th February, 15th May,
27th July and 3rd November. Oddly enough this
happens when the difference between the time of day as
indicated by the Sun and the time as shown by any
well-regulated clock is at its maximum. It occurs, or
most nearly occurs, when the Sun, having for a space
continuously gained (or lost) on the clock, begins at
last to take the opposite course and to let the clock gain
(or lose) on it. Thus during the months of September
and October, the Sun day by day gains on the clock,
and by, say, the 3rd of November it is about 16m. 21s.
before the clock. It then begins to fall behind, but the
loss is so gradual that on the following date it is still
about 16m. 21s. before the clock. This being so, the
interval between the Sun's appearances on the meridian
on these two successive days is exactly twenty-four
hours, or, at any rate, is not more than half-a-second
different therefrom.

Of course the cause of the irregularity in the Sun's
return to the meridian, and the consequent diversity in
the length of the apparent solar day, lies in the variation
in the Earth's daily movement in its orbit. Altitough
we speak of the Sun's return to the meridian, it would
be more correct to speak of the meridian's return to
the Sun. The Earth in its orbital course is continuously
travelling around the Sun and its velocity varies with
its distance from the Sun, being most rapid when
nearest and least rapid when farthest away from the

SOLAR AND SIDEREAL TIME. 95

Sun. As the rate of the rotation of the Earth on its
axis is constant throughout the year, and is quite un-
affected by the variation in the Sun's distance or the
orbital velocity of the Earth, it is evident that, the
greater the space the Earth advances between successive
returns of the Sun to the meridian, the more must the
Earth rotate on its axis in order again to direct the
same meridian to the Sun.

It must be remembered that the Earth, although
revolving around the Sun and rotating on its axis,
maintains its axis directed to the same region in space.
We may conceive of the axis as an iron rod passing
through the centre of the Earth from one pole to the
other, around which the Earth rotates, while the axis
itself remains unaffected by the rotation or by the
changing relations of the Earth to the Sun. The com-
parative steadfastness of direction of the terrestrial axis
during the rotational and revolutionary movements of
the Earth may be very simply represented by a person
walking around a table, and continuing while doing so,
to face towards the same side of the room. A person so
doing will typify the Earth as it moves around the Sun,
in so far as the preservation of the direction of the
terrestrial axis during the rotational and revolutionary
movements is concerned.

Iij^view of the virtual constancy in the direction of
the Earth's axis, whereby, notwithstanding the con-
tinual change in the situation of the Earth, the axis
remains parallel to itself during the orbital revolution,
we may, in so far as regards the diurnal return of the
Sun to the meridian, almost consider the advance made
by the Earth in its orbit between two consecutive

96 SOLAR AND SIDEREAL TIME.

appearances of the Sun, as if the Earth were moving in
a straight course. Thus when the axial rotation is
completed the raeridian is directed to the same region
in space as it was when the rotation began, but the Sun
is no longer there. Supposing the Sun to have been on
the meridian when the rotation began, then, as the
Earth must, in relation to the Sun, pass an appreciable
distance onward in the interval between the commence-
ment and the end of the rotation, the meridian must
necessarily, in order to be directed towards the Sun at
the end of the rotation as at the beginning, be directed
backward to an extent sufficient to compensate for the
Earth's forward movement during the rotation. In fact
the excess of the apparent solar day over the time of
the Earth's rotation (this excess being a consequence of
the direction of the Earth's rotational movement) is an
almost exact indication of the angular distance in its
orbital course travelled by the Earth between the suc-
cessive returns of the Sun to the same meridian.

There is, however, another cause which affects the
return of the Sun to the meridian. We have viewed
the matter as if the Earth in its orbital progress were
constantly moving in the same geographical plane. The
plane of the ecliptic, however, as the Earth's path
around the Sun is called — from the fact that it is the
centre of the region in which eclipses occur — does not
coincide with any plane having a natural relation to
the Earth's axis of rotation. The ecliptic forms an
angle with the Earth's equator of about 23° 27'. Thus the
Earth, as it travels around the Sun, follows a course
which, in relation to the terrestrial equator, may be
described as an alternately rising and falling one. If

SOLAR AND SIDEREAL TIME. 97

The Earth in its orbital progress is moving from right to left. AB is a
terrestrial diameter, and, vrith the Earth in the first position, it is noon at
the geographical point B. The Earth's rotation is completed when AB is
again parallel to its original direction, as is the case with the Earth in the
seoond position. The solar day, however, is not completed until the geo-
graphical point B is once again directed to the Sun as is the case with the
Earth in the third position. Of course the relative proximity of the Sun is
grossly exaggerated.

Movement of the Earth in an arc of its orbit, iUustratiug
diflercnce between period of rotation and length of solar day.

98 SOLAR AND SIDEREAL TIME.

we consider the north pole as the "top" of the Earth
and the south pole as the "bottom," we may say that
from December to June the Earth follows a "descend-
ing" course, whereby the Sun is brought upward to the
regions north of the equator, while from June to Decem-
ber it follows an " ascending " course, whereby the Sun is
brought downward to the regions south of the equator.

It is clear that this "ascending" and "descending"
movement of the Earth will not in itself affect the
interval between two returns of the Sun to the meridian.
It will effect only the place of the Sun on the meridian
— that is to say the elevation of the Sun above the
horizon. Were the Earth merely to "ascend" and

The relation between the plane of the Earth's equator
and the plane of the Earth's orbit.

" descend " in relation to the Sun the solar day would
coincide in length with the time of the Earth's rotation.

It is, of course, the case that each of these forms of
movement — the movement around the Sun and the
" ascending " and " descending " movement in relation to
the Sun — enter into the Earth's daily orbital progress.
They are, however, both variable in character, and the
difference between the length of the mean solar day
and of the apparent solar day varies correspondingly.

At the winter solstice of the northern hemisphere,
which occurs about the 22nd of December, the Earth,
after its long-continued movement of " ascent," whereby

SOLAR AND SIDEREAL TIME. 99

the Sun lias been caused apparently to pass southward,
begins to reverse its movement and to " descend " in its
orbit, causing the Sun apparently to pass northward.
Just as a locomotive which comes to a stand-still and
then restarts on an opposite course moves at first very
slowly, so the Earth at first moves very slowly in its
"descending" course. Gradually the movement of
"descent" quickens until at length, at the vernal
equinox, about 2lRt March, it attains its maximum. It
then begins to lessen in rapidity very gradually until,
at the June solstice, it comes again to a stand-still.
The reverse movement commences about 21st June.
Slowly it rises to its maximum, which is attained at the
September equinox, and it then as slowly decreases in
velocity until the jnovement terminates at the December
solstice.

As may be supposed, the movement of the Earth
around the Sun lengthens and shortens with the
decrease and increase respectively in the movement of
"'ascent" and "descent." When, at the solstices, the
movement of change of plane is most nearly absent the
movement of advance in the orbital journey is greatest. ■
When, at the equinoxes, the movement of change of
plane is most rapid, the length of the day's progress
around the Sun is at a minimum.

Thus at the equinoxes the length of the apparent
solar day, or the interval which actually elapses between
the successive returns of the Sun to the meridian, differs
least from the time of the Earth's axial rotation, while
at the solstices the difference is greatest. Of course in
every case the apparent solar day is longer than the
time of rotation, as, evidently, even when the change of

100 SOLAR AND SIDEREAL TIME.

plane is gi-eatest, orbital progress is far from being
altogether absent. The effect, however, of the two
characteristics of the orbital movement which we have
described is to give us the shortest solar days at the
equinoxes and the longest at the solstices.

Now comes in the effect of the variation of the
distance of the Earth from the Sun. We are nearest to
the Sun in the beginning of January, and farthest from
the Sun in the beginning of July, the difference between
our distance from the Sun on these occasions being
about three millions of miles. As the mean distance of
the Earth from the Sun is about 92,897,000 miles, we
may take it that we are about 1^ million miles less than
the mean at the beginning of January, and about the
same amount more than the mean in the beginning of
July. This efiect therefore acts in opposite ways at the
two solstices. At the December solstice our 'orbital
progress is further increased by our nearness to the Sun
and our consequent greater velocity. At the June
solstice the opposite is the case. Thus it comes about
that while at both solstices the character of the orbital
movement conduces to the lengthening of the apparent
solar day as compared with the movement at the
equinoxes, the variation in the distance of the Sun
conduces to the further lengthening of the apparent
solar day at the December solstice, but to its shortening
at the June solstice.

Of course the actual difference in length between the
apparent and the mean solar day is really inconsiderable.
Yet it is the case that when the difference is greatest —
which occurs, as we have seen, at and near the December
solstice — the apparent solar day is for some weeks in

SOLAR AND SIDEREAL TIME. 101

succession about half-a-minute longer than the mean
length, a difference which by accumulation becomes
very appreciable. The correction which is necessary to
bring the apparent solar day into harmony with the
mean length of the solar day is called " the equation of
time," and is given in many almanacs as the difference
between the Sun and the clock.

The sidereal day is usually accepted as the time of
the Earth's rotation on its axis. This, as we have
seen, is invariably shorter than the apparent solar
day. The Earth, in fact, makes an axial rotation in
23h. 56m. 4"09054s. of mean solar time. As the time
of a planet's rotation is really the measure of its day,
this period is the true length of the terrestrial day. As,
however, mundane matters depend more on the rotation
in relation to the Sun than on the rotation period itself
the solar day is, of necessity, of chief importance in all
civil affairs.

The difference between the mean solar day and
the sidereal day, or time of the Earth's rotation, is
Sm. 55*90946s. This is the time-measurement of the
angular distance which the Earth advances in its orbital
course in the period of rotation, taking the latter to
represent 360°. If we divide 3m. 55'90946s. into the
time of rotation — 23h. 56m. 4*09054s. — we get the length
of the solar year in days.

A little consideration will make it clear that in a
complete orbital revolution — with the terrestrial axis
remaining parallel to itself during the revolution — the
difference between the number of mean solar days and
the number of sidereal days must amount to exactly
one. Just as a traveller around the Earth in an easterly

102 SOLAR AND SIDEREAL TIME.

direction changes his position in relation to the Sun to
such an extent that on the completion of his journey he
will have gained exactly one day, while a traveller in the
opposite direction loses exactly the same period, so the
Earth in travelling around the Sun loses exactly one day
on the sidereal year; or, conversely, the sidereal year
gains exactly one day on the solar year, the number of
mean solar days in the year being precisely one less
than the number of sidereal days. Supposing that, with
the terrestrial axis steadfast, the Sun and a certain star
were, to begin with, on the meridian together, the Sun,
in consequence of the Earth's movement around it, will
every day reach the meridian a little later than the
star, so that in the course of a year it will have
apparently fallen behind by the complete circuit of the
heavens. Consequently the Sun and the star will then
again appear to occupy the same position in the heavens.
Thus the Sun, on the average, falls behind the star
daily by 3m. 55*90946s. — being the difference between
the mean length of the solar and the sidereal day — and
at the end of a year this difference amounts to the time
of the Earth's rotation (being 23h. 56m. 409054s.), that
being of course the time occupied by a meridian in
making (through the rotation) the circuit of the heavens.
It follows that if we divide the time of rotation by the
difference between the solar and the sidereal day we
shall get the length of the year. As in this proceeding
the diference between the solar and the sidereal day
represents one day so the time of rotation proportionally
represents in days the period of the complete orbital
movement. It will be found that the time of rotation
contains the difference specified 365"2422 times. Thd

SOLAR AND SIDEREAL TIME. 103

A**-

The Earth is moving in its orbit in the direction shown by the lai^e
arrows, while rotating as indicated by the small arrows. AB is a terrestrial
diameter. With the Earth in the position marked 1, the geographical point
B has both the Sun and a certain star on its meridian at the same time. It
is evident that B will not again have the Sun and star on the meridian
together until the orbital revolution of the Earth is completed. It will be
noticed that when the orbital revolution is half completed, the Sun is on the
meridian at B when the star is on the meridian at A, there being thus an
interval of about twelve hours between the Sun's and the star's appearance
on the meridian. A corresjionding difference in the subsequent semi-revolu-
tion makes up exactly one day.

Diagram illustrating how there is a diflference of exactly one day
betweeu the solar year and the sidereal year.

IM SOLAR AND SIDEREAL TIME.

fractional part represents 5h. 48m. 46s., making the
complete period 365d. 5h. 48m. 46s. This is, in fact, the
length of the tropical year, or the interval between two
successive returns of the Sun to the same tropic or to
the equator.

It has now to be noticed that the return of the Sun
to the same relation to the Earth — for instance, to the
same tropic, or to the equator at the same season of the
year — does not exactly coincide with the Sun's return
to the same relation to a star. We have spoken of
the Earth's axis as remaining parallel to itself during
the orbital revolution. We have now slightly to qualify
this statement. The axis of the Earth has in reality an
exceedingly slow conical motion around the pole of the
Earth's orbit. The period of the complete revolution
(that is, 360 degrees) of the axis in this conical move-
ment is about 25,868 years. It follows that during the
time of the Earth's orbital revolution the angular dis-
tance travelled by the axis is about 50'1". This move-
ment of the axis is in such a direction as to hasten the
return of the Sun to any particular portion of the Earth,
thus shortening by a slight extent the tropical or solar
year — or it would, perhaps, be more correct to say that
the conical movement described by the Earth's axis has
the effect of slightly lengthening the sidereal year by
retarding the return of the relations between the Earth
and any particular star on or near the plane of the
Earth's orbit.

But for this movement of the axis the completion of
the solar year would coincide with that of the sidereal
year. As it is, however, the Sun returns to a tropic, or
to the equator at the same season of the year, about

SOLAR AND SIDEREAL TIME. 105

20m. 236s. before the completion of the sidereal year,
the length of the latter being 365d. 6h. 9m. 963. Now
it will be remembered that on dividing the difference
between the solar and sidereal day into the time of the
Earth's rotation, we obtained the length of the tropical
year. It would naturally seem that the period obtained
should be the length not of the tropical but of the
sidereal year. This would certainly be the case were it
not for the fact that in the determination of the length
of the sidereal day, and, consequently, also in the deter-
mination of the period constituting the difference between
the length of the solar and that of the sidereal day, the
effect of this movement of the terrestrial axis in relation
to the return of a star to the meridian is allowed for.

This conical motion of the terrestrial axis is known
as the "precession of the equinoxes," the name having
reference to the fact that, in consequence of the move-
ment, the Sun's return to the equinox celestially, or to
the equator terrestrially, precedes the completion of the
sidereal year.

Although the sidereal day — using the name as im-
plying the period of the Earth's rotation — is not affected
by precession, the star day — as we may term the interval
between successive returns of a star to the meridian-
is affected by it, and the effect is of a variable character.
As there is no matter which more profoundly influences
the relations between solar and sidereal time than pre-
cession, it is necessary in dealing with these relations to
give special attention to this matter.

It was, indeed, through its apparent effects in chang-
ing the positions of the stars that the precession of the
equinoxes was first discovered. Hipparchus, the greatest

106 SOLAR AND SIDEREAL TIME.

astronomer of ancient times, who lived in the second
century before our era, in constructing a catalogue of
the stars, compared his own observations with observa-
tions made by others about one hundred and fifty years
earlier. He found that the distance of the stars from
the point at which the Sun crosses the celestial equator
at the spring equinox — the position known as the First
point of Aries — was quite appreciably different from
that previously noted. Thus the bright star Spica, in
Virgo, was found to be about two degrees farther away
from the equinoctial point, measuring eastwards, than
it had been at the earlier date. Assuming the sub-
stantial accuracy of the two observations, this inferred
an eastward change of position of about 48" per annum.
Hipparchus satisfied himself that the changes which
appeared to have occurred could not be accounted for on
the supposition of erroneous observation. The changes
were too numerous and too consistent for this. He
found that they could be explained most satisfactorily
by an actual change of place, in a westward direction
on the ecliptic, of the equinoctial point itself. Now the
equinoctial point, or the position occupied by the Sun
at the vernal equinox, is simply the point at which the
celestial equator — or the celestial plane of the terrestrial
equator — intersects the Sun's apparent path at that
season. Hipparchus found that the angular distance of
the stars from the ecliptic was unchanged. The con-
clusion was inevitable that the celestial equator was
itself inconstant, and, as the celestial equator is simply
an extension of the Earth's equator, it was clear that
the latter must have changed its plane in relation to the
heavens. As the intersection at the spring equinox of

SOLAR AND SIDEREAL TIME. lOT

these two great circles of the heavens — the ecliptic and
the celestial equator — is the point in relation to which
the position of the stars is determined, it is clear that
any change of position of this point necessarily produced
an apparent motion of the stars.

Now as the position of the celestial equator is deter-
mined by the position of the terrestrial equator, and as
the position of the latter and the positions of the terres-
trial poles are mutually dependent — every point on the
Earth's equator being necessarily ninety degrees from
each pole — it is clear that the movements of the celestial
equator could be explained by a movement of the terres-
trial poles. Again, as the movement of either pole
necessarily infers a converse movement of the other, any
movement of the terrestrial poles suggested a shifting of
the Earth's axis in its relation to the heavens as a reason-
able explanation of the apparent stellar displacement.

It was not, however, until long after the time of
Hipparchus that it was concluded that the explanation
of precession was to be found in a swaying or gyratory
movement of the terrestrial axis as the Earth rotated
and revolved, just as a spinning top may sway from
side to side while at the same time it spins around.

This explanation was put forward by Copernicus in
1543, but was only physically accounted for in 1685,
when Newton made public his discovery of the law of
gravitation. The precession of the equinoxes was then
explained as being occasioned by the gravitational in-
fluence of the Sun and the Moon on the equatorial
bulge in the figure of the Earth. The force of gravity
operating on this irregularity of figure would necessarily
cause the axis to sway, and would consequently result

108

SOLAR AND SIDEREAL TIME.

in the axis gradually describing a circle around the pole
of the ecliptic. The radius of this circle would corres-
pond with the inclination of the Earth's axis to the
ecliptic, being about 23^ degrees. The period of this
revolution is, as we have seen, about 25,868 years, the
annual angular extent being about 50"1 seconds of arc.

>-^

Conical circuit described by the Earth's axis in about 25,868 years through the
precession of the equinoxes, showing how the Earth sways like a spinning top.

Tlie ellipses above and below the figures of the Earth represent the circle which the
terrestrial axis describes in the heavens, while the arrows indicate the direction of the
movement of the axis in its precessional swing.

The effect of precession in relation to the Sun is to
restore it to its place terrestrially before it has returned
to its original position celestially. The Sun yearly
makes an apparent journey around the Earth in an
easterly direction. Starting, as we may suppose, from
the First point of Aries in the heavens and from the
equator on the Earth in its northward progress it
returns to the equator in its next northward journey,
in about 365d. 5h. 48m. 46s., but it is still a little to the
west of the celestial point from which it started. It
requires about 20m. 23'6s. in order to regain its celestial
starting point. We, however, name the celestial position
at which it is now again on the equator "The First
point of Aries," transferring the name from the celestial

SOLAR AND SIDEREAL TIME. 109

point at which the Sun's apparent journey began. As
we have the equinox — or day and night equal to each
ether — when the Sun is on the equator, the annual
return of the equinox precedes the return of the Sun to
its former place in the heavens. Hence the name " the
precession of the equinoxes," and the use of the term
'•'equinox" in reference to the points at which the
ecliptic intersects the celestial equator.

Of course the same reasoning applies exactly to the
autumnal as to the vernal equinox; and the Sun's
return to the equator in September, just as in March,
precedes its return to the point which it occupied in the
heavens when it was on the equator a year previously.
The vernal, or March, equinox is, however, the point of
prime importance in astromonical observations. From
this point right ascension and longitude are measured
around the heavens in an easterly direction.

It will be seen that this continuous change of posi-
tion of the First point of Aries is a little peculiar in its
effect on star observation. Celestial latitude and longi-
tude are measurements in relation to the ecliptic and
the poles of the ecliptic. Right ascension and declina-
tion, on the other hand, are measurements in relation to .
the celestial equator — or, as it is alternatively called, the
equinoctial — and its poles. The longitude of a star is its
angular distance from the First point of Aries measured
in an easterly direction on the ecliptic; the right ascen-
sion of a star is its angular distance from the First
point of Aries measured in an easterly direction on the
celestial equator. The latitude of a star is its angular
distance from the ecliptic measured in a northerly or
southerly direction towards the pole of the ecliptic

110 SOLAR AND SIDEREAL TIME.

The celestial equator or equinoctial forms an angle with the ecliptic (or
Sun's apparent annual course) of about 23° 27'. The two points of inter-
section of these great circles of the heavens are known as the equinoctial
points and are indicated in the diagram by the signs '"p (Aries) and zC^ (Libra).
Right ascension and declination have reference to the celestial equator and
its poles. The former is the angular distance measured on the celestial
equator eastward (being towards the left in the diagram) from the inter-
section marked T* — which is called the First Point of Aries, The latter is
the angular distance north or south of the celestial equator. Celestial
latitude or longitude, on the other hand, have reference to the ecliptic and
its poles, the former being the angular distance north or south of the ecliptic,
and the latter the angular distance measured on the ecliptic in an easterly
direction from the First Point of Aries.

Celestial latitude and longitude, and right ascension
and declination.

SOLAR AND SIDEREAL TIME. Ill

nearest to the star; the declination of a star is its
angular distance from the celestial equator measured
in a northerly or southerly direction towards the pole of
the heavens nearest to the star.

As precession does not effect the ecliptic — which is
simply the Earth's course in its orbit or, as it appears to
us, the Sun's annual course in the heavens — it, of course,
does not affect any star's relation to the ecliptic. Con-
sequently the latitude of a star, being its angular dis-
tance north or south of the ecliptic, is totally unaffected
by precession. The longitude of a star, however, is
affected by precession but in a purely artificial manner.
Longitude, being the angular distance along the ecliptic
measured in an easterly direction, must necessarily
change with any change in the position of the point
from which the measurement is made; and we do
actually shift this point by oO'l" westward every year.
It is exactly the same as if we should, to commence
with, measure terrestrial longitude from the meridian
of Greenwich, and, in the following year, from 50"1" to
the west of that meridian, and should each year reckon
from a position 50*1" to the west of that from which we
reckoned in the preceding year. We should also, to
preserve the analogy, have invariably to call this shift-
ing position the meridian of Greenwich.

Supposing now, that instead of measuring terrestrial
longitude to the amount of 180 degrees in both an
easterly and a westerly direction from Greenwich
meridian, we measured continuously around the Earth
in an easterly direction, until we arrived again at the
meridian of origin, it is clear that Dublin, for instance,
instead of being reckoned about 6° 20' 15" (Dunsink

112 SOLAR AND SIDEREAL TIME.

Observatory) west of Greenwich meridian, would be
reckoned 353' 39' 45" (being 360" -6° 20' 15") east of
that meridian. Then as the meridian shifted westward,
and we still measured from it in an easterly direction,
the longitude of Dublin, instead of being lessened by the
approach of the meridian, would be increased, until at
last, when the meridian actually reached Dublin, the
longitude would have attained 360°, upon which it
would at once fall to 0° 0' 0". It would then again very
slowly mount up as the meridian passed farther to the
west. This exactly corresponds with the effect of preces-
sion in relation to celestial longitude. Were the measure-
ment made from a fixed point instead of from a constantly
shifting point, the longitude, like the latitude, of a star
would be unaffected by precession. Unfortunately the
most convenient point of measurement is this shifting
position which is named the First point of Aries.

In connection with the right ascension and the
declination of a star, the effect of precession is much
more important. The celestial equator being, as we
have noticed, simply an extension of the plane of the
terrestrial equator, the celestial poles are necessarily
merely the vanishing points of extensions of the terres-
trial axis. Every change which the precessional move-
ment of the axis causes terrestrially, in relation to the
direction indicated by the axis, and in relation to the
plane of the equator, is therefore evidently reproduced
in the heavens. In fact it is simply through their
reproduction in the heavens that these changes are made
apparent to us. As the diurnal and annual movement
of the stars is an apparent one, arising from our real
movement of rotation and revolution, so any real change

SOLAR AND SIDEREAL TIME. 113

attaching to our movements of rotation and revolution
is made evident to us by its effects on the apparent
stellar movements.

Through our movement of rotation around the
terrestrial axis, the stars appear to us to describe daily
revolutions around the pole of the heavens. As the axis
of the Earth sways in its precessional movement, the
pole of the heavens is displaced and the apparent stellar
revolution takes place around a new point. As the pre-
cessional movement is never-ceasing, so there is a con-
tinuous change in the position of the celestial poles, and
as the position of the equator is dependent on that of
the poles, there is necessarily a corresponding change in
the position of the celestial equator. Thus both the
right ascension and the declination of stars are affected
by precession.

The effect can be conveniently illustrated by imagin-
ing a similar state of matters to occur geographically.
Let us suppose, for instance, that the geographical north
pole described, in a prolonged period, a circle on the
Earth having an angular diameter of about 47 degrees.
We may conceive of the pole as being situated on the
Arctic Circle, and as describing this movement on that ■
circle. As all parts of the Arctic Circle are at a dis-
tance of about 23 J degrees from the existing north
pole, the angular diameter of the Arctic Circle is about
47 degrees. Let us see how, in these circumstances,
London, for instance, would be affected geogi'aphically.
We may imagine the pole to be at present at the point
on the Arctic Circle nearest to London. The angular
distance of London from the pole would therefore be
23| degrees less than it at present is, and the latitude,

114 SOLAR AND SIDEREAL TIME.

or angular distance from the equator, would conse-
quently be 23^ degrees more than it is. As the latitude
of London is 51° 30' north, its latitude, in the given
circumstances, would be 75 degrees north. It is evident
that when the pole had accomplished one-half of its
revolution its angular distance from London would be
increased by 47 degrees, and as the equator must neces-
sarily conform to the movement of the pole — as it must
in all its parts always remain exactly ninety degrees
from the pole — the equator would, with the recession of
the pole, be drawn nearer to London. Thus the latitude
of London would have lessened by 47 degrees, being
therefore changed from 75 degrees north to 28 degrees
north. These are evidently the extremes in the varia-
tion of the latitude of London which would result in the
circumstances we have described. Thus such a move-
ment of the pole would cause the latitude of London
to vary slowly through every angular distance between
75 degrees north and 28 degrees north.

Now let us suppose, in the circumstances described,
longitude to be measured from one of the two points at
which the shifting equator (depending on the movements
of the pole along the Arctic Circle) would intersect the
existing equator. It is clear that these points of inter-
section would occur at the positions where the existing
equator would be exactly ninety degrees from the
moving pole. We have conceived that at the commence-
ment the moving pole is 23| degrees nearer to London
than the present pole. Thus, at the meridian of London,
the moving equator would be 23| degrees south of the
existing equator, while on the opposite meridian it
would evidently be 23^ degrees north of the existing

SOLAR AND SIDEREAL TIME. 115

equator. Thus the points of intersection would, it is
clear, be midway between these positions, being ninety
degrees to the east and ninety degrees to the west of the
meridian of London. Let us suppose that we measure
longitude in an easterly direction from the point of inter-
section situated (as measured on the existing equator)
ninety degrees to the east of the meridian of London.
Then the longitude of London, to start with, will be 270
degrees. Supposing the pole to describe its circuit in
a westward direction, it is clear that the points of
intersection will also pass westward at a corresponding
rate. Thus when the pole had completed ninety degrees
of its circuit, the point from which we have assumed
longitude to be measured would have passed ninety
degrees to the west along the existing equator, so that
the longitude of London, which until then would have
been slowly increasing, would have attained 360 degrees
and consequently would have returned to zero. Thus,
evidently, in the stated circumstances, the longitude
would pass through every point from zero to 360
degrees, and this evidently would be the case whatever
the position of the place.

Thus the latitude would vary between certain limits,
differing according to the position of the place, while
the longitude would vary through every arc of the
circle.

If, now, for the moving pole, we substitute the pole
of the heavens; for the existing pole, the pole of the
ecliptic; for the moving equator, the celestial equator
or equinoctial; for the existing equator, the ecliptic;
for London, a star; and for latitude and longitude,
declination and right ascension, we shall have a reason-

116 SOLAR AND SIDEREAL TIME.

able conception of the effect of precession on the appear-
ance of the stars.

In applying this illustration it is important to
observe that it is the moving pole and not the fixed
pole which really corresponds with the Earth's geo-
graphical pole. Its movement, however, is in relation
to the heavens and is not a geographical movement.
The pole moves but the whole Earth moves with it, so
that geographically there is no displacement, although, in
relation to the heavens, a circular movement is described.
As, then, it is the moving pole which represents the
actual terrestrial pole, it is evident that it must be the
moving meridian which really indicates the true time of
the Earth's axial rotation.

It will be remembered that the meridians in relation
to the moving pole are represented as having, in con-
iaequence of the movement of the pole, a very slow
westward movement on the Earth's surface, while at the
same time the Earth as a whole is rotating eastward.
Let us conceive of a portion of one of these moving
meridians as being represented by an arc in the heavens^
and let us imagine a star as marking the original posi-
tion of this arc. We shall suppose the arc of meridian
and the star to be directly over London at the com-
mencement of a rotation of the Earth. As the Earth
rotates, the arc of meridian will, we may suppose, pass
slowly westward to the extent required to maintain its
relation to the moving pole, the star on the other hand
retaining its fixed position. It is evident, in these
circumstances, that the rotation of the Earth will bring
London back to the arc of meridian before it brings it
back to the star. The time by which its return to the

SOLAR AND SIDEREAL TIME. 117

arc of meridian will precede its return to the star will,
in fact, be a measure of the velocity of the meridian's
westward progress.

As we have noticed, the moving pole and moving
meridians represent the geographical pole and meridians
of the Earth, the movement being simply relative to the
heavens. Thus the return of London to the arc of
meridian represents the completion of the Earth's axial
rotation, while its return to the star represents the com-
pletion of the "star-day." The period of rotation is
therefore slightly less than the time required to bring a
etar back to the meridian.

Thus although "sidereal day" is the name universally
applied to the time of the Earth's rotation, the use of
the term in this connection is not absolutely correct.
We should naturally accept the name as signifying the
interval between two successive appearances of a star on
the meridian, but this interval is not quite synonymous
with the time of the Earth's rotation. However, the
use of the name as descriptive of the period of the
Earth's rotation is now probably beyond correction, so
that another term should be made use of when the
period indicated is that between two successive returns'
of a star to the meridian, the name frequently used for
this purpose being " star-day."

Thus we see that " sidereal day " and " star-day " are
not merely different names for the same phenomenon,
but that they actually signify different periods of time.
We shall see also that the star day is itself variable —
that the interval between two consecutive returns of a
star to the meridian depends to some extent on the
star's place in the heavens.

118 SOLAR AND SIDEREAL TIME.

We have already noticed that the difference between
the time of the Earth's rotation — or the sidereal day —
and the mean interval between two consecutive returns
of a star to the meridian depends, in general, on the
amount of precession, or, to employ our illustration, on
the amount of westward movement of the meridian,
during the rotary period. The difference between the
sidereal day and the mean star day, applicable to the
greater part of the heavens, is indicated by the propor-
tion which the precessional movement of the Earth's
axis during a rotation-period bears to the complete pre-
cessional movement, taking the latter as corresponding
to one complete sidereal day. If we accept the complete
precessional period as 25,868 years, and the sidereal day
as 23h. 56m. 4'09054s., we find that the portion of the
precessional movement accomplished in the sidereal day
is "0000001056, on the ratio of the complete movement
being represented by 1. The same proportion of the
sidereal day is '009 of a second. This, therefore, is the
difference in time between the sidereal day and the
mean or average star day, the latter being the longer.
It will be noticed, in fact, as our. illustration indicates,
that in the complete precessional period the sidereal
days number exactly one more than the star days,
just as in the period of the Earth's orbital revolution
the sidereal days are exactly one more than the solar
days. Sir John Herschell computes the number of
sidereal days in the precessional period as 9,448,300,
and the number of star days as 9,448,299, and propor-
tionally he finds the length of the star day as equivalent
to 1 00000011, on the ratio of the sidereal day being
equivalent to 1.

SOLAR AND SIDEREAL TIME. 119

The foregoing statements apply to the portion of
the celestial sphere lying outside the circles described
by the respective extremities of the Earth's axis in the
processional period. As we have noticed, the interval
between the successive returns of a star to the meridian
is variable ; but, in this respect, a distinction falls to be
drawn between the stars within and those outside the
precessional circle. For stars outside the precessional
circle the interval varies with the star's position in the
heavens in relation to the ever-changing plane of the
celestial equator. Still, whatever the position of the
star, as long as that position is outside the precessional
circle, it is the case that the mean interval between its
successive returns to the meridian, throughout the pre-
cessional period, will bear the ratio mentioned to the
length of the sidereal day.

A distinct difference arises in regard to stars situated
within the precessional circuit described by the Earth's
axis, that is to say in regard to stars whose latitude is
greater than that of the celestial pole. In this case the
interval between two consecutive returns of a star to
the meridian varies according to the position of the star,
for the time being, in relation to the Earth and the pole
of the ecliptic.

Let us make a circle representing the precessional
circuit described in the heavens by the Earth's axis and
let us make dots at different positions on this circle
representing the geographical pole for the time being,
or, rather the point in the heavens towards which the
geographical pole is for the time directed. The arrows
at these dots indicate the direction of the Earth's
rotation, and the arrows at the circle indicate the direc-

120

SOLAR AND SIDEREAL TIME.

tion of the precessional circuit. We shall suppose the
point marked S to represent a star.

With a star in the position indicated, the interval
between two consecutive returns to the meridian will,
when the Earth's axis is directed towards the point
marked A, be almost precisely the period of rotation.
The star, it will be noticed, will then be on the meridian
at the same moment as the pole of the ecliptic As the

The centre of the circle (P) is the pole of the ecliptic.

Diagram illustrating the variation in length of star day
in relation to stars situated within the precessional circuit
described by the Earth's axis.

Earth's axis describes the movement from A to B (the
movement being by way of C), the successive returns
of the star to the meridian will gradually become more
and more delayed, until, when the axis is directed
towards a point on the circuit about midway between
C and B, the interval between the appearance on the

SOLAR AND SIDEREAL TIME. 121

meridian of the pole of the ecliptic and of the star will
attain its maximum.

Thereafter the interval will gradually lessen, the
fall, it is evident from the diagram, being rather more
rapid than the rise. When the terrestrial axis is at
last directed towards B, the pole of the ecliptic and the
star will once again come to the meridian together,
and then again the interval between the consecutive
appearances of the star on the meridian will be almost
exactly the time of the Earth's rotation. As the terres-
trial axis swings onward from B to D, the interval
between the successive returns of the star to the
meridian will continue to fall, until the point towards
which the axis is directed is about midway between B
and D, in which situation the star must appear on
the meridian an appreciable time before the pole of
the ecliptic. Then again the interval will gradually
lengthen until the axis returns to A, when once more
the star and the pole of the ecliptic will come to the
meridian together, and the interval between the suc-
cessive appearances of the star on the meridian will be
practically indistinguishable from the period of rotation.

It will be observed that the nearest approach to
absolute conformity between the star day and the
sidereal day, as indicated in our diagram, takes place
when the star has the same right ascension as the pole
of the ecliptic, while the extreme difference — whether
greater or less than the sidereal day — occurs when the
pole of the ecliptic and the star are most widely
separated in right ascension.

It is evident that every star within this circuit must
be similarly affected. Thus the interval between the

122 SOLAR AND SIDEREAL TIME.

consecutive returns to the meridian of all stars situated
within the circle, excepting only any star situated
exactly at the pole of the ecliptic, is constantly varying.
But yet it comes about, from the regularity of the
fluctuation, that the mean interval, throughout a com-
plete precessional period of about 25,868 years, between
the successive returns to the meridian of every star
situated in this region of the heavens, is practically an
exact measure of the sidereal day.

We find, therefore, that the length of the mean
sidereal day, or the time of the Earth's rotation on its
axis coincides with or at least is indistinguishable from : —

(1) The interval between two consecutive appear-
ances of the pole of the ecliptic on the meridian.

(2) The mean interval, during a precessional period,
between successive appearances on the meridian of any
star within the precessional circle described by the
Earth's axis.

(3) The mean interval, during a precessional period,
between successive appearances on the meridian of any
star outside the precessional circuit of the Earth's axis
minus (as nearly as may be) '009 of a second (being,
in time-measurement — on the basis of a sidereal day,
or 360°, to the precessional period — the portion of the
precessional circuit described in one rotation).

We have spoken of the precessional circuit as if it
were a regular and uniform circle described by the axis
of the Earth. Owing, however, to the variability in
the influence of the Sun and Moon on the irregularity
of the Earth's figure through their changing relations
to the Earth — a variability which is made evident in a
Tiodding movement of the terrestrial axis as it describes

SOLAR AND SIDEREAL TIME. 123

its precessional circuit and which is therefore called
nutation — the precessional circuit is really of a wavy
character, and this waving or nodding of the terres-
trial axis has an appreciable effect in relation to the
stars. This matter is, however, of subordinate import-
ance, and does not materially affect our subject.

It is manifest also in regard to the star day or the
successive returns of any particular star to the meridian.

The Earth is moving in its orbit in the direction shown by the arrow,
while light-rays are being received from a star situated at S^. The move-
ment of the light-rays and the movement of the Earth have jointly the eflfect
of apparently displacing the star in the direction of the Earth's progress so
that the star is seen in the direction S^. The angle made at the Earth by the
lines from S^ and S^ respectively is about 20 '5 seconds of arc.

Sketch illustrating effect of aberration of light.

that aherration—vfYnali. is the result of the combined
velocities of the Earth in its course and of the light
transmitted from the star — must have an effect on the
star's appearance and re-appearance. In fact through
the aberration of the transmitted light every star
appears to describe a very small ellipse each year,
having as its centre the point at which the star would
actually be seen were the Earth at rest. The general
effect of aberration is to make the star appear to be

124 SOLAR AND SIDEREAL TLME.

slightly in advance of its true place, the displacement
being in the direction in which at the time the Earth
is moving in its orbit.

Like aberration, prober motion, or the actual physical
change of place on the part of the stars themselves, as
well as on the part of the solar system, must have an
effect on the star day. These effects are, however, of a
variable character, and, as regards proper motion, of an
uncertain amount. While they are both of appreciable
importance in relation to the variable star day, they do
not affect our consideration of the sidereal day.

It is interesting to notice the similarities between
the solar and the sidereal day. We have seen that the
apparent solar day differs from the mean solar day, and
is itself variable. This we now see is also the case as
regards the sidereal day. As the apparent solar day is
subject to the equation of time to convert it into the
mean solar day, so the apparent sidereal day is subject
to the equation of the equinoxes to convert it into the
mean sidereal day. The mean sidereal day like the
mean solar day does not seem to be determinable by
direct observation, but is obtained by calculation based
on continued observations. In the case of both solar
and sidereal days the variation between the apparent
and the mean period is dependent on the obliquity of
the ecliptic.

We may conceive of a further similarity in these
relations, a similarity applicable to the complete cycle in
each case — solar and precessional. Thus we might speak
of the solar year and the precessional year; and, is it
not possible that the latter may be characterised by
changes analogous to those pertaining to the former?

SOLAR AND SIDEREAL TIME. 125

The time of the Earth's axial rotation — the mean
sidereal day — is probably the most unchanging pheno-
menon connected with the Earth, and yet the consensus
of scientific opinion is that even this is not invariable.

Sir G. H. Darwin in studying the effects of tidal
friction found strong evidence that in the remote geolo-
gical ages the length of the sidereal day was only a
little over two hours. But even during the period of
the existence of civilized life on the Earth, there is some
evidence of a slight lengthening of the sidereal day.
Thus a comparison of ancient and modern eclipses
suggests that the period of the Earth's rotation is
lengthening by about one two-hundreth of a second of
mean solar time in a century. If this be so then
at the beginning of our era (say 2000 years ago)
the length of the mean sidereal day must have been
about one-tenth of a second less than it now is, the
difference in time in the course of the 366 sidereal
days constituting a year being thus about 37 seconds.
Of course, also, a change in the length of the sidereal
day infers a corresponding change in the length of the
mean solar day.

After all, however, in any consideration of change in ■
the length of the sidereal day the point which appeals
to the imagination is not the variation but the amazing
constancy of the period. The history of the human
race — in so far at least as it is not founded on a purely
geological basis — goes back at furthest to about eight
thousand years before the commencement of our era, its
earliest beginnings being thus about ten thousand years
ago. Throughout these ten millenniums the Earth during
every recurring period of about 23h. 56m. 4s. has accom-

126 SOLAR AND SIDEREAL TIME.

plished a rotation on its axis, the variation in the time
of rotation at the end as compared with the beginning
being about half a second! What a profound lesson we
have here on the brevity of human existence. Dynasties
have come ; dynasties have gone. For a moment only,
comparatively speaking, have the great conquerors of
history strutted about on a small portion of the surface
of this great moving planet. They have appeared ; they
have disappeared — swallowed up in a little of the
surface dust as the Earth goes on ever rotating.

We can conceive of a time when the Earth rotated
in about two hours as it now rotates in nearly twenty-
four, a change in the ever-lengthening time of rotation
of no less than about twenty-two hours, and of this
change the recognized history of mankind applies to
only half a second. Yet how old is human history —
written and unwritten! How old even is civiliza-
tion compared, for instance, with national or dynastic
life, not to mention that of the individual! The
contrast between the evanescence of human existence
and the enduring constancy of the Earth's rotational
movement is sublime and awe-inspiring. And yet even
the rotation of the Earth itself, emblem of constancy
though it be, is subject to change. Marvellously slow as
the change is, it is present and is continuous. As
regards the rotation, indeed, the Earth may be compared
to a clock which is running down, though at an infinitely
slow rate. And man, with all his frailties and tran-
sience, is nevertheless endowed with faculties capable
of estimating the changes occurring in this great Globe
which he inhabits, and of judging of their periodic effects.
The creature of a day, confined in narrow bounds, man

SOLAR AND SIDEREAL TIME. 127

is capable of mentally grasping the ages of the past,
the aeons of the future; of estimating the magnitude
not only of this Earth, but of the system to which it
pertains, and even the distance of stars from which this
Globe is absolutely indistinguishable. This strange
contrast between frailty and aspiration may well be
summed up in the eloquent words of the Psalmist —
*' Behold Thou hast made his days as an handbreath ;
and his age is as nothing before Thee. . . ." "Thou
hast crowned him with glory and honour. Thou madest
him to have dominion over the works of Thy hands."

sryopsis.

Irregularities of the calendar — Rule as to
variation in length of month — Mnemonic rhjones
for determining relatively the week-day opening
the respective months in the same year — Fixing
day of week of given date in another year, past
or future — Hypothetical question in general know-
ledge — "Perpetual Calendars" — Bules regulating
the calendar — Adoption of New Style in British
Dominions — Facts simplifying ascertainment of
day of week of any given datfr — Determination of
day of the week of given date in current month or
current year — Application of the method to other
periods — Fractional portion of leap-year period —
Practical examples — Allowance for change of
Style — Application of system to foreign countries —
Diversities in regard to adoption of New Style —
Date-difference between Old and New Styles and
its cause — Julian chronology — Changes in date of
commencement of year — Additional examples of
calculations illustrating the system.

A SIMPLE MEANS OF ASCERTAIN-
ING THE DAY OF THE WEEK
OF ANY GIVEN DATE IN
THE CHRISTIAN ERA.

A SIMPLE MEANS OF ASCERTAIN-
ING THE DAY OP THE WEEK
OP ANY GIVEN DATE IN THE
CHRISTIAN ERA.

Amongst the minor ills of civilization we may fairly
place the irregularities of the calendar. The length of
the calendar-month, or the day of the week on which
any particular date falls, is — like the weather — delight-
fully uncertain. The calendar in this respect would
seem to uphold the old proverb, "Changes are lightsome,"

The first and simplest of the difficulties arising from
the vagaries of the calendar is the recollecting of the
number of days in any particular month. We have a
limited choice, ranging from 28 to 31. Have we not all in
our childhood been irritated by this uncertainty, although
with advancing years our difficulties may be forgotten ?

The rule in this matter is that the months consist of 31
days and 30 days alternately — commencing with 31 days
— subject to only two exceptions. The exceptions are
February and August. The former, according to the rule,
should consist of 30 days,but it,of course, actually has only
28 days in common years and 29 in leap years. August,
the second exception, with its 31 days, is, as it were, in-
terpolated, and in following the rule must be disregarded.

Greater difficulty is occasioned by the calendar's
irregularities when, in the absence of any appropriate
calendar, we wish to ascertain the day of the week of

32 THE DAY OF THE WEEK

either a past or a coming date. Any method which
enables us readily to fix the relative connection of the
opening day of the respective months is of assistance in
quickly overcoming this difficulty in its relation to the
current year. The late Rev. James Gall, of Edinburgh,
suggested a short rhyme as a convenient mnemonic for
this purpose. It enables us to determine the day of the
week with which any month begins when the first day
of any one month is already known : —

Let April and July have one,
September, December, one more ;
Let June for the third stand alone,
Feb., March, and November take four;
Five to August, six to May, must be given,
October and January count the full seven.

Thus if, for instance, Sunday is the Jlrst day of
April, it is also the Jlrst day of July. It is, however, not
the first but "one more" than the first — that is to say
it is the second — of both September and December. The
same day of the week is the third day of June; the
fourth day of February, March and November; the fifth
day of August; the sixth day of May, and the seventh
day of both January and October.

The following rhyme, though longer, is perhaps
more easily remembered. It is based on the idea of
making a week commence with the first day of January
— whatever day of the week that date may really be —
the relative position of the first day of every other
month in the same year being fixed therefrom : —

With January 1, our toeek begins,

October the 1st repeats ;
The second day commences May,

The third day Attgust greets.

OF ANY GIVEN DATE. 33

The fourth a trio ushers in,

March, February, Novemher ;
The fifth brings June, the sixth brings both

September and December.

Of week-days only one remains,

And it we must apply
Unto the two remaining months.

Viz. : — April and Jtdy.

But in leap-year, 'tis very clear.

Each month moves one day on,
Excepting only January

And February alone.

It is, however, when we come to deal with other
years than that current, and endeavour, in the absence
of a calendar for the specified year, to determine the
day of the week of any given date, past or future, that
the irregularities of the calendar are most in evidence.
The difficulties, of coui-se, increase with the separation
of the given year from that presently running its
course.

There would probably be some resentment on the part
of the candidates if the following question were sub-
mitted at an examination in general knowledge: —

The 1st of January of the year 2000 is a Saturday.
King John signed the Great Charter on 15th June
1215. On what day of the week was the Charter
signed?

Yet, after all, the question is not really difficult, and
the information supplied in the question itself should
enable any student of reasonable education to work out
the answer.

There are, of course, various ingenious tables which
form what are called "Perpetual Calendars," by which

34 THE DAY OF THE WEEK

the day of the week of any date in our era can readily
be calculated. Perhaps the simplest and most compact
of these "Perpetual Calendars" is that devised by Col.
F. W. M. Spring, late of the Royal Artillery, which
appears annually in a well-known almanac. Usually,
however, if one has any occasion to enquire as to the
day of the week upon which any event occurred, a
" Perpetual Calendar " is not immediately available.
In any case such information as may be desired can
generally be worked out without any such calendar
being consulted.

It is necessary only to keep in mind the ordinary
and well-known rules which regulate the calendar.
These are as follows: —

1. Every year exactly divisible by 4 is a leap year,
excepting only the century years, which are leap
years only if exactly divisible by 400.

2. The New Style was adopted in the British
Dominions in 1752, the 3rd September of that
year being called 14th September — eleven days
being thus omitted.

3. Prior to 1752, all the century yeai-s (being exactly
divisible by 4) were leap years.

If the period dealt with does not comprise a century
year not exactly divisible by 400 (or the date of the
introduction of the New Style), the calendar is repeated
after an interval of twenty-eight years, or any multiple
thereof. Thus the calendar for 1801 was applicable to
1829, 1857, and 1885. It would not be applicable,
however, to 1913, although the interval between 1885
and 1913 is twenty-eight years, as the year 1900 is
comprised in the period. That year not being a leap-

OF ANY GIVEN DATE. 35

year, throws the repetition a year forwai-d, so that the
1885 calendar is not repeated until 1914. This repetition
of the calendar, in the usual case, after an interval of
twenty-eight years arises from the fact that 28 is the
product obtained when 4 is multiplied by 7, the former
number applying to successive leap-years and the latter
to the recurrence of the same day of the week.

The ascertainment of the day of the week of any
given date is simplified by the fact that, in all cal-
culations, complete multiples of 7 (being the number of
days in the week) drop out of reckoning, remainders
only being of importance. Thus in a common year the
number of days is 365, and if this number is divided by
7, the remainder is 1. It follows that if the period
intervening between any two given dates is 365 days
these dates as week-days will differ only by 1. If our
calculation is from an earlier to a later period, that is to
say forward in date, then the day which it is desired
to find will be one day later in the week than the day
of the earlier date. If, on the other hand, our cal-
culation is backward, then the day which it is desired
to find will be one day earlier in the week than the
day of the date from which we are reckoning. The
weekly reckoning, therefore, is always in the same
direction as the calculation. If we calculate forward
we go forward in the week, and if we calculate back-
ward we go backward in the week. Thus 1st May
1909, being Saturday and the interval between that
date and Ist May 1910 being 365 days, the later date
must necessarily be Sunday — that is to say, the re-
mainder of 365 when divided by 7 being 1, and the
calculation being forward, the later date is one day later

* See pajjcfi 144 and 145.

36 THE DAY OF THE WEEK

in the week than the earlier. Conversely, 1st June
1910 being Wednesday, and the period intervening
between that date and 1st June 1909 being 365 days,
the last-mentioned date must be Tuesday. Similarly, if
leap-year separates the two dates, there is a change of
two days, which arises from the fact that when 366 is
divided by 7 the remainder is 2.

The same principle of ignoring complete multiples
of 7 and reckoning only tlie remainders applies in
estimating the change of week-day resulting from
change of date in the same year. Thus the number of
days in January is 31, which, if divided by 7, has 3 as
remainder. It follows that corresponding dates in
January and February differ by 3 as days of the week.
If Ist January is a Sunday, 1st February, therefore,
must be Wednesday. Conversely, if 1st February is
Sunday, 1st January must be Thursday — being the third
day earlier. The same relation must necessarily apply
to any other corresponding dates in these months.

If, now, we go a step further, we find in dealing with
the common year of 365 days that corresponding dates in
the various months will as week days vary as follows: —

January to February

(31 days-

-remainder 3) 3 days later

February to March . .

. (28

>»

0) same day

March to April

.(31

3) 3 days later

April to May

. (30

>»

2)2 „ ,

May to June

.(31

>»

3) 3 „ ,

June to July

(30

>»

2) 2 „ ,

July to August

.(31

}»

3) 3 „ ,

August to September

(31

>»

3)3 „ ,

September to October

(30

2) 2 „ ,

October to November

(31

>»

3)3 „ ,

November to December (30

»>

2) 2 „ ,

December to January

(31

3)3 „ ,

OF ANY GIVEN DATE. 37

Thus from any date in January to the same date in
December the variation of week-day can be got by
adding up the various particulars, omitting only the
last — being the number of week-days (3) separating a
date in December from the corresponding date in
January. If so added the figures amount to 26, which,
if divided by 7, leaves 5 as remainder. Consequently
any date in December in a common year is the fifth
week-day after (or, which is, of course, the same thing,
the second week-day before) the day of the week of the
corresponding date in January. If 1st January were
Sunday, 1st December would, therefore, be Friday, and
conversely, if 1st December were Sunday, 1st January
would be Tuesday.

The same principle necessarily applies in fixing the
day of the week of different dates in the same month.
From the 1st of any month to the 25th of the same
month is 24 days. If 24 be divided by 7 the remainder
is 3, so that the 25th of the month must occur on the
third day of the week after that on which the 1st of
the month occurs. If the 1st is Wednesday, the 25th
must be Saturday. If the 25th is Wednesday, the 1st
must be Sunday. A similar state of matters applies,
of course, to any two dates in the same month.

In calculating from one period to another, whether
in the same year or in different years, the earliest day
of the period is not included while the latest day of the
period is included. This sounds a little complicated and
difficult to keep in mind, but in practice it is not so, as it
is the course invariably adopted without thought. Thus
if we reckon the variation of week-day between 10th
January and 27th August we should proceed as follows: —

THE DAY OF THE WEEK

Day*

Jany. (31-10 = 21. Divide by 7, remainder 0)

Feby. (28 days. „ „ 0) —

Mar. (31 „ „ „ 3) 3

April (30 „ „ „ 2) 2

May (31

June (30

July (31

Aug. (Tothe27th = 27days.

3)
2)
3)
6)

7)19(2
14

Remainder.

Consequently 27th August in common years falls on the
fifth day of the week after that on which 10th January
falls. If the latter falls on Sunday, the former must
fall on Friday.

If. again, we reckon backwards between two dates,
desiring to know the day of the week on which, say,
the 19th of March fell, from our knowledge of the day
on which 29th December has fallen, we should proceed
as follows : —

Dec.

(29

days included.

Divide by 7, remainder

Nov.

(30

days.

» )f

Oct.

(31

»j >i

Sept.

(30

>i >»

Aug.

(31

>> >>

July

(31

>> It

June

(30

>t »i

May

(31

» »

April

(30

)i i>

Mar.

(31-

-19 = 12.

Remainder

Days
1
2
3
2
3
3
2
3
2
5

7)26(3
21

Thus the 19th of March (being earlier than 29th

OF ANY GIVEN DATE.

December) occurs on the fifth daj'^ of the week before
(or the second after) that on which 29th December
occurs. If 29th December is Saturday, tlie 19th of
March must necessarily be Monday.

In reckoning the number of leap yeai"S occuring
between different dates, due recognition must be given
to the fractional portion of a leap-year period remaining
over after allowing for the complete number of these
periods. A leap-year period is four years; but though
a period of four years cannot possibly comprise more
than one leap year — although, of course, it may by the
addition of a single day be made to comj)rise two leap-
years — yet three years, or two years, or one year, iniay
comprise a leap-year, or on the other hand, may not.
This allowance can only be made by observation of the
terminal dates of the period dealt with.

Let us now come to practical examples.

(1). 10th December 1911 being Sunday, what day of

the week was 1st February 1901 ?
Find, to begin, the day of the week of 1st February 1911.

December (10 days)
November

3
2

October

September
August

2
3

July ...
June ...

May ...

April

March

3
2
3

February (28 - 1 = 27)

7)32(4
28

Rcmai

nder

40 ' THE DAY OF THE WEEK

Therefore 1st February 1911 is the fourth day of the
week hefore that on which 10th December 1911 falls.
As the latter is Sunday the former is, therefore,
Wednesday.

From this find 1st February 1901.

Subtracting 1901 from 1911 the remainder

is 10

Dividing this by 4 to get leap-years, we get 2
and the remainder is 2.

This shows that the period comprises 2 complete
leap-year periods with 2 years over. Adding
2 to 1901 we find we have gone back only to
1903 in getting these two leap-year periods and
as the date dealt with is 1st February, the exact
time to which we have gone back is 1st Feb-
ruary 1903. We have, therefore, to notice
whether any leap-year day is included in the
two years not reckoned, being those between
1st February 1901 and 1st February 1903.
It is clear that such a day is not included,
neither 1902 nor 1901 being divisible by 4,
The total, therefore, comes to 12 (being one
day for each of the ten years and one day
additional for each of the two leap-years com-
prised in the period), which divided by 7 has 5 7)12(1
as remainder. Consequently the 1st of Feb- 7

ruary 1901 was the fifth day of the week 5

before that of 1st February 1911. As the
latter is Wednesday the former was, therefore,
Friday.

It will readily be seen that, in order to ascertain
whether a leap-year day is comprised in the fractional
part of a leap-year period remaining over after the
years intervening between the dates are divided by 4,
it is necessary only to notice whether the remaining
period includes any j'^ear divisible by 4 (omitting the

OF ANY GIVEN DATK 41

century), and, if so, whether the last day of February
of that year is included in the period dealt with. It
will also be observed that in dealing with the various
figures prior to division by 7 it is quite unnecessary to
add them up formally, as all that is desired is to find
the remainder. In the process of addition whenever
7 is attained that sum can be dropped and only the
balance, if any, carried on in the addition. Thus the
remainder will immediately be ascertained without
ascertainment of the full amount, or any process of
division. Keeping these points in mind let us take
another simple example.

(2). 5th May 1902 was a Monday. What day of the

week is 12th August 1919?
Find to begin with the day of the week of 12th August
1902.
May (31 - 5 = 26. Divide by 7, remainder 5) 5
June (30 „ „ „ 2) 2

July (31 „ „ „ 3) 3

Aug. (To the 12th =12 days. „ „ 5) 5

Remainder 1_^

Tims 12th August 1902 was the first day of the week
after 5th May 1902, and as the latter was Monday the
former was, therefore, Tuesday.

If then, 12th August 1902 was Tuesday, what is 12th
August 1919? 1919

1902

Divide by 4 to get leap-years. 4)17

4—1

As there is a remainder of 1 year we have
reckoned only to 12th August 1918. Is there
any leap-year day between 12th August 1918
and 12th August 1919?

As there is not, the total is 21

42 ' THE DAY OF THE WEEK

As there is no remainder on dividing by 7, the 12 th of
August 1919 must fall on the same day of the week as
12th August 1902, being, therefore, Tuesday.

We may now take one or two examples which
include century years.

(3). 24th April 1908 being Friday, what day of the week
was 9th November 1841 1

April (30 - 24 = 6) 6

June ...

July

August

September

October

November (9 days) ...

Remainder ...

Therefore 9 th November 1908 was the third day after
Friday, that is Monday. 1908

1841

4)67
16—3
Remainder, 3 years, consisting of the period
from 9th November 1841 to 9th November
1844.

The year 1844 being a leap-year, and 29th
February of that year being included in the
period, we have to add 1 1

Total 84

As there is a century year included (1900)
not divisible by 400 and therefore not a leap
year, we now subtract 1 1

7)83(11
77
Remainder 6

OF ANY GIVEN DATE. 45

Thus 9th November 1841 was the sixth day before 9th
November 1908. The latter being Monday, the former
was, therefore, Tuesday.

(4). The 9th of November 1841 being Tuesday, what
day of the week is 25th December 2135 1
November (30 -9 = 21, no remainder) ...
December (25 days, remainder 4) ... 4

Remainder ... ... 4

Therefore 25th December 1841 was the fourth day
after Tuesday, that is, Saturday. 2135

1841
4)294
73—2

The fractional part of the leap-year period
is two years, being, therefore, the interval

between 25th December 2133 and the same

date in 2135, which does not include leap-year, 367

Subtract for century j'ears not divisible by
400, and, therefore, not leap-years, being 1900
and 2100 2

7)365(52
364

Remainder.

The remainder being 1, 25th December 2135 is the
first day of the week after that on which 25th December
1841 occurs. The latter being Saturday, the former is,
therefore, Sunday.

(5). The 25th of December 2135 being Sunday, what da j'
of the week was 1st January 1799 1
As 25th December 2135 is Sunday, the 1st of January
2136 will also be Sunday. 2136

1799
4)337
84—1

44 THE DAY OF THE WEEK

The fractional leap-year period is one year, 4)337
being, therefore, the year from 1st January 84 — 1
1799 to 1st January 1800, which does not
include a leap-year day. 421

Subtract in respect of the century years
included, not being leap-years, 1800, 1900 and
2100. 3

7)418(59
413

Remainder.

The remainder being 5, 1st January 1799 was the
fifth day of the week before that on which the same date
occurs in 2136. The latter being Sunday, the former
was Tuesday.

(6). 31st January 1909 was Sunday. What day of the
week was 31st January 1780?

1909
1780
4)129
32—1
The fractional part of the leap-year period
is one year, being, therefore, the interval
between 31st January 1780 and 3l8t January

1781, which includes a leap-year day. Add

162

Subtract for century years 1800 and 1900

7)160(22

154

Remainder

The remainder being 6, 31st January 1780
must have been Monday.

It will be noticed that as the New Style was adopted
throughout the British Empire only in 1752, the utmost

I OF ANY GIVEN DATE. 45

which falls to be subtracted in respect of century years
in reckoning backwards from the twentieth century is
2. The subtraction falls to be made only for the years
1900 and 1800, the previous century years having been
I'eckoned as leap-years.

In the ascertainment of the day of the week of any
date earlier than the 3rd September 1752 in any part of
the British Dominions, allowance has to be made in
respect of the eleven days which were omitted when the
change was made from the Old Style to the New Style.
The omitted dates were the 3rd to the 13th of September
1752 both dates inclusive. The allowance required on
this account can be made in different ways. Thus we
might, if we wished to ascertain the day of the week of,
for instance, the 16th of April 1746, work out the result
in the same manner as we have already done for later
dates, ignoring the change of Style. On this result being
obtained, we subtract eleven from the date, and accept
the week day found as being the 5th, not the 16th, of
April 1746. We have then to work out the result from
the 5th to the 16th of the month. As the difference
between the 5th and 16th of the month is eleven days,
we have in this final step — according to our rule of
ignoring multiples of 7 — merely to add 4, being the
remainder when 11 is divided by 7. This would give us
tlie day of the week of the 16th of April, Old Style.
Thus if we found the answer according to the previous
method of reckoning to be, s&y, Monday, the correct
answer, on allowing for the eleven days, would be
Friday — being four days later than Monday.

Another method is to allow for the eleven omitted
daj'^s to commence with. In this case we would ascer-

46 ' THE DAY OF THE WEEK

tain the day of the week of the 27th of April in the
year from which we are calculating, and the answer
would, therefore, apply to the 16th of April 1746, being
eleven days earlier in the month. One coui-se is just as
easy as the other, but probably, there is less risk of
overlooking the requisite allowance in respect of the
eleven days if it is made to start with, rather than left
over as the final step in the calculation.

We may illustrate both processes for the sake of
clearness.

(7). The 25th of April 1908 being Saturday, what day
of the week was 16th April 1746 (being the date of
the Battle of Culloden) ?
(a) As 16 from 25 leaves 9, which when
divided by 7, has 2 as remainder, 16th
April 1908 was 2 days before Saturday,
that is, it was Thursday. 1908

Subtract 1746 from 1908. 1746

Divide by 4 to get leap-year periods. 4)162

40—2
As there is a remainder of 2 years we
have to notice whether any leap-year
day occurs between 16th April 1746 and
16th April 1748 — which is, of course,
the case. We therefore add 1 1

^03
Subtract for the century years which

were not leap-years, being 1800 and 1900 ^

Divide by 7 7)201(28

196
Remainder 5

The remainder being 5, 16th April 1746, according to
the New Style, was the fiftli day of the week before that
on which 16th April 1908 fell. As the latter was
Thursday, the former was, therefore, Saturday. To

OF ANY GIVEN DATE. 47

change from the New Style to the Old St3'le we add
four days as above explained, which makes the 16th of
April 1746 Wednesday.

(b) By the second method we account for
the eleven days to start with. Tiius to
find the day of the week of 16th April
1746 we work back, not from the 16th
but from the 27th of April 1908.

25th April 1908 being Saturday, 27th
April 1908 was Monday. 1908

Subtract 1746 from 1908. 1746

4)162
Add for leap-year periods, r 40 — 2

and for leap-year included in the frac-
tional period of two years, 1

203

Subtract for the century years 1800 and

1900 2

Divide by 7 7)201(28

196

Remainder 5

The remainder being 5, 27th April 1746, according to
the Now Style (which was 16th April 1746 according
to the Old Style) was the fifth week-day before Monday,
that is, it was "Wednesday.

We can now deal with the question submitted in an
early part of the paper.

(8). 1st January 2000 being Saturday, what day of the
week was 15th June 1215, the date of the signing
of the Magna Charta ?

January (31 - 1 =30; divide by 7, remainder 2) 2

February (Leap-year, 29 days, remainder 1) 1

March, 3; April, 2; May, 3; June, 1 (remainder 2) 2

Remainder 5

48 THE DAY OF THE WEEK

15th June 2000 is, therefore, the fifth day after
Saturday, that is Thursday. 2000

1215

4)785
The fraction of the leap-year period being 196 — 1
1 year, we see that the division by 4 takes
us back only to 15th June 1216. As
1216 was leap-year (16 being divisible by
4), and as the period in question includes
the 29th of February we have to add 1

~982
Subtract for century years 1800 and 1900 2

7)980(140
980

There is no remainder, so that, according to the New
Style, 15th June 1215 was the same day of the week
as 15th June 2000, which we have found to be Thursday.
To change to the Old Style, we add four days so that
the day was really Monday.

By the alternative method we work from 26th June
2000 to get to 15th June 1215.

1st January 2000 being Saturday we ascertain the day
of the week of 26 th June in the same year as follows : —
January, 2 ; February, 1 ; March, 3 ; April, 2 ; May, 3 ;
June (26 divided by 7, remainder 5) 5 ; in all sixteen.
Divide by 7, remainder 2. Therefore 26th June 2000 is
the second day after Saturday, that is
Monday. 2000

1215

The deduction of 2 in respect of the century 4)785
years can be made mentally when adding. 196 — 1

As there is no remainder, the 15th of June 7)980

1215 (Old Style) was the same day of the 140

week as 26th June 2000 (JSTew Style), that ~
is Monday.

OF ANY GIVEN DATE. 49

In the application of this system of reckoning past
dates to countries outside the British Dominions whose
calendar, like our own, is based on the Roman calendar,
the date of substitution of the New Style for the Old
has to be carefully noticed and the requisite allowance
made in the calculation. In the British Dominions, as
we have seen, the New Style was adopted on 3rd
September 1752, that date being changed to 14th Sep-
tember. As the British change of Style preceded the
American War of Independence, the adoption of the
New Style also took effect in the United States at the
same time, through the same Act of Parliament — 24
George II, Cap. 23.

In other countries the change was effected at various
dates and the correction made at the date of the change
thus also varied. In Spain, Portugal, and part of Italy,
including Rome, the change was made in 1582 by the
omission of ten days in that year, October 5th being
called the 15tli. In France the change was made in
December of the same year. In the Protestant States
of Germany and in Denmark and Sweden the New
Style was adopted in or about the year 1700. When
the change was made in the British Empire it was
necessary to omit eleven days, instead of, as in Rome
and elsewhere, ten days, as the year 1700 had intervened
and had been reckoned a leap-year under the Old Style
but a common year under the New Style.

In Russia and Greece and the smaller Eastern States
which adhere to the Greek Church, the New Style has
not yet been adopted, although its adoption in Russia
is at present (Autumn 1909) under consideration by
the Council of the Empire, and the Duma. In these

50 THE DAY OF THE WEEK

countries the date is now thirteen days behind that
recognized in the countries which have adopted the
New Style, the 1st January, for instance, in the former
being the 14th of January in the latter. The difference
between the number of days (eleven), which had to be
omitted when the New Style was adopted in the British
Dominions, and the number (thirteen), now separating
the dates under the New Style and the Old Style, arises
from the years 1800 and 1900 having been leap-years
under the Old Style and common years under the New
Style. The separation of the respective dates will not
further increase until the year 2100 as, under both
Styles, the year 2000 is a leap-year. The difference
between the Styles arises from the fact that by the Old
Style every century year is a leap-year, while by the
New Style no century year is a leap-year unless exactly
divisible by 400 — and it is proposed that century years
exactly divisible by 4000 (although they are necessarily
also exactly divisible by 400) should be common years.

The system of calculation now submitted would not
apply without further correction to dates preceding the
commencement of the Christian Era by more than ten
years, as up till then the cJilendar was on an exceed-
ingly unsettled basis. The Julian system of chronology
came into complete operation about the year 10 B.C.,
and as this is the basis of our calendar the method of
calculation of week-day now suggested would operate,
without further modification than has been indicated, as
far back as that date. As regards the future no limit
can be fixed to its application.

The only special qualification which has to be noticed
as regards ancient calculations is that the beginning of

OF ANY GIVEN DATE. 61

the year has to some extent been variable. Although
w hen the Julian system was first instituted, the be-
ginning of the year was made the 1st of January, this
feature of Julian chronology has not always been
adhered to by the nations whose calendars are based
on the system. In Britain from the seventh century
to the fourteenth century the year legally began at
Christmas. In the twelfth century the beginning of
the ecclesiastical year was changed from Christmas to
the 25th of March, and in the fourteeenth century the
same date came to be generally recognized as the be-
ginning of the year. By degrees, however, the first of
January afterwards came to be popularly accepted as
the beginning of the year, as this was the date of
general observance in other European countries. In
Scotland the beginning of the year was legally changed
from the 25th of March to the 1st of January in the
year 1600. The same change was made for the rest of
the British Dominions by the Act which introduced the
New Style. Previous to the enactment that the year
should begin on 1st January instead of 25th March, it
had come to be the practice to make use of two dates
in legal documents, one for the civil year beginning on
25th March and the other for the historical year
beginning on 1st January, thus — " 10th February 1679-
80." In the retracing of past dates, however, it is
generally assumed that the year began on 1st January,
tlie adoption of that date being thus accepted as
operative since the beginning of our era. This method
conduces in general to the prevention of confusion.

We may now conclude with a few additional
examples applicable to various dates.

THE DAY OF THE WEEK

(9). At the time of the Council of Nice in 325 A.D., the
vernal equinox fell on 21st March, having retro-
graded four days since the Julian calendar was
introduced. Give the day of the week on which the
equinox fell in 325 a.d„ assuming that Christmas
Day 1912 is Wednesday.
December (31 - 25) 6

January (31 divided by 7, remainder 3) 3

February (28 „ „ 0)

March (21 „ „ 0)

9 divided by 7 — remainder 2

Thus 21 March 1913 is the second day after Wednes-
day, that is Friday. 1913

325

(Subtract
and 1900)

2 for the century years

4)1588
1800 397

7)1983

283—2

We find that were it not for the omission of the
eleven days at the time of the introduction of the New
Style, the day would be ttoo days before Friday. As
that omission involves, as we have seen, a progression of
four days, the actual day was two days after Friday,
that is Sunday.

The same result, of course, is arrived at by allowing
for the eleven days to start with. Instead of working
back from 21st March 1913 we work back from eleven
days later, being Ist April 1913. We shall then, without
further allowance for the month or day of the month,
obtain the answer applicable to 21st March 325 A-D.
December ... ... ... ... 6

January
February
March
April ...

Remainder

OF ANY GIA^EN DATE. 53

Tlierefore 1st April 1913 is the sixth day after
Wednesday, being Tuesday. 1913

325

4)1588

(Subtract 2 for the century years 1800 397

and 1900 7)1983

~283— 2

The remainder being 2, the day required was the
second day before Tuesday, being Sunday.

(10. King Charles I. was beheaded at Whitehall on 30th
January 1649. Find the day of the week of his
execution from 21st March 325, a.d. — being Sunday.
March ... ... ... ... ...

February ... ... ... ... ...

January (31-30 = 1) 1

The 30th of January 325 was, therefore, the first day
before Sunday, being Saturday. 1649

325

4)1324
331

7)1655

236—3

30th January 1649 was thus the third day after
Saturday, being, therefore, Tuesday.

In this case the eleven days omitted at the change of
Style do not occur in the period dealt with.

(11). Find the day of the week of the same date (30th
January 1649) from 1st January 1909 — being Friday.

Allow to begin with for the eleven daj's omitted when
the New Style was adopted.

January (31 - 1 = 30; divide by 7; remainder 2) 2
February (10 days; „ „ 3) 3

THE DAY OF THE WEEK

February 10th 1909 is, therefore, the fifth day after
Friday — being "Wednesday. 1909

1649

4)260

We subtract 2 in respect of the century 65

years. As the remainder is 1, the day wanted 7 )323

was the first day before Wednesday, being,
therefore, Tuesday. '

46—1

(12). The battle of Waterloo was fought on 18th June
1815. Find the day of the week from 31st Decem-
ber 1908 — ^being Thursday.
January ... ... ... ... ... 3

February

March

April

May

June

Remainder

18th June 1909
being Friday.

was the first day after Thursday —

1909
1815

The fractional part of the leap-year period 4)94
being two years, which includes 29th February 23 — 2
1816, we add 1. We have, however, to sub- 1
tract 1 for the year 1900. As the remainder
is 5, Waterloo was fought on the fifth day 7)117
before Friday — being Sunday. 16 5

(13). We have found that 21st March of the year 325
A.D. fell on a Sunday. Find from that on what day
of the week the 1st of January of the year 2345 a.d.
will fall.

March ...

February ... ... ... ... ...

January (31 -1=30) __2

~~2~

OF ANY GIVEN DATE. 55

Thus 1st January 325 fell on the second day before
Sunday — being Friday. 2345 '

325

4)2020

505

Subtract for the century years 1800, 1900, 2525

2100, 2200 and 2300 not being leap-years. 5

7)2520
360—0

As the period includes the date at which the New
Style was adopted, and as that circumstance involves
an advance of four days in reckoning backward, it must
involve a retrogression or subtraction of four days in
reckoning forward. As there is no remainder the day
would, but for this, fall on Friday. As it is, it must fall
on the fourth day before Friday (or, which is, of course,
the same, the third day after Friday). Consequently the
1st of January of the year 2345 must be Monday.

We have now illustrated every kind of calculation
in relation to the system, and it is evident that no great
complication arises in any case. Thus the system in a
simple manner enables the day of the week of any date
in the Christian Era to be very readily ascertained from
the slightest data in the complete absence of any Table,
and this with comparatively little risk of error. From
the extreme facility with which the result obtained can
be checked by other similar calculations it also offers
means of easily and conveniently discovering any error
which may possibly occur in the working.

SYNOPSIS,

Civil calendar based on that of the Romans —
Ancient confusion in civil year — Julian reform of
calendar — Why August has 31 days — Why the
year begins on 1st January — The introduction of
the Julian system — The use of the system in
ancient chronology — Defects of the system — Pope
Gregory's reform — Incompleteness of the altera-
tion — The dat« of the vernal equinox — ^The real
author of the New Style — Adoption of New Style
in Britain and America — Change of commence-
ment of year in British Dominions — ^Where "Old
Style" still retained and present difference in
Styles— Day of week unaffected by change of
Style — The excellence of the Gregorian reform —
Proposed slight amendment — Practical disadvan-
tages of calendar — Want of harmony between week
and year — Proposed remedy by "non-counting"
days — The re-arrangement of the length of the
months — The desirability of year being an exact
multiple of week — How this might be attained —
Application to existing circumstances — Rules
securing harmony between week and year —
Precessional correction — Desirability of changing
commencement of year to Christmas Day — Applic-
ability of new system to months and quarters —
Its applicability to the moveable Feasts — Import-
ance of international agreement in any future
reform of calendar.

THE KEFORM OF THE CALENDAR.

.If MIS ("KSAll. 1 1. 1 ;{•_'.

Tlie itiaugiutitor of the Juliiin System of Cliiouiilogy.
{I'rOiH the bust i,t the liritUli Miuxuni).

THE REFORM OF THE CALENDAR.

The civil calendar of all European countries is based
on that of the Romans.

In ancient times the greatest confusion prevailed in
the reckoning of the civil year. Sir John Herschel
very aptly compares the history of the calendar, with
reference to chronology or to the calculation of ancient
observations in astronomy, to a clock " going regularly
when left to itself, but sometimes forgotten to be wound
up, and when wound, sometimes set forward, sometimes
backward, either to serve particular purposes and private
I interests, or to rectify blunders in setting."

In the reign of Julius Caesar, in the first century
before Christ, it was foimd that there was such extreme
divergence between the civil year and the solar year
that the vernal equinox, which according to the civil
calendar should have occurred on or about the 25th of
March, actually occurred about three months earlier, the
; calendar winter months being thus carried back into
[■autumn and the calendar autumn months into summer.
In order to put an end to the confusion thus brought
about, Caesar sought the assistance of Sosigenes, an
eminent Alexandrian astronomer and mathematician.
It is to the advice of Sosigenes that we owe the simple

131

132 THE REFORM OF THE CALENDAR.

and convenient arrangement of the introduction of »
leap year of 366 days after every three common years
of 365 days each.

In order to rectify the errors of the preceding years,
Caesar, in the 46th year before Christ, decreed that the
year then current should have two additional months
thrown in between November and December, the first
consisting of thirty-three days and the second of thirty-
four days, being together sixty-seven days ; and that in
the future every fourth year should consist of 366 days
and the remaining years of 365 days. Prior to this
change the Roman year consisted of 355 days, but in
every second year an additional month was intercalated
which consisted of twenty-two and twenty-three days
alternately. It happened that the year 46 B.C. was one
of the years containing this additional month, and that
on that occasion the month consisted of twenty-three
days. Thus the year would, independent of Caesar's
order, have consisted of 378 days, and, as he added 67
days to its length, the year 46 B,c. actually consisted of
445 days. This, in all probability, was the longest
calendar year in human history, and thus although to
a great extent it put an end to the accumulated con-
fusion of ages, it has been specially named " the year of
confusion."

When thus introducing an orderly arrangement in
regard to the year, Caesar did not overlook the import-
ance of placing the months on a convenient basis. He
instructed that every month whose place in the year
was indicated by an odd numeral — that is the first, third,
fifth, &c. — should consist of thirty-one days and that
each of the remaining months, excepting only February in

THE REFORM OF THE CALENDAR. 133

common years, should have thirty days. In leap years
February fell into the ordinary arrangement, but in
common years it was to have twenty-nine days only.
As the commencement of the year was then fixed for
the 1st of January — instead of as previously in March —
this arrangement would in leap years work as follows —
February having one day less in common years : —

January. ..31 days. May 31 days. September... 31 days.

February.. 30 „ June 30 „ October 30 „

March 31 „ July 31 „ November. ..31 „

April 30 „ August.. .30 „ December. ..30 „

Total 366 „

This convenient, simple and orderly arrangement of
the length of the various months was upset in the reign .
of Augustus, the grand-nephew of Julius Caesar, as a i|
gratification of personal vanity. The month of July
had been named in honour of Julius Caesar and August •
was called after Augustus. By the systematic method
of the length of the months introduced by Julius it
came about that July had thirty-one days, while August
had only thirty. This was clearly an indication that
Julius was greater than Augustus. It was in another
form the insinuation which vexed King Saul when the
women shouted "Saul hath slain his thousands and
David his ten thousands." Was not Augustus as
Emperor of Rome as great a man as Julius had been ?
It was absolutely necessary that August should be
made equal to July. Consequently August was given
thirty-one days to gratify the monarch's vanity.

As this change could not be made without taking
ii day from another month a further alteration was
requisite. It was decided that the required day should

134 THE REFORM OF THE CALENDAR.

be taken from February which, consequently, was left
with only twenty-eight days in the common year. It
was then seen that the addition of a day to August
would result in there being three months in succeasion
— July, August, and September — having each thirty-
one days. Ultimately September and November were
reduced to thirty days, and October and December
increased to thirty-one. The simple and convenient
plan devised by Julius Cassar was thus utterly destroyed
for the sake of flattering his successor.

It is interesting to notice how it came about when
Julius Caesar thus amended the calendar that the 1st of
January was fixed upon as the commencement of the
year. It must have been evident to Sosigenes as
to astronomers generally that the solar year may
truly be said to commence at one or other of the
solstices. Most naturally in the northern hemisphere
the winter solstice, which now occurs about 22nd
December annually, may be considered as the com-
mencement of the Sun's year. At that period the Sun,
after its prolonged journey southward, apparently turns
in its course and retraces its path northward. Thus
any astronomer arranging for the improvement of the
calendar on the basis of the Sun's movements, might
reasonably be expected to fix the beginning of the year
at the solstice, and the particular solstice selected might
reasonably in the northern hemisphere be expected to be
that which occurs in our winter. Evidently Sosigenes
was of opinion that the winter solstice was the true
period of the commencement of the year in Nature. It
is, however, equally evident that the ancient idea of
attaching importance to the state of the Moon was ever

THE REFORM OF THE CALENDAR. 135

in the mind of Caesar's advisers. The Sun might be,
and no doubt was, the prime factor to be considered in
the amendment of the calendar, but the Moon must not
be overlooked. It happened that the new Moon im-
mediately following the winter solstice at the time of
the introduction of the Julian system fell on the 1st of
January. Consequently the 1st of January was fixed
as the beginning of the year, due weight being, we may
suppose, given to the relative importance of both Sun
and Moon. Thus it came about that the beginning of
the year has been permanently separated from the
winter solstice by a mere matter of nine or ten days,
although the date of the solstice might naturally be
accepted as the true date of the commencement of the
year.

The new system introduced by Julius Caesar came
into full operation in the beginning of 45 B.C., being the
708th year after the foundation of Rome, or, according
to the old form, 708 Ah Urhe Condita (A.U.C.) — that is
** From the Building of the City." It is supposed that
CsBsar, by way of securing the intercalation, as a matter
of precedent, of the extra day which pertained to the
leap years made the first year of the new system a leap
year. Consequently, as he in all probability ordered
that " every fourth year " should be leap year, the next
leap year would be the year 41 B.C., the intervening
years (44, 43, and 42 ac, or 709, 710 and 711 A.U.C.)
being common years. In March of the year 43 B.C.,
however, Caesar waa assassinated, and this had the
unfortunate efiect o^ again throwing the calendar
into some confusic^^ A misunderstanding arose as to
the interpretation of his order, and it was agreed —

136 THE REFORM OF THE CALENDAR.

according to the priestly mode of reckoning — that in
fixing every fourth year, the preceding leap year itself
fell to be enumerated as the first of the four. Con-
sequently, as the year 45 B.C., being leap year, was the
first of the series, the year 42 B.c. would be the fourth, and,
therefore, also leap year. Thus it came about that only
two years, instead of three years as intended, intervened
between succeeding leap years. This went on for thirty-
six years in which, consequently, twelve years instead
of nine were counted as leap years. The error was then
recognized, and, in order to rectify it, Augustus ordered
that the ensuing twelve years should all be counted as
common years and that thereafter there should be three
common years between succeeding leap years. This cor-
rection restored the Julian system in its completeness.

In calculating the dates of occurrence of very ancient
events, chronologists and astronomers reckon backwards
according to the Julian chronology as if the system had
never suffered interruption, and had subsisted from all
time, although it is evident that in thus fixing the date
of any historical occurrence many tangled mazes and
obscurities have often to be encountered. Still the
Julian system has been of no little aid in the unravelling
of many entanglements of ancient date.

We have noticed that in the employment of Julian
chronology every fourth year, without exception, is given
366 days, and every other year 365 days. The number of
days in four years is, therefore, 1461, the mean length of
the year being thus fixed at 365J days — that is 365
days 6 hours. Unfortunately Nature is not so accom-
modating as this. The solar or tropical year, which is
the basis of the calendar, does not consist of 365 days 6

THE REFORM OF THE CALENDAR. 137

hours, but of — as nearly as possible — 365cl. 5h. 48m.

i6s. The Julian year is, therefore, about 11m. 148.

onger than the tropical year — a trifling error, but yet

)ne which by accumulation becomes of importance.

rhis error in the course of centuries forced itself on the

ittention of the authorities, and it was agreed that some

'urther rectification of the calendar was necessary.

I The new reform was carried into effect by Pope

Gregory XIII. in 1582. The period intervening between

. corresponding date in the years 45 B.C. and 1582 A.D.,

3 1626 years — not as might be supposed 1627 years,

ince the year 1 B.C. is immediately followed by the year

A.D. As the mean annual error or excess in the Julian

hronology is about 11m. 14s., it follows that the correc-

ion necessary in 1582, to restore the Julian system to

armony with the Sun's apparent movements, would be

626 times 11m. 14s.— that is 12d. 16h. 25m. 24s. Thus

16 best possible correction would have been the omission

f thirteen days, being the number nearest to the error

the fractional part is omitted.

The next best course to adopt would have been to
ave made the correction such as would have had the
feet of dating back, to the time of the introduction of
le Julian system, the system adopted in 1582 as appli-
kble to the future.

Pope Gregory, acting on the advice of the astro-
Dmers and mathematicians, decided that for the future
le Julian system should be amended by the application
■ a new rule in relation to the century years. Caesar
id made every fourth year — or, as it happens in our
a, every year exactly divisible by four — a leap year,
id had admitted no exception. Gregory in effect

138 THE REFORM OF THE CALENDAR.

ordered that the century years should be leap years only
if exactly divisible by 400. Thus if Gregory's rule were
dated back to the time of Caesar's amendment of the
calendar it would have resulted in changing the follow-
ing years, all of which had been reckoned as leap-years,
into common years, viz.:— a.d. 100, 200, 300, 500, 600,^
700, 900, 1000, 1100, 1300, 1400, and 1500. As the
century years changed in this way are twelve in all,
the number of days which fell to be omitted in 1582
in order thus to antedate the Gregorian system was
twelve.

Gregory, however, did not make his amendment so
complete as he should have done. He was, doubtless, a
churchman first, and a chronologist only subject to hi&
priestly bias. Instead of going back to the time of
Julius Caesar in estimating the necessary correction, he
thought it sufficient to go back to the Council of Nice,
which was held in 325 A.D. From 325 to 1582 is 1257
years, and the error in that time at 11m. 14s. per annum
is 9d. 19h. 20m. 18s. Gregory, therefore, made his cor-
rection ten days, that period being omitted in 1582 by
the 4th of October being immediately followed by the
15th of October. Csesar, when he amended the calendar,
had restored the vernal equinox to the date which had
been fixed for its occurrence in the reign of his pre-
decessor, Numa Pompilius, in the fourth century B.C.
Through the inherent defects of the Julian system,
however, the equinox had gradually and continuously
retrograded. At the time of the Council of Nice in 325
it fell on the 21st of March, and in 1582 it fell on the
11th of that month. As Gregory, in his reform of the
calendar, went back only to the date of the Council of

THE REFORM OF THE CALENDAR. 139

Nice, he restored the equinox to about the 21st of
March, and not to the date upon which Caesar had
proposed it should fall.

There is some reason, indeed, to believe that Caesar

while intending to restore the equinox to the 25th of

March failed in his purpose from the start, through

making the first year of the Julian system a leap year

nstead of a common year. The weight of evidence

jertainly indicates that the initial year of the Julian

lystem had 366 days, and this is in accordance with the

lates which even to this day happen to be leap years.

Chus it may have been the case that even in the first

^ear of the Julian system the equinox was thrown back

o the 24th of March; otherwise there should have been

•nly three days of retrogression by 325 a.d. — being the

ays arising through the century years 100, 200, and

00 being reckoned as leap years instead of common

ears — whereas there seems to be no doubt that by 325

hie equinox had retrogressed to the 21st of March,

eing four days of difference from the 25th.

I The real author of the "New Style," as the Gregorian

I p^stem is called, was, of course, not Pope Gregory him-

slf, who was necessarily guided by skilled advisers. It

as Aloysius Lilius, or Luigi Lilio Ghiraldi, a learned

jtronomer and physician of Naples, who, however, died

jfore the system came into operation. The necessary

Jculations in verification of the system were under-

Iken by Christopher Clavius, a German Jesuit and
athematician, who more than any other contributed to
ve the ecclesiastical calendar its present form, and who
^ed to see the system widely adopted, as he survived
I 1612. Gregory himself, with whose name the New

140 THE REFORM OF THE CALENDAR.

Style is most closely identified, only survived the in-
auguration of the system by three years, his death
occurring in 1585.

When Gregory introduced this most useful reform of
the calendar, England and Scotland were still rejoicing
in their recently completed revolt from the yoke of
Rome. Consequently they would have none of it. They
recalled the words of Virgil, " Timeo Danaos et dona
ferentes." Be the reform never so good, it was proposed
by the Pope, and they distrusted him. Thus it came
about that this beneficial change in the calendar was not
adopted by the British until 1752, In the interval
another century year which under the Julian system
was a leap year and under the Gregorian system a
common year had occurred. This was the year 1700.
Therefore, the change which Gregory had brought about
by omitting ten days in the year 1582 necessitated the
omission of eleven days in the year 1752. The change
was made by the Act 24 George II. Chapter 23, which
applied not only to the British Isles, but to "all His
Majesty's dominions and countries in Europe, Asia,
Africa, and America, belonging or subject to the crown
of Great Britain." As the United States of America
were then included in the British Colonies, this Act
eflTected the change of system in that country also. The
Act declared that "the natural day next immediately
following the second day of September (1752) shall
be called, reckoned, and accounted to be the fourteenth
day of September, omitting for that time only the
eleven intermediate nominal days of the common
calendar."

This Act of Parliament, with its wide application.

THE REFORM OF THE CALENDAR 141

not only eflfected the rectification of the error between
the civil year and the solar year, in accordance with
Gregory's reform, but it also changed the date of the
commencement of the year throughout the British
Dominions. Up to 1752 the British civil year, excepting
only in Scotland, had legally commenced on the 25 th
3f March. It was now enacted that it should commence
on the Ist of January. In Scotland this change of date
jf the beginning of the year from 25th March to 1st
January had been effected long previously, the year
1600 having in that country been the first year which
egally began on the date which we still recognize as
lommencing the year.

Thus in all countries subject to the British crown

except Scotland) the year 1751 consisted only of 282

lays, the months of January and February, and twenty-

our days of March having been subtracted from the

isual length of the year, while the year 1752, which

v&s leap year, had notwitlistanding this, only 355 days.

The "Old Style" is still retained in Russia and

Jreece and the other states which adhere to the orthodox

rreek (or Eastern) church, although Russia is at present

Autumn 1909) considering the advisability of adopting"

18 New Style. In all other Christian countries the

Few Style is now observed. The difierence between

le Old Style and the New now amounts to thirteen

ays — the 1st of January, for instance, by the Old

tyle, being the 14th by the New Style. As the next

intury year (2000 A.D.) is a leap-year by both Styles,

lis difference will not be further increased until the

3ar 2100 A.D.

Neither in the case of the change made by Gregory

142 THE REFORM OF THE CALENDAR

himself nor in the adoption of the Gregorian system in
Britain or elsewhere was the day of the week interfered
with. The 4th of October 1582 was a Thursday, and
though the next day, in the countries subject to the
papal order, was made the 15th of October, it neverthe-
less was still Friday. The 2nd of September 1752 in
the British dominions was Wednesday, and the following
day was Thursday although it was called the 14th of
the month instead of the 3rd.

In the British Isles, as in Rome and elsewhere, the
reform of the calendar went back merely to the year
325 A.D. and not to the year of the institution of the
Julian system which was the subject of reform. Con-
sequently the date of the equinox was restored to about
the 21st of March and not to the 25th of that month as
intended by Caesar.

The Gregorian reform of the calendar reaches a high
degree of excellence. As we have noticed, the true
length of the solar year is as nearly as possible 365d. 5h.
48m. 46s. Under the Gregorian system there are in a
period of 400 years 97 leap years of 366 days each, and
303 common years of 365 days each, the total number
of days in the period being, therefore, 146,097. By
dividing this number by 400 we get the mean length of
the year in the reformed calender, which is, therefore,
365d. 5h. 49nL 12s. The mean annual error is thus only
about 26s. of excess, as compared with 11m. 14s. in the
Julian system. Thus it would take about 3323 years for
the error to sum up to one day. Even this insignificant
error, however, it is proposed to overcome by a very
slight reform of the Gregorian system, a reform which
is in strict accordance with the system. The suggestion

THE REFORM OF THE CALENDAR 143

is that every year divisible exactly by 4000 should be
made an exception to the rule whereby century years
divisible exactly by 400 are leap years, and should,
therefore, be a common year. The proposed amendment
of the Gregorian system so exactly meets the defect
that the remaining error, being the amount by which
the calendar period would exceed the astronomical
period, would be barely five hours in four thousand
years and would not amount to a day until about 21,557
A.D. It is evident, therefore, that it is scarcely possible
bo suggest any reform of the calendar which would
bring our civil reckoning into greater harmony with
astronomical chronology than that already attained.

While this is so, it cannot be overlooked that the
calendar under our present system has certain practical
iisadvantages of a character distinct from the harmon-
zing of astronomical and civil chronology. These
irawbacks are the want of harmony between the week
md the year, and the complete absence of systematic
irrangement in relation to the months. As the year
onsists of either 365 or 366 days it is never a multiple
»f the week, and the months follow each other in such
lelightful irregularity as regards length that even the
taid man of business has often to recall the old nursery
hyme in order to satisfy his mind as to the duration
i any particular month : —

Thirty days hath September,

April, June, and November;

All the rest have thirty-one

Excepting February alone,

Which hath but twenty-eight days clear,

And twenty-nine in each leap-year.

144 THE REFORM OF THE CALENDAR.

The want of harmony also between the week and
the month is certainly a defect, no month except
February being a multiple of the week, and February
itself having the same defect every leap year.

The irregularity of the system is well illustrated in
connection with the recuiTcnce of the calendar for any
particular year. When will the calendar for the present
year again apply ? As there are only seven days on
which the year can commence, and as identity of opening
day means identity of calendar so long as both years
dealt with are of the same description — either common
years or leap years — it would seem that the determina-
tion of the next year of repetition of this year's
calendar should be a very simple matter. The • follow-
ing statement shows how greatly this question is
complicated : —

1. The calendar for the first year after leap year is

repeated six years later and is itself a repetition of
that for the eleventh year earlier. Exception : — The
calendar for the year '97 prior to a century year not
exactly divisible by 400 is not repeated until the
twelfth year later. Thus the 1897 calendar is not
repeated until 1909.

2. The calendar for the second year after leap year is

repeated eleven years later and is itself a repetition
of that for the eleventh year earlier. Exceptions : —
The calendars for the years '90 and '98 prior to a
century year not exactly divisible by 400 are not
repeated until the twelfth year later ; while that for
the year '94 prior to such a century year is repeated
in the sixth year later. Thus the calendar for 1890
is not repeated until 1902, and the calendar for 1898
is not repeated until 1910; while that for 1894 is
repeated in 1900.

THE REFORM OF THE CALENDAR 145

3. The calendar for the third year after leap year is

repeated eleven years later and is itself a repetition
of that for the sixth year earlier. Exceptions : — The
calendar for the year '91 prior to a century year not
exactly divisible by 400 is not repeated until the
twelfth year later, while the calendars for the years
'95 and '99 prior to such a century year are each
repeated in the sixth year later. Thus the calendar
for 1891 is not repeated until 1903, while the
calendars for the years 1895 and 1899 are repeated
in 1901 and 1905 respectively.

4. The calendar for leap year is repeated twenty-eight

years later and is itself a repetition of that for
the twenty-eighth year earlier. Exceptions: — The
calendars for the years '72, '76, '80, '84 and '88,
prior to a century year not exactly divisible by 400
are respectively not repeated until the fortieth year
later; while those for the years '92 and '96^rior to
such a century year are respectively repeated in the
twelfth year later. Thus the calendars for 1872>
1876, 1880, 1884 and 1888 are not repeated until
1912, 1916, 1920, 1924, and 1928 respectively;
while those for the years 1892 and 1896 are
repeated in 1904 and 1908 respectively,

6. The calendar for a century year, when not a leap
year, and that for the year following are respectively
repeated six years later. Thus the calendars for
1900 and 1901 are applicable to 1906 and 1907
respectively.

6. The calendars for the years '02 and '03 following a
century year not exactly divisible by 400 are
respectively repeated eleven years later. Thus the
calendars for 1902 and 1903 are applicable to 1913
and 1914 respectively.

If a century year not exactly divisible by 400 is not

146 THE REFORM OF THE CALENDAR.

comprised in the period the calendar is repeated as
follows : —

1st year after leap year repeats in 6 years,
2nd „ » 11 „

3rd „ „ 11 „

Leap year „ 28 „

Adding up the intervals between the repetition of
the calendar in so far as it applies to common years — 6
years, 11 years, and 11 years — we get 28 years, being
the period required to bring about a repetition of the
leap year calendar. This is the time required to pro-
duce a sequence of the calendar's repetition, provided
always the period does not cover a century year not
exactly divisible by 400.

It has been proposed that the want of harmony
between the week and the year might be overcome by
having New Year's Day and (in leap year) the 29th of
February as days apart. These two days would not be
reckoned as belonging to any week, but would be
observed as separate and unallocated days, and treated
as public holidays. They would be neither Sunday nor
Saturday nor any of the five intervening days, but
would be simply " New Year's Day " and " Leap Day."
This would leave exactly and invariably 364 days as the
" counting " days of the year, being precisely fifty-two
weeks.

The proposal has such a charming simplicity and
boldness about it that criticism is well-nigh disarmed,
while at the same time one's breath is almost taken
away. It is refreshing to think that every year we may
be presented with one day, and in leap year with two
days, which shall not "count" against us in life's journey

THE REFORM OF THE CALENDAR. 147

— days which, as it were, are " thrown into the bargain "
in our lives. We are to be no older at the end of each
of these days than at the beginning, while yet we shall
have had a pleasant holiday. It seems like an exquisite
game of " make believe," and we feel that it would be
nice to go on in this way and to throw a few more
" non-counting " days into each year so that we might
remain young. One realizes that the desire for such
days would become stronger and stronger as the
'counting" days accumulated against one.

After all, however, the proposal though novel and
startling has much which can seriously be urged in its
:avour, and it would evidently bring about a very
lesirable relation between the week and the year. Un-
ortunately it cannot be overlooked that it is subject to
)ne serious objection, which in all probability must be
atal to its chances of adoption. The objection is that
t runs counter to the religious susceptibilities of the
)eople, of whatever sect or denomination.

It is clear that the occurrence at the beginning of
ach year of a day having no place in the week then
undent would have the eflFect of throwing the ensuing
ays one place backward. If, for instance, the 31st of
)ecember were Wednesday, the following day would
ot be Thursday, but "New Year's Day," while the next
ay, which but for the change would have been Friday,
ould through the change be Thursday. Thus the
J)Ilowing Sunday would fall on the day of the week,
hich but for the change would have been Monday. A
inilar displacement would in leap years occur at "Leap
ay." Thus the weekly day of rest would in a very
m years be associated with every day of the week.

148 THE REFORM OF THE CALENDAR.

The Jews would find themselves observinor what had
previously been the first day of the week as their
Sabbath, and Christians would be observing the seventh
day. It cannot be doubted that Churchmen, whether
high or low, would combine with Nonconformists ; Jews
with Gentiles ; and Roman Catholics with Presbyterians
to resist any system involving such a complete upsetting
of their most cherished convictions. It would seem,
therefore, that any reform of the calendar depending on
the institution of " New Year's Day " and " Leap Day "^
as days apart from the week, has no reasonable prospect
of adoption at any rate within the next few centuries.

The re-arrangement of the duration of the months is
a minor matter, which does not present any insuperable
difficulty, and which can readily be given effect to in con-
nection with any projected amendment of the calendar,
if merely by reversion to the sj^^stem instituted, as
already explained, when the Julian chronology was
inaugurated.

The practical advantages which would accrue from
the year being made an exact multiple of the week
suggest the enquiry whether this might not be brought
about by some means free from the objection inseparable
from the proposal to have "New Year's Day" and "Leap
Day" as days apart, while yet maintaining, as is
essential, as close a relation as possible between the
civil and the astronomical year.

As the astronomical j^^ear consists of 365d. 5h. 48m, ;
46s., and as the civil year must necessarily consist of a ]
certain number of days free from any fractional part,
it is evident that the closest approximation which can \
exist between the civil and astronomical year in any 1

THE REFORM OF THE CALENDAR 149

]»articular year is secured when the civil year is reckoned
<is 365 days. It is also evident that the fraction remain-
ing over can only be dealt with periodically when it
Attains the amount required for at least one day.
lender our present system the diflficulty is dealt with
in the simplest possible manner, and consequently the
relation between the week and the year is sacrificed.

In the present industrial age the week has come to
be of prime importance in our division of time, and the
harmonizing of the week with the year is, therefore,
more urgent than it has ever been in the past, and,
probably, will become still more desirable as time goes
on. Were the year an exact multiple of the week and
a very simple re-arrangement of the months effected,
there would never be any difficulty in fixing a date in
association with the day of the week. The year would
invariably begin on the same week-day, and this would
be the case also with every month. It is well known
that an incalculable number of errors have arisen
through the want of correspondence in this respect.

If, however, such a change is to be effected in a
manner free from any insuperable objection — such as
would attach to the periodical changing of the day of
the week, by omitting a day, or otherwise — it can only
be done by some slight amount of sacrifice of the close-
ness of approximation of each separate civil year to the
astronomical year. This is apparent from the fact that
the nearest correspondence between these two periods —
being 365 days — is not a multiple of the week. If some
sacrifice is allowable in this respect in view of the
accruing benefits, then it is quite possible to secure
harmony between the week and the year — that is, to

150 THE REFORM OF THE CALENDAR.

secure that the year shall be invariably an exact
multiple of the week.

All that is necessary to effect the reform is that the
common year should be 364 days — being exactly 52
weeks — instead of 365 days; and that the leap year
should be 371 days — being exactly 53 weeks — instead
of 366 days, the frequency of leap year being fixed so
as to establish as near a coincidence as possible with
the duration of any definite number of astronomical
years.

If every fifth year — or, we may say, every year
exactly divisible by 5 — with the exception of the
century and half -century years, were made a leap year,
and every other year a common year, it would be
necessary to make only a single qualification in 400
years — as is done under the present system — to obtain
perfect uniformity with the existing 400 year period.
It would be desirable that this qualification should take
effect about the middle of the period.

Let us apply these suggestions to existing circum-
stances.

We may take it that it would be desirable that the
year should commence on a Sunday as that is the day
with which the week begins. Thus, if adopted, the
proposed system might with great advantage take efiect
on the first day of some coming year which under the
present system begins on Sunday. In 1928 the 1st of
January falls on a Sunday, and that year is otherwise
in many respects suitable for the inauguration of such a
change of calendar. Let us then suppose that the
change should come into operation on the Ist of January
1928, and let us, on this supposition, trace out year by

THE REFORM OF THE CALENDAR. 151

year the discrepancies between the present system and
the proposed system. On this basis a convenient time
for the occurrence of the qualified year, to which refer-
ence has been made as occurring once in 400 years —
and which is merely the substitution of a common year
of 364 days for a year which would otherwise under our
rule be a leap year of 371 days — is the year 2175, being
the I75th year after the year divisible by 400. This
date can very simply be remembered from the fact that
twenty-five years is necessarily the longest period by
which any year is separated from the ordinary half-
century occurrence of an exception to the quinquennial
rotation of leap year, and that this period — twenty-five
years — multiplied by seven (being the number of days
in the week, which forms the basis of the system), is
175. Thus the 175th year after every year divisible by
400, might very conveniently be made the exceptional
year occurring once in 400 years, and which is thus a
common year instead of a leap year.

On this understanding the following Table shows the
relation between the new system and the present system
for a period of 400 years, commencing with the begin-
ning of the year 1928.

152

THE REFORM OF THE CALENDAR.

Table showing divergence between the proposed New System
of the Calendar and the present system, year by year
for a period of four hundred years, commencing Ist
January 1928 and ending 31st December 2327 (both
inclusive).

riod.

Last date

comprised

period.

New Syitem.

Pretent

Syitem.

No. of days by
which New
System is
longer (+) or
shorter ( - )
than (iresenb
system.

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

Days

added to

period

by year to

date
specified.

Number

days

comprised

period.

1 Year

31st Dec. 1928

364

366

- 2

2 Years

„ 1929

364

728

365

731

- 3

„ 1930

371

1099

365

1096

+ 3

„ 1931

364

1463

365

1461

+ 2

«)

„ 1932

364

1827

366

1827

„ 1933

364

2191

365

2192

- 1

„ 1934

364

2555

365

2557

- 2

„ 1935

371

2926

365

2922

+ 4

„ 1936

364

3290

366

3288

+ 2

))

„ 1937

364

3654

365

3653

+ 1

„ 1938

364

4018

365

4018

„ 1939

364

4382

365

4383

- 1

„ 1940

371

4753

366

4749

+ 4

» 1941

364

5117

365

5114

+ 3

„ 1942

364

5481

365

5479

+ 2

„ 1943

364

5845

365

5844

+ 1

„ 1944

364

6209

366

6210

- 1

„ 1945

371

6580

365

6575

+ 5

>•

„ 1946

364

6944

365

6940

+ 4

))

„ 1947

364

7308

365

7305

+ 3

„ 1948

364

7672

366

7671

+ 1

))

„ 1949

364

8036

365

8036

))

„ 1950

364

8400

365

8401

- 1

„ 1951

364

8764

365

8766

- 2

))

„ 1952

364

9128

366

9132

- 4

„ 1953

364

9492

365

9497

- 5

„ 1954

364

9856

365

9862

- (>

„ 1955

371

10227

365

10227

„ 1956

364

10591

366

10593

- 2

„ 1957

364

10955

365

10958

- 3

„ 1958

364

11319

365

11323

- 4

„ 1959

364

11683

365

11688

- 5

„ 1960

371

12054

366

12054

„ 1961

364

12418

365

12419

- 1

„ 1962

364

12782

365

12784

- 2 .

THE

REFORM OF THE CALENDAR.

153

eriod.

Last date

comprised

period.

.ATew Syitem.

Pruent SytUm.

No. of days by
which New
System is
longer (+) or
shorter (-)
than present
system.

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

Days

added to

period

by year to

date
specified.

Number

days

comprised

period.

Years

31st Dec. 1963

364

13146

365

13149

- 3

))

1964

364

13510

366

13515

-.5

1965

371

13881

365

13880

+ 1

1966

364

14245

365

14245

1967

364

14609

365

14610

- 1

1968

364

14973

366

14976

- 3

1969

364

16337

365

15341

- 4

1970

371

15708

365

15706

+ 2

1971

364

16072

365

16071

+ 1

1972

364

16436

366

16437

- 1

1973

364

16800

365

16802

- 2

»>

1974

364

17164

365

17167

- 3

»>

1975

371

17535

365

17532

+ 3

1976

364

17899

366

17898

+ 1

• >»

1977

364

18263

365

18263

1978

364

18627

365

18628

- 1

1979

364

18991

365

18993

- 2

1980

371

19362

366

19359

+ 3

1981

364

19726

365

19724

+ 2

1982

364

20090

365

20089

+ 1

1983

364

20454

365

20454

1984

364

20818

366

20820

- 2

1985

371

21189

365

21185

+ 4

> n

1986

364

21553

365

21550

+ 3

. »

])

1987

364

21917

365

21915

+ 2

»»

1988

364

22281

366

22281

1989

364

22645

365

22646

- 1

1990

371

23016

365

23011

+ 5

»>

1991

364

23380

365

23376

+ 4

1992

364

23744

366

23742

+ 2

»>

1993

364

24108

365

24107

+ 1

1994

364

24472

365

24472

1995

371

24843

365

24837

+ 6

1996

364

25207

366

25203

+ 4

1997

364

25571

365

25568

+ 3

1998

364

25935

365

25933

+ 2

1999

364

26299

365

26298

+ 1

2000

364

26663

366

26664

- 1

2001

364

27027

365

27029

- 2

2002

364

27391

365

27394

- 3

2003

364

27755

365

27759

- 4

2004

364

28119

366

28125

- 6

»>
«

2005

371

28490

365

28490

154

THE REFORM OF THE CALENDAR.

Period.

79 Yrs.

80 „

81 „

82 „

83 „

84 „

85 „

86 „

87 „

88 „

89 „

90 „

91 „

92 „

93 „

94 „

95 „

96 „

97 „

98 „

99 „

100 „

101 „

102 „

103 „

104 „

105 „

106 „

107 „

108 „

109 „

110 „

111 „

112 „

113 „

114 „

115 „

116 „

117 „

118 „

119 „

120 „

121 „

T.ia8tdata

comprised

period.

31st Dec. 2006

„ 2007

„ 2008

„ 2009

„ 2010

„ 2011

„ 2012

„ 2013

„ 2014

„ 2015

„ 2016

„ 2017

„ 2018

„ 2019

„ 2020

„ 2021

„ 2022

„ 2023

„ 2024

„ 2025

„ 2026

„ 2027

„ 2028

„ 2029

„ 2030

„ 2031

„ 2032

„ 2033

„ 2034

„ 2035

„ 2036

„ 2037

„ 2038

„ 2039

„ 2040

„ 2041

„ 2042

„ 2043

„ 2044

„ 2045

„ 2046

„ 2047
2048

A'eir Syitem.

Days
added to

period
by year to

date
specified.

364
364
364
364
371
364
364
364
364
371
364
364
364
364
371
364
364
364
364
371
364
364
364
364
371
364
364
364
364
371
364
364
364
364
371
364
364
364
364
371
364
364
364

Number
of

days

comprised

period.

28854
29218
29582
29946
30317
30681
31045
31409
31773
32144
32508
32872
33236
33600
33971
34335
34699
35063
35427
35798
36162
86526
36890
37254
37625
37989
38353
38717
3.9081
39452
39816
40180
40544
40908
41279
41643
42007
42371
42735
43106
43470
43834
44198

Prctent Sytem.

Days
added to

l)eriod
by year to

date
specified.

365
3<)5
366
365
365
365
366
365
365
365
366
365
365
365
366
365
365
365
366
365
365
365
366
365
365
365
366
365
365
365
366
365
365
365
366
365
365
365
366
365
365
365
366

Number

days

comprised

period.

28855
29220
29586
29951
30316
30681
31047
31412
31777
32142
32508
32873
33238
33603
33969
34334
34G99
35064
35430
35795
36160
36525
36891
37256
37621
37986
38352
38717
39082
39447
39813
40178
40543
40908
41274
41639
42004
42369
427.35
43100
43465
438.30
44196

THE REFORM OF THE CALENDAR.

155

Pwiod.

Last date
comprised

period.

yeu Syatem.

Pretent System.

No. ofdnysbjr
which New
System is
longer (+) or
shorter (-)
than present
system.

Days

added to

period

by year to

date
specified.

Number

days

comprised

period.

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

J2Yrs.

31st Dec. 2049

364

44562

365

44561

+ 1

J.3

2050

364

44926

365

44926

24 „

2061

364

45290

365

45291

- 1

J5 „

2052

364

45654

366

45657

- 3

56 „

2053

364

46018

365

46022

- 4

57 „

2054

364

46382

365

46387

- 5

58 „

2056

371

46753

365

46752

+ 1

59 „

2056

364

47117

366

47118

- 1

JO „

2057

364

47481

365

47483

- 2

n „

2058

364

47845

365

47848

- 3

»^ „

2059

364

48209

365

48213

- 4

J3 „

2060

371

48580

366

48579

+ 1

W „

2061

364

48944

365

48944

55 „

2062

364

49308

365

49309

- 1

56 „

2063

364

49672

365

49674

- 2

«7 „

2064

364

50036

366

50040

- 4

w „

J»

2065

371

50407

365

50405

+ 2

J9 „

2066

364

50771

365

50770

+ 1

iO „

2067

364

61135

365

61135

n »

2068

364

51499

366

51501

- 2

12 „

2069

364

51863

366

51866

- 3

t3 „

2070

371

52234

365

52231

+ 3

14 „

2071

364

52598

365

62596

+ 2

15 „

2072

364

52962

366

62962

16 „

2073

364

53326

365

63327

- 1

17 „

2074

364

53690

365

53692

- 2

18 „

2075

371

54061

365

54057

+ 4

19 „

l»

2076

364

54426

366

54423

+ 2

K) „

2077

364

54789

365

54788

+ 1

il „

2078

364

55153

365

55153

+ •

i2 „

2079

364

55517

365

55518

- 1

i3 „

2080

371

55888

366

55884

+ 4

'4 „

2081

364

56252

365

56249

+ 3

i5 „

2082

364

56616

366

56614

+ 2

)6 „

2083

364

56980

365

56979

+ I

»7 „

2084

364

57344

366

67345

- 1

>8 „

2085

371

87715

366

67710

+ 6

•)9 „

2086

364

58079

365

68076

+ 4

50 „

2087

364

58443

365

68440

+ 3

51 „

2088

364

68807

366

58806

+ 1

52 „

2089

364

69171

365

69171

)3 „

2090

371

59542

365

59536

+ 6

54 „

2091

364

59906

365

59901

+ 5

156

THE REFORM OF THE CALENDAR

iod.

I^astdate

comprised

period.

New System. |

Present

6'y((«m.

No. of days by

wliich New

System is

longer (+) or

shorter ( - )

than present

system.

Per

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

Days

added to

period

by year to

date
specitied.

Number

days

comprised

period.

165 Yrs.

31st Dec, 2092

364

60270

366

60267

+ 3

166

2093

364

60634

365

60632

+ 2

167

2094

364

60998

365

60997

+ 1

168

))

2095

371

61369

365

61362

+ 7

169

2096

364

61733

366

61728

+ 5

170

2097

364

62097

365

62093

+ 4

171

2098

364

62461

365

62458

+ 3

172

))

2099

364

62825

365

62823

+ 2

173

2100

364

63189

365

63188

+ 1

174

2101

364

63553

365

63553

175

2102

364

63917

365

63918

- 1

176

2103

364

64281

365

64283

- 2

177

2104

364

64645

366

64649

- 4

178

2105

371

65016

365

65014

+ 2

179

2106

364

65380

365

65379

+ 1

180

)}

2107

364

65744

365

65744

181

2108

364

66108

366

66110

- 2

182

))

2109

364

66472

365

66475

- 3

183

2110

371

66843

365

66840

+ 3

184

))

2111

364

67207

365

67205

+ 2

185

2112

364

67571

366

67571

186

2113

364

67935

365

67936

- 1

187

2114

364

68299

365

68301

- 2

188

2115

371

68670

365

68666

+ 4

189

2116

364

69034

366

69032

+ 2

100

))

2117

364

69398

365

69397

+ 1

191

2118

364

69762

365

69762

192

2119

364

70126

365

70127

- 1

193

2120

371

70497

366

70493

+ 4

194

J»

2121

364

70861

365

70858

+ 3

195

2122

364

71225

365

71223

+ 2

19G

2123

364

71589

365

71588

+ 1

197

»>

2124

364

71953

366

71954

- 1

198

2125

371

72324

365

72319

+ 6

199

>»

2126

364

72688

365

72684

+ 4

200

2127

364

73052

365

73049

+ 3

201

2128

364

73416

366

73415

4- 1

202

2129

364

73780

365

73780

203

2130

371

74151

365

74145

+ 6

204

2131

364

74515

365

74510

+ 5

205

2132

364

74879

366

74876

+ 3

206

2133

364

75243

365

75241
75606

+ 2

207

2134

364

75607

365

+ 1

THE

REFORM OF THE CALENDAR.

157

iBriod.

Last date

coiuprised

period.

Jfw System.

Pretent

Syitem,

No. of days by
which New-
System is
longer (+) or
shorter ( - )
than ])resent
systeiu.

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

SYra.

31st Dec. 2135

371

75978

365

76971

+ 7

9 „

})

2136

364

76342

366

7a337

+ 5

„

))

2137

364

76706

365

76702

+ 4

1 „

2138

364

77070

365

77067

+ 3

2 »

2139

364

77434

365

77432

+ 2

3 „

)]

2140

371

77805

366

77798

+ 7

i „

2141

364

78169

365

78163

+ 6

5 »

2142

364

78533

365

78528

+ 5

6 „

2143

364

78897

365

78893

+ 4

7 „

2144

364

79261

366

79259

+ 2

8 „

2145

371

79632

365

79624

+ 8

9 »

2146

364

79996

365

79989

+ 7

„

2147

364

80360

365

80354

+ 6

1 »

2148

364

80724

366

80720

+ 4

2 „

2149

364

81088

365

81085

+ 3

3 „

2150

364

81452

365

81450

+ 2

4 „

2151

364

81816

365

81815

+ 1

5 „

2152

364

82180

366

82181

- 1

6 „

2153

364

82544

365

82546

- 2

7 „

2154

364

82908

365

82911

- 3

8 „

2155

371

83279

365

83276

+ 3

9 „

2156

364

83643

366

83642

+ 1

„

2157

364

84007

365

84007

1 »

2158

364

84371

365

84372

- 1

2 „

2159

364

84735

365

84737

- 2

3 ,,

2160

371

85106

366

85103

+ 3

4 »

>»

2161

364

85470

365

85468

+ 2

5 „

2162

364

85834

365

85833

+ 1

6 „

2163

364

86198

365

86198

7 „

2164

364

86562

366

86564

- 2

S „

2165

371

86933

365

86929

+ 4

„

2166

364

87297

365

87294

+ a

„

2167

364

87661

365

87659

+ 2

1 »

2168

364

88025

366

88025

2169

364

88389

365

88390

- 1

3 „

2170

371

88760

365

88755

+ 5

4 „

2171

364

89124

365

89120

+ 4

5 „

2172

364

89488

366

89486

+ 2

« „

2173

364

89852

365

89851

+ 1

7 „

2174

364

90216

365

90216

« „

2175

364

90580

365

90581

- 1

9 „

2176

364

90944

366

90947

- a

„

2177

364

91308

365

91312

- 4

158

THE REFORM OF THE CALENDAR.

T^astdate
comprised

period.

Nno System.

Present

System.

No. of days by

which New

System is

longer (+) or

shorter ( - )

than present

system.

Period.

Days

added to

period

by year to

date
specified.

Number

days

comprised

period.

Days

added to

period

by year to

date
specified.

Number

days

comprised

period.

251 Yrs.

3l8t Dec. 2178

364

91672

365

91677

- 6

252 „

})

2179

364

92036

365

92042

- 6

253 „

2180

371

92407

366

92408

- 1

254 „

2181

364

92771

365

92773

- 2

255 „

))

2182

364

93135

365

93138

- 3

256 „

2183

364

93499

365

93503

- 4

257 „

2184

364

93863

366

93869

- 6

258 „

2185

371

94234

365

94234

259 „

))

2186

364

94598

366

94599

- 1

260 „

2187

364

94962

365

94964

- 2

281 „

))

2188

364

95326

366

95330

- 4

262 „

))

2189

364

95690

365

95696

- 5

263 „

2190

371

96061

365

96060

+ 1

264 „

))

2191

364

96425

365

96425

265 „

))

2192

364

96789

366

96791

- 2

266 „

))

2193

364

97153

365

97156

- 3

267 „

))

2194

364

97517

365

97521

- 4

268 „

2195

371

97888

365

97886

+ 2

269 „

)}

2196

364

98252

366

98252

270 ;;

)}

2197

364

98616

365

98617

- 1

271 „

2198

364

98980

365

98982

- 2

272 „

2199

364

99344

365

99347

- 3

273 „

2200

364

99708

365

99712

- 4

274 „

2201

364 ■

100072

365

100077

- 5

275 „

)}

2202

364

100436

365

100442

- 6

276 „

2203

364

100800

365

100807

- 7

277 „

2204

364

101164

366

101173

- 9

278 „

2205

371

101535

365

101538

- 3

279 „

2206

364

101899

365

101903

- 4

280 „

2207

364

102263

366

102268

- 5

281 „

2208

364

102627

366

102634

- 7

282 „

2209

364

102991

365

102999

- 8

283 „

2210

371

103362

365

103364

- 2

284 „

2211

364

103726

365

103729

- 3

285 „

2212

364

104090

366

104095

- 5

286 „

2213

364

104454

365

104460

- 6

287 „

2214

364

104818

365

104825

- 7

288 „

2215

371

105189

366

105190

- 1

289 „

2216

364

105553

366

105556

- 3

290 „

2217

364

105917

365

105921

- 4

291 „

2218

364

106281

365

106286

- 5

292 „

2219

364

106645

365

106651

- 6

293 „

2220

371

107016

366

107017

- 1

THE REFORM OF THE CALENDAR.

159

Last date

oomprised

period.

31st Dec. 2221

„ 2222

„ 2223

„ 2224

„ 2225

„ 2226

„ 2227

„ 2228

„ 2229

„ 2230
„ - 2231

„ 2232

„ 2233

„ 2234

„ 2235

„ 2236

„ 2237

„ 2238

„ 2239

„ 2240

„ 2241

„ 2242

„ 2243

„ 2244

„ 2245

„ 2246

„ 2247

„ 2248

„ 2249

„ 2250

„ 2251

„ 2252

„ 2253

„ 2254

„ 2255

„ 2256

„ 2257

„ 2258

„ 2259

„ 2260

„ 2261

„ 2262
2263

Kere System.

Present Systetn. \

Days

added to

period

by year to

date
specified.

Number

days

comprised

period.

Days

added to

period

by year to

date
specified.

Number

days

comprised

period.

364

107380

365

107382

364

107744

365

107747

364

108108

365

108112

364

108472

366

108478

371

108843

365

108843

364

109207

365

109208

364

109571

365

109573

364

109935

366

109939

364

110299

365

110304

371

110670

365

110669

364

111034

365

111034

364

111398

366

111400

364

111762

365

111765

364

112126

365

112130

371

112497

365

112495

364

112861

366

112861

364

113225

365

113226

364

113589

365

113591

364

113953

365

113956

371

114324

366

114322

364

114688

365

114687

364

115052

365

115052

364

115416

365

115417

364

115780

366

115783

371

116151

365

116148

364

116515

365

116513

364

116879

365

116878

364

117243

366

117244

364

117607

366

117609

364

117971

365

117974

364

118335

365

118339

364

118699

366

118705

364

119063

365

119070

364

119427

365

119435

371

119798

365

119800

364

120162

366

120166

364

120526

365

120531

364

120890

365

120896

364

121254

365

121261

371

121625

366

121627

364

121989

365

121992

364

122353

365

122357

364

122717

365

122722

No. of days by

which New

System is

longer (+) or

shorter ( - )

than )>re«ent

system.

- 2

- 3

- 4

- 6
+

- 1

- 2

- 4

- 5
+ 1
+

- 2

- 3

- 4
+ 2
+

- 1

- 2

- 3
2
1

+
+

- 1

- 3
+ 3
+ 2
+ 1

- 1

- 2

- 3

- 4

- 6

- 7

- 8

- 2

- 4

- 5

- 6

- 7

- 2

- 3

- 4

- 6

160

THE REFORM OF THE CALENDAR.

Period.

337 Yrs.

338 „

339 „

340 „

341 „

342 „

343 „

344 „

345 „

346 „

347 „

348 „

349 „

350 „

351 „

352 „

353 „

354 „

355 „

356 „

357 „

358 „

359 „

360 „

361 „

362 „

363 „

364 „

365 „

366 „

367 „

368 „

369 „

370 „

371 „

372 „

373 „

374 „

375 „

376 „

377 „

378 „

379 „

TiOst date

comprised

period.

31st Dec. 2264

„ 2265

„ 2266

„ 2267

„ 2268

„ 2269

„ 2270

„ 2271

„ 2272

„ 2273

„ 2274

„ 2275

„ 2276

„ 2277

„ 2278

„ 2279

„ 2280

„ 2281

„ 2282

„ 2283

„ 2284

„ 2285

„ 2286

„ 2287

„ 2288

„ 2289

„ 2290

„ 2291

„ 2292

„ 2293

„ 2294

„ 2295

,^ 2296

„ 2297

„ 2298

„ 2299

„ 2300

„ 2301

„ 2302

„ 2303

„ 2304

„ 2305
2306

Ifeie SysUm.

Days
added to

Ijeriod
by year to

date
sx>ecified.

364

371

364

371

364

371

364

371

364

371

364

371

364

371

364

371

364

Number

days

comprised

period.

123081

12.3452

123816

124180

124544

124908

125279

125643

126007

126371

126735

127106

127470

127834

128198

128562

128933

129297

129661

130025

130389

130760

131124

131488

131852

132216

132587

132951

133315

133679

134043

134414

134778

135142

135506

135870

136234

136598

136962

137326

137690

138061

138425

Present Syxlem.

Days

added to

period

by year to

date
specified.

366

365

366

365

366

365

366

365

366

365

366

365

366

365

366

365

366

365

366

365

Nunit)er

days

comprised

period.

123088

123453

123818

124183

124549

124914

125279

125644

126010

126375

126740

127105

127471

127836

128201

128566

128932

129297

129662

130027

130393

130758

131123

131488

131854

132219

132584

132949

133315

133680

134045

134410

134776

135141

135506

135871

136236

136601

136966

137331

137697

138062

138427

THE REFORM OF THE CALENDAR.

161

Last date

comprised

period.

Hew Syttem.

Prtunt Syt(«»i.

No. of days by
wbiuh New
System is
longer (+) or
shorter ( - )
than present
system.

Period.

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

Days
added to

period
by year to

date
specified.

Number

days

comprised

period.

WYrs.

31st Dec. 2307

364

138789

365

138792

- 3

<1 „

„ 2308

364

139153

366

139158

- 5

^2 „

„ 2309

364

139517

365

139523

- 6

« „

„ 2310

371

139888

365

139888

54 „

„ 2311

364

140252

365

140253

- 1

15 „

„ 2312

364

140616

366

140619

- 3

16 „

„ 2313

364

140980

365

140984

- 4

17 „

„ 2314

364

141344

365

141349

- 5

« „

„ 2315

371

141715

365

141714

+ 1

•9 „

„ 2316

364

142079

366

142080

- 1

o »

„ 2317

364

142443

365

142445

- 2

1 „

„ 2318

364

142807

365

142810

- 3

2 „

„ 2319

364

143171

365

143175

- 4

3 „

„ 2320

371

143542

366

143541

+ 1

4 „

„ 2321

364

143906

365

143906

5 »

„ 2322

364

144270

365

144271

- 1

6 „

„ 2323

364

144634

365

144636

- 2

7 „

„ 2324

364

144998

366

145002

- 4

8 „

„ 2325

371

145369

365

145367

+ 2

9 „

„ 2326

364

145733

365

145732

+ 1

» »

„ 2327

364

146097

365

146097

Thus in the period of 400 years there is absolute
eorrespondence between the present system of the
calendar and the system now submitted. It follows —
since the 400 year period exactly repeats itself under
the Gregorian style until the year 4000 A.D., when the
slight amendment of the style proposed by making that
year a common year instead of a leap year takes effect
— ^that a corresponding coincidence will subsist until the
year 4000 A.D. It will be noticed that the extreme
divergence between the two systems which occurs in
the period dealt with in the Table is + 8 and — 9
being practically a week on each side, a divergence

162 THE REFORM OF THE CALENDAR.

which is unavoidable in a system based on the har-
monizing of the week and the year. In general,
however, the divergence is almost a negligible quantity.
We see, therefore, that the observance of the follow-
ing rules would be suflScient for over 2000 years to
come to secure harmony between the week and the
year, so that the latter should be an exact multiple of
the former, viz. : —

1. Let the common year be 364 days (52 weeks) and the
leap year 371 days (53 weeks).

2. Let every year which is exactly divisible by 5 but not
by 50 be a leap year, excepting only the 175th year
after every year exactly divisible by 400.

We have noticed that the present system results in
an error as compared with astronomical reckoning of
one day in 3323 years. As the system may be said, in
view of the value of Pope Gregory's correction of the
error which occurred prior to his reform, to have been
commenced in the year 325 A.D., the error of the system
will, therefore, amount to one day by the year 3648 A.D.
The system now submitted, being founded on the exist-
ing system, would, of course, have a corresponding mean
error, and as the error is a plus one — the calendar
period of 3323 years being one day longer than the
astronomical period — the individual plvs errors would
by the year 3648 A.D. be one day more than our Table
would indicate, and the individual minVyS errors one
day less — this of course in comparing the system with
solar time, not merely with the existing system. We
have to consider how this mean error could be rectified.
It is, as we have seen, met under the present system by

THE REFORM OF THE CALENDAR. 163

making the year 4000 a.d. a common year. Could
the difficulty be got over in the same way under the
amended system ?

It will at once be noticed that to make an amend-
ment of one day would necessitate the retrogression in
the week of the day of commencement of the year, or
else the recognition of one day not as a " non-counting "
day but as a " double counting " day. We have already
indicated that the latter course is very undesirable.
The alternative mentioned would involve that for the
ensuing 4000 years or thereabout the year should com-
mence not with Sunday but with Saturday, a similar
retrogression being made in every period of 4000 years.
This would be distinctly objectionable and would de-
prive the system of the element of permanency which
should, at least on the face of it, pertain to any system
of time-reckoning.

Another plan more in consonance with the system
is available, which is in accord with its merits and also
with its inseparable defect as regards the departure of
the individual civil year from the closest approximation
to the true length of the tropical year. This is that in
applying this long-period correction the week should be
accepted as the basis for the correction just as in the
shorter periods. We have found that in the year 3648
a minus correction on the calendar of one day is
required, and that every 3323 years thereafter call
for a further minus correction of one day. Thus by
the year 11,955 a.d. the error would, in the absence
of prior long-period corrections, amount to about
three-and-a-half days. This date is arrived at as
follows : —

164 THE REFORM OF THE CALENDAR.

325 A.D.
3,323 yeara

3,648 A.D.
3,323 year*

6,971 A.D.
3,323 years

Virtual date of institution of New Style
according to Pope Gregory's correc-
tion in 1582

Period requiring correction of one day . . .

Date at which first correction of one day
is required

Period on the expiry of which a further
correction of one day is requisite ...

Date at which correction of two days is
required in the absence of prior long-
period corrections ...

Period on the expiry of which a further
correction of one day is requisite , . .

Date at which correction of three days is
required in the absence of prior long-
period corrections

Period on the expiry of which a further
correction of half-a-day (12 hours) is
requisite

Date at which correction of 3| days is
required in the absence of prior long-
period corrections

If then, the year 11,955 A.D., or any other year near
that date, which in the ordinary course would be a leap
year of 371 days, were converted into a common year
of 364 days we would be making a correction of seven
days — being exactly double the amount required. The
error would thus, though unaltered in amount, be
thrown to the other side, the calendar period being then
on the mean Sh days short of the solar period. The
effect of this would be that for the next three-and-a-half
periods of 3323 years the error would be getting counter-
acted and would in that time — say in 11,631 years — be
completely nullified. In a further period of 11,630
years the error of 3| days would again recur and the

THE REFOHM of the calendar. 165

correction of one week could then again be repeated.
Thus the long-period correction requisite would be
simply the conversion of a single year, which in the
ordinary course would be a leap year, into a common
year, and this would be desirable in or near the year
11,955 A.D., and about every 23,261 years — being seven
times 3323 years — thereafter. But even this long-period
inaccuracy would at the utmost involve a mean error of
only three-and-a-half days.

Another point may be mentioned in connection with
this long-period difference. It is quite probable that in
process of time our knowledge of the exact length of
the solar year may become still more accurate and that
the infinitesimal variations, which are known to occur
in ages in the length of the solar year, will become more
fully understood. Thus the system will allow of the
long-period correction — converting what in the ordinary
course would be a leap year into a common year — being
applied exactly at the time necessary to secure the
greatest possible harmony between the calendar and the
Sun. The correction would be required only once in
about 23,000 years, and the year in which the correction
should be given effect to — whether a few hundred years
earlier or later — would be a matter for the astronomers
and chronologists of the time. Thus the system would
possess absolute completeness and permanency in so far
as procurable in connection with the basis of the week.
It will be noticed that the interval between the recur-
ring long-period corrections bears some anology to the
precessional period (25,868 years). This correction
might, therefore, be conveniently referred to as the
" precessional correction."

166 THE REFORM OF THE CALENDAR.

Before referring to the amendment of the length of
the months there is one other matter in connection with
the year as a whole which falls to be noticed as of im-
portance in relation to any reform of the calendar.

When the Julian system was instituted CaBsar, as we
have seen, shifted the beginning of the year from March
to January — from, we may say, the time of the vernal
equinox to the time of the winter solstice. The names
of certain of the months still, oddly enough, bear record
to this change. The last four months of the year, being
the ninth, tenth, eleventh, and twelfth months, are
still named the seventh, eighth, ninth, and tenth —
September, October, November, and December. Un-
fortunately, as we have seen, Csesar, in fixing the time
of the beginning of the year under the Julian system,
was not guided by the movements of the Sun only.
The Moon also was considered. He fixed the season by
the Sun, and the exact day of the commencement of the
system by the Moon. The day thus selected was the
date of the new Moon which immediately followed the
winter solstice of the year 45 B.C., and this date was
about a week or ten days later than that of the solstice.
Thus a lamentable mistake was made at the initiation
of the system whose consequences affect us unto this
day. Thus it comes about that we recognize as the first
day of the year a day which in Nature's year is the
tenth or eleventh day.

Julius Caesar, however, lived and died before the
Christian era. Although we now refer our chronology
to the birth of Christ, Csesar referred his to the founda-
tion of Rome. It may, therefore, reasonably be argued
that although in Julian chronology the year should

THE REFORM OF THE CALENDAR. 167

begin with the winter solstice, it should in our
chronology commence with the date accepted by the
early fathers as that of the birth of our Lord. It is
quite unnecessary to recall that the weight of evidence
goes to show that not only can little reliance be placed
on the date recognized as Christmas Day, but that even
the year named as the first of our era is evidently
erroneous. It is sufficient for our purpose to notice that
one day in the year is accepted by Christendom as the
anniversary of the birth of Christ, that we nominally
commence our era with the birth of Christ, but that the
accepted anniversary is not selected as the date of the
commencement of the year in our so-called Christian
era. Were Christmas Day long separated from the first
day of the year less might be thought of the dis-
crepancy. But when the difference is only one week
the absurdity of the separation is almost self-evident.

There are thus two days, both in the immediate
neighbourhood of the date at which we begin the year,
either of which might most reasonably be adopted as
the first day of the year, while there is no reason, other
than the right of prescription, why the present date
should be retained. Nature, undoubtedly, points to the
solstice as the time of the year's beginning, while the
very name of our era and the very reckoning of our
dates year by year point to Christmas Day as the most
appropriate date.

It will be noticed in relation to the system now
submitted, that the argument in favour of the mean date
of occurrence of the winter solstice being accepted as
the first day of the year, does not apply with such force
as it applies under the present system. The week being

168 THE REFORM OF THE CALENDAR.

the basis of the new system, the beginning of the year
would in relation to the Sun necessarily swing backwards
and forwards to the extent of a few days. If, there-
fore, the mean date of the winter solstice were, to start
with, adopted as the beginning of the year, we would,
in a very few years, have the solstice occurring some-
times a few days before the old year had expired, and
sometimes a few days after the new year had begun.
The argument in favour of the solstice can, therefore,
under the new system be left out of account.

It is different with the argument in favour of
Christmas Day being accepted as the first day of the
year, and there would be no great complication in the
adoption of that day. At present Christmas Day occurs
exactly one week before the first day of the year, and
it falls on the same day of the week as the latter.
Thus all that would be necessary to effect the change
would be that in the final year under the present system
the closing week should be omitted and that this week
should form the first week under the new system. If
the new system were accepted as coming into operation
in the beginning of 1928, then the date which under the
present system would be recognized as the 25th day of
December 1927, being Sunday, should be accepted as
the Isfc day of January 1928. By this means we
would, in view of the varying divergence of date in the
new as compared with the existing system, also secure
any advantage associated with the year beginning at
the solstice, as the solstice would fluctuate to and fro in.
the immediate neighbourhood of the first day of the
year.

In a year of 364 days there would be 91 days in

J'OI'K (iltKCOHV XIII., HKl-OUMKIi Ol Til K C'AI.KNDA II. p. l(i.S.
I'roui lih ,„(,,ui„i<,il 111 SI. /'. ^,■.«. /tftiii,.

THE REFORM OF THE CALENDAR. 169

;ach quarter — being exactly thirteen weeka The first
nonth of each quarter might conveniently be given five
j^eeks, or 35 days, and the remaining two months
'our weeks, or 28 days, each. In leap years the extra
veek might be given to December, which would then
lave 35 da,js, while in the common years it would have
58 days. The length of each month would, therefore,
)e as follows : —

Jany. 35 days (5 weeks)

April

35 days (5 weeks)

Feby. 28 „ (4

» )

May

28 „

(* „ )

March 28 „ (4

« )

June

28 „

(4 „ )

91 „ (13

» )

H "

(13 » )

July 35 days (5 weeks)

Oct.

35 day

s (5 weeks)

Aug. 28 „ (4

» )

Nov.

28 „

(4. » )

Sept. 28 „ (4

» )

Dec.

28 „

(^ ^» )

91 „ (^3

» )

91 »

(13 „ )

First Quarter 91 days, being 13 weeks

Second „

])

Third

Fourth „

Total length of

Common Year

364

Leap Year.

December 35 days

(Making Fourth

Quarter 98 days

or 14 weeks) add

Total length of

Leap- Year

371

In this way every month would invariably commence
m the same day of the week and end on the same day
Df the week, and the same dates in every month would
Jways occur on the same day of the week. If the

170 THE REFORM OF THE CALENDAR.

system were, as suggested, introduced so as to have the
year commencing on Sunday, then the year would
always end on Saturday ; while every month also would
begin on Sunday and end on Saturday. Also, the 1st,
8th, 15th, and 22nd of every month would invariably
be Sunday, the 2nd, 9th, 16th, and 23rd invariably
Monday, and so on. Similarly, in the additional week
of the longer months the dates would correspond with
the days of the week in each of these months alike.

As regards the observation of the Moveable Feasts
of the Church — being Easter Day and the " high days "
whose date is dependent on Easter — it is necessary
only to notice that these would be in no way affected by
the adoption of the change of calendar now explained.
Probably, as it is, the fixing of Easter — the date of
which is made to depend on the full Moon in association
with the vernal equinox, or rather with a tabular, as
distinguished from the true, full Moon in association
with a fixed, as distinguished from the naturally
variable, equinox — is as complicated, artificial, and
inconvenient as could be devised. The proposed system
does not touch this matter. The relation between the
suggested system and that now existing is at all times
capable of easy ascertainment and consequently the
dates of the Moveable Feasts could, notwithstanding the
adoption of the reform, be fixed just as conveniently,
or as inconveniently, as they now are.

It is, of course, evident that no important alteration
of the calendar could conveniently be made in these
days except as a matter of international agreement, and
it is equally evident, as has already been indicated,
that any feasible method of making the year an exact

THE REFORM OF THE CALENDAR. 171

multiple of the week must necessarily, with its
advantages, have some disadvantages, especially in the
direction of lessening in individual years the closeness
of approximation to the true length of the solar year.
Thus the subject is one requiring careful and anxious
consideration and well-balanced judgment before action
can be decided upon. These, however, there is no doubt,
are and have ever been invariable accompaniments in
the determination of a matter which so closely touches
the lives of the people of all classes as the reform of
the calendar.

THE MAGNETISM OF THE EARTH.

SYI^OPSIS.

Chinese discovery of the properties of the lode-
stone — Early use of the magnetic needle in China
— Discovery of the deviation of the compass —
Chinese method of preparing the magnetic needle
— Distinction between needles pointing south and
needles pointing north — Lord Brougham on
Chinese stagnation — Magnetic needle introduced
into Europe — Magnetic variation, geographical
and periodic — "True as the needle to the Pole"
— Magnetic variation in London and Paris — The
secular variation in London, its period and its
extent — The return of the needle in London to
the true north — The needle's swing in Paris and
New York — The magnetic dip — Magnetic equator
and poles — Early records of dip in London — The
secular period of the magnetic dip in London —
Magnetic equator and poles not definitely fixed —
Magnetic intensity — Daily movement of needle in
London — Geographical difference in amplitude of
daily movement — Seasonal differences in needle's
daily movement — "Magnetic storms" — Difference
between daily movement in northern and southern
hemispheres — The causes of the needle's move-
ments — Early investigations — Gilbert's conjecture
— Barlow's hypothesis — Discoveries by Arago and
Ampere — Application of these discoveries to
terrestrial magnetism — Sun-spots and aurorae —
The Sun as a distributor of electricity — Superficial
character of the Earth's magnetization — Increase
of temperature with descent — Magnetization lost
by heating — Conclusions suggested by the argu-
ment — Magnetism and gravity.

THE MAGNETISM OF THE EARTH.

According to Eastern tradition the properties possessed
by the lodestone, or natural magnet, of attracting iron
filings, and of taking up its position — if freely sus-
pended — in a northward and southward direction, were
first discovered by the Chinese. Although tradition is
very vague and uncertain as regards the period of these
discoveries, there is no reason to doubt that Chinese
knowledge had so far advanced as to enable use to be
made of the magnetic needle as a guide in travelling in
the far East long before the commencement of our era*

The Chinese, in very early times, had a contrivance
which they called " tchi-nan," that is " south-indicating."
The term is supposed to have reference to an appliance
by which, through the action of a magnetic needle
floating freely on water, the arm of a diminutive
human figure was caused to point constantly towards
the south. This mechanism was made use of in con-
nection with a coach or waggon in travelling, being fitted
into the upper part of the vehicle. Hence the name
*' tchi-nan " — " south-indicating " — was applied, not to the
mechanism itself, but to the vehicle fitted with it.

This invention is traditionally imputed either to
Hoang-ti, who is supposed to have reigned in the

175

176 THE MAGNETISM OF THE EARTH,

twenty-seventh century B.C., or to Prince Tcheon-Kong,
who lived in the eleventh century B.C. Thus, if we
accept even the later date, it would appear that the
directive tendency of the magnetic needle in relation to
the cardinal point was known and utilized by the
Chinese three thousand years ago.

To the Chinese also is due the discovery that the
direction of the magnetic needle cannot be accepted as
being geographically exactly north and south. This
fact, however, does not appear to have been ascertained
until towards the close of the eleventh century of our
era, or more than two thousand years after the later
date assigned by tradition as that at which the directive
tendency of the needle was made practical use of. It is
the case, indeed, that in China the direction of the
needle differs very little from the true north and south.

The Frenchman, Biot, who was a most eminent
mathematician and scientist at the time of Waterloo,
quotes the following interesting passage from a Chinese
author of the eleventh century as descriptive of the
ancient Chinese method of preparing the magnetic
needle : —

" Those who perform the trick rub the needle with a
magnet-stone; then it will mark the south; it will,
however, always decline a little towards the east. It
does not exactly indicate the south. When such a
needle floats on water, it is very much agitated. If one's
finger-nails simply touch the edge of the basin where it
floats, they throw it into agitation. It is better to
suspend it in order to manifest its virtue as much as
possible. This is the method : — Take a thread out of a
new skein of cotton and stick one end of the thread to
the exact middle of the magnet with a piece of wax as

THE MAGNETISM OF THE EARTH. 177

big as a mustard seed. Then hang it in a place free
from draughts. The needle will then point steadily to
the south. Among these needles obtained by rubbing
there are always some which mark the north. Our
conjurers always have some which point south, and some
which point north."

The strange distinction which this Chinese writer
draws between needles which point south and needles
which point north is accepted as showing that the
Chinese were not aware of the fact that every magnet
has two opposite poles. It seems also to show that the
magnetic needles described must have had only one
pointed end, and that the position towards which that
eTid was directed was accepted as the direction indicated
by the needle.

To us, with our western notions, it is also remarkable
to observe the complete confidence with which the
Chinese accept the prime tendency of the needle as
being towards the south, just as we on our part are
disposed to consider the compass-needle as pointing
towards the north. Seeing that the needle necessarily
indicates two opposite directions, it is evident that if
one end turns towards the north the other end must
turn towards the south. The fact that the Chinese
regard the needle as pointing southward, while we
accept it as pointing northward — the diverse view being
a mere unimportant detail — would seem to be symbolical
of the essential contrariety of eastern and western ideas.

It is not a little surprising that with their early
knowledge of the directive power of the magnetic
needle the Chinese have made so little practical use
of the discovery. "They afford," according to Lord

178 THE MAGNETISM OF THE EARTH.

Brougham, "a singular instance of a nation early
making some progress, and then stopping short for
ages." He continues, " Possessed of the mariner's com-
pass twelve hundred years before it was known in
Europe, they have scarcely ever put it to the use which
it really can best serve, but creep along their coasts
from headland to headland, like the most ignorant of
the South Sea Islanders, and rather employ it on shore,
where other marks might better serve to guide them."

The magnetic needle, or compass, was introduced
into Europe by the Arabs in the eleventh or twelfth
century, in its original form of a needle floating on
water, and its introduction was the forerunner of the
great exploring expeditions of the Middle Ages. It was
was not until the latter half of the fourteenth century
that the balanced needle came into use, this important
improvement being, it is supposed, the invention of
Flavio Gioja, a native of Amalfi in Italy.

Although the Chinese, as we have noticed, discovered
that the direction taken up by a magnetic needle is not
exactly due north and south, it is uncertain whether
they also discovered that the direction of the needle is
subject to variation. It is known that Christopher
Columbus discovered independently that the direction
of the needle varies from place to place, but whether he
was actually the first to notice this geographical varia-
tion is not known. The fact that the variation in
direction is not only geographical but also periodic
must have been made evident only by careful and
long-continued observation.

It is, indeed, the case that the directive tendency of
the masrnetic needle is in a state of continuous fluctua-

THE MAGNETISM OF THE EARTH. 179

tion. "True as the needle to the Pole," is a saying
which even to this day is sometimes made as a declara-
tion of constancy, yet even King Henry the Eighth wafi
not more fickle in his affections than is the magnetic
needle in its steadfastness to the true north.

In 1657 the compass needle in London pointed to
the true north, but this was not the case in the preceding
years nor has it been the case since then. Previous to
1657 the needle pointed to the east of north, its direction
in 1580 — the date of the earliest reliable observation —
having been 11° 15' to the east of north. Since 1657
the needle in London has continuously pointed to the
west of north.

Although the matter is somewhat uncertain, as the
records do not show any reversal of the needle's move-
ment during the period when it pointed to the east of
north, there are reasons for thinking that at or near the
date of the earliest London record the needle was
pointing to its extreme easterly position. This is
suggested by a comparison with certain early records
applicable to Paris. Thus in 1580 when the needle in
London was pointing 11° 15' to the east of north, the
needle in Paris was pointing 11° 30' to the east of north.
In 1622 the direction indicated in London was 6° 0' EL,
while in Paris it was 6° 30' E. ; in 1634 the respective
directions were 4° 6' E., and 4° 16' E. In 1657, as we
have seen, the needle in London pointed to the true
north, but this was not the case in Paris until 1666.
Thus it would seem that the direction indicated by the
needle in Paris is somewhat to the east of the direction
indicated in London, although the records certainly go
to show that the difference is by no means constant.

180 THE MAGNETISM OF THE EARTH.

This applies at the present time just as it applied at
the time with which we are dealing, the direction
indicated by the needle in Paris at present — when the
needle in both cities is pointing to the west of north —
being an appreciable amount less to the west of north
than is indicated by the needle in London. This is, of
course, equivalent to saying that the needle in Paris is
now pointing rather more to the east than is the case
in London.

Now, although 1580 is the date of the earliest record
for London it is not so for Paris. We have a record for
the latter city for the year 1550. The direction then
indicated by the needle in Paris was 8° 0' to the east of
north. If we compare, as noted above, the records for
London and Paris for the dates prior to 1657, being the
records applicable to the years 1580, 1622, and 1634, we
find that the mean difference in the direction of tlie
needle in these two cities was 18"3'. It will be seen
that the difference in 1580 was 15' ; in 1622, 30' ; and
in 1634, 10' ; the needle in Paris in each case pointing
more to the east than the needle in London. If, now,
we apply this mean difference to the Paris record
for 1550 — being 8° 0' E. — we find that the probable
direction of the needle in London in 1550 was about
7* 42' E. Although this estimate can only be put forward
as roughly approximate, it yet goes to show very strongly
that the direction indicated by the needle in London in
1550 must have been distinctly less to the east of north
than the direction which was indicated in 1580. This
shows that some time about 1580 the direction of the
needle's movement in London must have been reversed.

We may endeavour to determine with somewhat

THE MAGNETISM OF THE EARTH. 181

greater precision the date of this reversal of movement
by considering the approximate mean annual movement
revealed by the records about the specified period.

Proceeding first on our calculation of the direction
indicated by the needle in London in 1550 — being 7' 42'
east of north — we find that between 1550 and 1580 —
when the needle pointed 11° 15' to the east of north —
the change in direction amounted to 3° 33'. As the
period is thirty years, the mean, annual change would
have been 71'. Between 1580 and 1622, a period of
forty-two years, the direction indicated by the needle
changed from 11° 15' E. to 6° 0' E., a change of 5° 15',
which is equivalent to 7*5' per annum. Between 1622
and 1634 the direction changed from 6° 0' E. to 4° 6' E.,
a change of 1° 54' in the intervening twelve years or of
95' per annum. In the final period, being 1634 to 1657»
the change was from 4° 6' east of north to the true
north, the change therefore amounting to 4° 6' in a
period of twenty-three years. This is equivalent to
10*7' per annum. The mean annual ratios of change in
the periods dealt with are, therefore, as follows : —

From 1550 to 1580

7-1'

From 1580 to 1622

7-5'

From 1622 to 1634

9-6'

From 1634 to 1657

... 10-7'

Now, it is in accordance with observed facts in other
natural phenomena, as well as with experience in mag-
netic phenomena, that, at or near the time of reversal
movement, the ratio of periodic change is least. We see
that the lowest rate of change occurred in the earliest
periods. This would suggest that the time of reversal
of movement was very shortly before the] year 1580,

182 THE MAGNETISM OF THE EARTH.

such an interval indeed before that date as would brin^
the two earliest ratios of change into exact correspond-
ence. The difference, however, is already very slight,
7*1' comparing with 7'5', and the data on which the
earliest computation is arrived at is rather uncertain.
The argument evidently suggests that the date of re-
versal must have been very near, and shortly before, the
year 1580, say about the year 1578.

About this date then, in all probability, the needle in
London pointed farthest to the east of north, and it then
began to move towards the west. Very slowly, year by
year, this westerly movement went on. In 1657, as we
have seen, the needle pointed to the true north, but the
westerly movement of the needle still continued. It
went on until the year 1818, when the needle pointed
24' 38' 25" to the west of north. The movement of the
needle was then reversed, and since 1818 it has con-
tinuously moved towards the east, the westerly deflection
of the needle having been reduced by 1909 to somewhat
rather less than sixteen degrees.

We see then that the needle very probably was in its
extreme easterly position in or about the year 1578, a
position in which its direction would have been about
11° 20' to the east of north, and we know that in 1818
the needle was in its extreme westerly position, its
direction being 24° 38' 25" to the west of north. This
would make the period occupied by the needle, in moving
from one extreme to the other, about 240 years, and the
arc covered by the movement about 36 degrees.

Supposing the reverse, or eastward, swing of the
needle to occupy the same time, the period intervening
between consecutive returns of the needle to the same

THE MAGNETISM OF THE EARTH. 183

extreme in London would be about 480 years. There is
some reason, however, to suppose that the eastward
swing of the needle is somewhat slower than the west-
ward swing. Thus the needle passed from its westerly-
extreme of 24" 38' 25" W. in 1818 to about 16° W. in
1907, that is to say in a period of eighty-nine years.
The corresponding arc, from 16° W. to 24' 38' 25" W.,
was covered between 1740 and 1818, the interval being
only seventy -eight years. The westward movement thus
occupied about eleven years less than the corresponding
eastward movement. These considerations would suggest
that the period intervening between consecutive returns
of the needle in London to the same extreme is really
somewhat longer than five hundred years.

The eastward movement of the needle from 1818
to 1907 was about 8' 38-6' (being from 24° 38*4' W.
to 15* 59*8' W.) which is equivalent to about 58' per
annum. If the eastward movement is continued at
the same mean annual rate it will proportionally be
about 165 years after 1907 before the needle in London
will again regain the true north. This takes us up
to the year 2072, which would make the interval be-
tween the consecutive returns to the true north, with
the westerly deflection intervening, about 415 years-
It would seem that with the easterly deflection inter-
vening the interval should be less than half the period
which is required when the westerly deflection in-
tervenes.

There is reason to believe that the period of the
needle's movement from one extreme to the same ex-
treme again has a wide range of difference according
to geographical situation, and that there is also a wide

184 THE MAGNETISM OF THE EARTH.

diversity both in the extent of the needle's movement,
and in tlie interval which elapses between successive
returns to the true north.

In Paris, as we have seen, the needle pointed to the
true north in 1666, being nine years later than the date
at which this was the case in London. As, however, the
direction indicated by the needle appears, as we have
noticed, to be consistently rather more easterly in Paris
than it is in London, it would seem, with the needle
now moving towards the east in both cities while still
pointing to the west of north, that the true north will
be regained by the needle in Paris some years earlier
than in London. Thus the interval between the
successive returns of the needle to the north, with a
westerly deflection intervening, will be considerably
less in Paris than in London. Naturally, however,
the opposite may be the case when an easterly deflec-
tion intervenes.

In many parts of the Earth such as, for instance.
New York, the movement of the needle is thought
not to cover the true north at all. Thus in New
York the deflection is believed to be constantly towards
the west, and the arc of the needle's movement to be
very much less than in either London or Paris. Al-
though for many years the movement of the needle
in both London and Paris has been towards the east,
its movement at New York during late years has
been towards the west. The diversity which has
characterized the needle's movement in these cities is
illustrative of what occurs all the world over, the action
of the needle in no two places on the surface of the
Globe being exactly similar in all respects to each other.

THE MAGNETISM OF THE EARTH. 185

Besides changing constantly in its relation to the
cardinal point — a change which is known as the
" declination " or the " variation " — the needle has
another and quite dissimilar movement. This is the
movement of " dip " or " inclination."

In the ordinary compass the needle is virtually
prevented from exhibiting the dip, through the method
in which it is balanced. In order that the dip may be
made evident, the needle has, of course, to have perfect
freedom of movement in respect of deviation from the
horizontal.

In the tropics there is an irregular belt around the
Earth, partly to the north and partly to the south of
the equator, in all parts of which the needle remains
horizontal. This belt is called the magnetic equator.

To the north of the magnetic equator the north-
indicating end of the needle dips, and to the south of the
magnetic equator the south-indicating end of the needle
dips, the dip in each case increasing, generally speaking,
with separation from the magnetic equator, although
even in this respect some diversity may result from
local and other causes. In two regions, which are not,
very far removed from the arctic circle and the antarctic
circle respectively, the dipping needle takes up a vertical
position. These regions — which cannot be described as
points on the Earth's surface as they each cover a con-
siderable area — are of course the magnetic poles. At
the north magnetic pole the north-indicating end of the
needle points directly downwards, while at the south
magnetic pole this is the case with the opposite end of
the needle.

The north magnetic pole was discovered by Sir

186 THE MAGNETISM OF THE EARTH.

James Clark Ross in 1831, in latitude 70' 5' north and
longitude 96° 43' west, the locality being in Boothia, in
Uie extreme north of the Dominion of Canada. A
Norwegian scientific expedition under Captain Roald
Amundsen visited the region in 1903 and made in-
vestigations during the following seasons. The geo-
graphical position of the south magnetic pole was
approximately determined many years ago by calcula-
tion based on observations made by antarctic expeditions

Similarity of action of the magnetic needle as regards dip on the surface
of the Globe and on a magnetic bar.

but this pole was not actually discovered until the
winter of 1908-9. It was then located by Professor
David, one of the scientific members of Lieutenant
Shackleton's expedition in the Nimrod, in 72° 25' south
latitude and 154° east longitude.

The action of the magnetic needle as regards dip
in difierent geographical situations, has been compared
to the corresponding action of a suspended magnetic
needle when moved along a magnetized bar. The com-
parison is convenient, but it is defective in so far as

* Since printing of above, Lieut. Shackletou's narrative has been pub-
lished. He gives the " mean position " of this pole as 72° 25' S., 155° 16' E.

THE MAGNETISM OF THE EARTH. 187

irregularity results terrestrially through the action of
local and other causes.

The earliest record of the dip of the needle in
London is for the year 1576, when the dip was 71* SCK.
Judging by the records immediately following — being
for the years 1600 and 1676 — the dip when first ob-
served was increasing. In 1723 the dip was 74° 42',
and, as far as can be determined from the available
records, this was the maximum dip in London. The
next observation of the dip in London after 1723 is for
the year 1773, and the dip was then 72° 19', being 2° 23'
less than it was fifty years earlier. Thereafter the

A.D. 1676— dip, 71° SO*. A.D. 1723— dip, 74" 42'. A.D. 1908— dip, 66° 56'.

Dip of magnetic needle in London at vaiious dates.

dip decreased gradually and with some irregularity until
1906, in which year it amounted to only 66° 55*17',
being nearly 7° 47' less than the dip recorded for the
year 1723. The dip appears to have reached its minimum
in 1906, the record for the year 1907 showing a slight
upward movement to 66° 56*0', which increased during
1908 to 66° 56-28'. Thus, if we accept the dip recorded
for 1723 as being the maximum, the period occupied by
the needle in London in passing from the maximum to
the minimum is evidently about 183 years. Supposing
the rate of increase in dip to be the same as the rate

188 THE MAGNETISM OF THE EARTH.

of decrease, this would make the period intervening
between consecutive returns to the same extreme about
366 years. Although this period is very uncertain, the
great difference which it shows from the estimated
period of the needle's movement in cardinal direction
would seem at least to indicate that these respective
periods do not coincide, but that, on the contrary, the
time intervening between the diverse extremes in dip is
much shorter than the time intervening between the
extremes in cardinal direction.

Similar changes in the dip of the needle to those
which we have noted as having occurred in London
occur all over the Globe, the changes in some geo-
graphical positions being more pronounced, in others
less pronounced.

It has been found indeed that the position of the
magnetic equator, or region of no dip, and even the
positions of the magnetic poles, or regions of vertical
dip, are not definitely fixed on the Earth's surface, but
are themselves subject to slight changes of position.

Besides the changes in declination and dip there is
another inconstant " element " as it is called, in connec-
tion with terrestrial magnetism. This is the intensity
or force of the magnetism which affects the needle.
It is indicated by the vibrations of the needle when
diverted from its true direction, or perhaps we should
say by the strength of its effort when diverted to return
to its natural position.

Thus the position taken up by the magnetic needle
is subject to change in horizontal direction, in dip and in
intensity, the cycle of which, in general, occupies
centuries, although the periods differ in every separate

THE MAGNETISM OF THE EARTH. 189

locality. Not only is this so, the needle has also a
corresponding daily movement, which is subject to
seasonal variation, and the characteristics of these
short-period movements are also locally diverse from
each other.

In London at present (1909) the needle points, let us
suppose, 15° 50' to the west of north, and the dip is
about 66° 57'. If we accept these figures as absolutely
correct for the mean position, it will be the case that
during certain hours daily the needle will point slightly
more than 15° 60' to the west of north, while at other
times the westerly variation will be slightly less than
the amount specified. Similarly, as regards dip, the
needle will fluctuate somewhat above and belovi' the
amount mentioned, the divergence varying according to
the time of day. Of course, although imperceptible, it
must be the case that at present the daily swing of the
needle in London is, on the average, infinitesimal ly
greater towards the east than towards the west, and
towards the vertical than towards the horizontal, seeing
that secularly the westerly variation is now decreasing
while the dip is increasing.

Between 7 and 8 a.m. the needle in London in its
small daily swing points nearest to the true north. As
the variation is at present westerly, this, of course, means
that the needle is then at its easterly extreme for the
day. The needle then begins to move towards the west,
and between 10 and 11 a.m. it attains its mean position.
Still moving westerly it reaches its extreme westerly
variation for the day between one and two o'clock in
the afternoon. The needle then retraces its course and
passes towards the east very slowly and somewhat

190 THE MAGNETISM OF THE EARTH.

irregularly until between 7 and 8 a.m. it is again in its
extreme easterly position.

The amplitude of the daily swing in declination
varies, generally speaking, with the latitude, or, perhaps
it would be more correct to say, with the distance from
the magnetic equator. In high latitudes it is as great as
15', in London and Paris it is only about 9' or 10', while
near the magnetic equator it is only 3' or 4'.

The daily movement in dip is of a similar character.
At London the dip is greatest about 10 a.m. when it
is about 1'3' more than the mean; while it is at its
minimum about 7.30 p.m., being about '6' less than the
mean. The dip is at its mean for the day about 6.30
a.m. and 3 p.m. The daily movement in dip, as in
declination, varies with geographical position.

Doubtless, similar fluctuations occur in relation to
the intensity, or force, although oscillations in this
element of terrestrial magnetism are peculiarly difficult
of observation.

All these movements are subject to seasonal differ-
ences being, in general, more pronounced in summer
than in winter. They are also subject to sudden and
violent disturbance of not infrequent occurrence, through
what are called " magnetic storms." These changes are
often very abrupt, and it may happen that even in half-
an-hour the needle may oscillate more than the mean
amplitude of its daily swing. Although it is rare during
a magnetic storm in Britain to have an oscillation in a
day in declination of IJ degrees, much larger movements
occur during these storms in the polar regions. At the
British station, Fort Rae, in the north of Canada, a
range of 11;^ degrees was observed in the declination

THE MAGNETISM OF THE EARTH. 191

in a day, and on ten days in one year the movement
exceeded 5 degrees. A magnetic storm of exceptional
severity occurred on 25th September 1909, which affected
the magnetic needle all over the Earth to such an extent
as seriously to interiere with the working of the tele-
graph. The aurora was at the same time conspicuous
in the temperate latitudes of both the northern and
the southern hemispheres.

In connection with the daily movement, it should be
noticed that in the southern hemisphere the needle
moves in the opposite direction to that in which it
moves at the same hour in the northern hemisphere.
Thus when the north-indicating end of the needle is
moving to the west in the northern hemisphere it is
moving to the east in the southern hemisphere, and
vice versa.

What, then, are the causes of these mysterious
movements of the magnetic needle with their daily,
seasonal, and secular characteristics ?

There can be little doubt that enquirers must, at a
very early date, have been forced to conclude that the
power of directive tendency possessed by the magnetic
needle must flow from the Earth itself. The whole
circumstances indicate that this must be the case. This
being accepted, the fact that the needle is liable to be
deflected by the proximity of any other magnet, whether
natural or artificial, and can be made to follow the
movement of any other magnet would naturally suggest
that the Earth, in causing the needle to take up a
certain position in relation to the cardinal points and
to vary in dip with geographical situation, is really
acting on the needle exactly as another magnet

192 THE MAGNETISM OF THE EARTH.

would act. The conclusion would thus be arrived at
that the Earth itself must either be a magnet or must,
in its structure, comprise a magnet or a combination
of magnets.

Thus a fascinating and puzzling problem would, by
a regular and simple train of reasoning, be presented
to those interested in the discovery of natural laws.

Dr William Gilbert, a native of Colchester and
physician to Queen Elizabeth, was one of the those who
interested himself in the study of terrestrial magnetism.
He published in 1600 a book on the subject, which is
one of the earliest as well as one of the most important
contributions ever made to the science of magnetism in
Britain. In this work, Gilbert conjectured that the
Earth must either be itself a great magnet, or that there
must inside the Earth be a large magnet whose poles lay
near the geographical poles.

Barlow, who was mathematical master in Woolwich
Academy from 1806 to 1847 and who made some
valuable contributions to the literature of this subject,
suggested the existence of electric currents in the Earth's
crust circulating around the Earth from east to west.
Barlow's hypothesis, although defective in not satis-
factorily accounting for the existence of the conjectured
electric currents, is noteworthy as a theoretical con-
ception.

Early in the nineteenth century Arago and Ampere,
two distinguished French scientists whose names are
linked together in connection with investigations in
relation to electricity, discovered that magnetism is
energetically induced in iron or steel if the iron or
steel substance is encircled by a spiral coil of wire

THE MAGNETISM OF THE EARTH. 193

and an electric current is caused to flow through the
The process adopted was to coil a wire con-

wire.

nected with an electric battery around a glass tube,
place the needle which it was proposed to magnetize
inside the tube, and then pass the current. It was
found that the needle was thus immediately magnetized,

Magnetization (rf a steel needle by right-handeil and left-handed
spiral coilB.

the magnetization being temporary in the case of iron,
but permanent in the case of steel.

The investigators also discovered the very remark-
able fact that the nature of the poles formed at the
exti"emities of a needle magnetized in this manner
depended on the direction in which the encircling wire
was spirally wound. Supposing the tube to be in a

194 THE MAGNETISM OF THE EARTH.

vertical position with the wire coiled from top to bottom,
and the current caused to pass through the coil from
the top downwards, it was found that if, in its descend-
ing spiral course, the wire passed from right to left on
the side of the tube turned towards the operator, the
south-indicating pole of the magnetized needle occurred
at the upper extremity, the north-indicating pole at the
lower. If, on the other hand, the wire passed around
the tube from left to right, the conditions otherwise
being unchanged, the north-indicating pole was formed
at the upper end of the needle and the south-indicating
pole at the lower end. It was thus found that the
deciding cause as to which end of the needle was to
point northward depended entirely on the apparently
insignificant detail whether the course of the encircling
electric current was right to left or left to right.

If, now, the Earth is magnetized by electric currents
passing from east to west, as suggested by Barlow, are
such currents of the right-to-left or the left-to-right de-
scription ? It will be seen at once by a glance at a
Globe or map that, as seen from outer space, assuming
that the north pole of the Earth is uppermost, an east
to west movement around the surface of the Earth is
exactly equivalent to a right-to-left movement. Thus
if we could look upon the Earth from a distance as
we can look upon the Moon, and if the northern ex-
tremity of the terrestrial axis were at the top to our
view, the direction which to us would seem to be right-
to-left, would, to dwellers on the Earth be simply east-
to-west — right-to-left and east-to-west being thus
synonymous descriptions.

If then, electric currents passing around a needle

THE MAGNETISM OF THE EARTH. 195

from right to left give rise to the formation of a south-
indicating pole at the upper extremity and of a north-
indicating pole at the lower extremity, it is evident
that, in the circumstances stated, if the Earth, or its
atmosphere, is magnetizable, the passage around it of
electric currents flowing from east to west must give
rise to the formation of a south-indicating pole in the
north, being the region corresponding to the upper
extremity, of the needle, and to the formation of a
north-indicating pole in the south, being the region
corresponding to the lower extremity of the needle.

Now, as is well known, one of the first principles of
magnetism is that unlike poles attract and that similar
poles repel each other. Consequently a south-indicating
pole at the north of the Earth would attract the north-
indicating end of the needle, and a north-indicating pole
at the south of the Earth would attract the south-
indicating end of the needle.

Thus such electric currents as Barlow surmised the
occurrence of, would, if the substance of the Earth or of
its crust, or the enveloping atmosphere, be magnetizable,
exactly meet the conditions required to induce the mag-
netic needle to take up its position, generally speaking,
in a northward and southward direction, and would
be consistent also with the dip of the needle both geo-
graphically and as regards its converse poles.

It was discovered in the latter half of the nineteenth
century that when the sensitive magnetic needles of the
observatories exhibit the tremulous agitation which is
recognized as indicating the occurrence of a magnetic
storm, there are usually, if not invariably, at or about
the same time, displays of aurora and conspicuous spots

196 THE MAGNETISM OF THE EARTH.

on the Sun. It has also been ascertained that the
period of these spots, being about eleven years, is re-
cognizable in connection with magnetic disturbances.
These facts have been accepted as proving what other
facts also indicate — that solar activity is intimately
associated with terrestrial magnetism.

The advances in electrical science in its relation to
Nature have, of late years, brought many to believe
that the Sun is our great store-house of electricity, and
that electricity itself may even be atomic in structure.

Thus the Sun, in pouring out light and heat, is
believed to be also pouring out electricity, or electric
energy, on all surrounding space. Whether in the form
of waves or vibrations induced in the ether of space, or
in the form of physical particles, this electrical stream
dashes against the Earth on the hemisphere which for
the time is exposed to the Sun. As the Earth in its
daily motion rotates eastward, causing the Sun appar-
ently to pass around the Earth daily in a westward
direction, this stream or flow of electricity is caused to
wind continuously around the Earth in a westward
direction, a coil so to speak, being completed in each
daily rotation, the coil extending from the northern to
the southern limit for the time being, of the terrestrial
exposure to the solar rays. Here then we have a
feasible explanation of the existence of the electric-
currents passing from east to west in the crust of the
Earth, or in the surrounding atmosphere, the occurrence
of which was surmised by Barlow about the middle of
last century as explanatory of the Earth's magnetism.

Thus the magnetization of the Earth may probably
result in a manner not dissimilar to that employed

THE MAGNETISM OF THE EARTH. 197

when a needle is magnetized by an electric current
through a right-handed spiral coil. The action of the
compass needle and of the dipping needle are both con-
sistent with such an hypothesis.

At the same time it is not necessary to conclude that
the Earth as a physical whole is subjected to magnetiza-
tion, that the Earth itself is really a great magnet as
Gilbert conjectured. The evidence would seem to in-
dicate rather that the magnetization is comparatively
superficial — that it is confined entirely to the crust of
the Earth, and, perhaps, mainly to the atmosphere.

We know that the temperature of the Earth is found
to rise as the surface is departed from, and that at a
distance from the surface quite insignificant in compari-
son with the size of the Earth the temperature is very
great. It has been calculated that the increase in
temperature with descent from the surface-level amounts
on the average to one degree Fahrenheit for every fifty
or sixty feet of descent. The rate, however, varies con-
siderably. Thus at South Balgray, near Glasgow, in a
coal mine, the temperature was found to increase on the
mean by one degree for every forty-one feet of descent,
while at Dukinfield, near Manchester, the descent re-
quired on the mean for each degree of rise of temperature
was found to be slightly over eighty-three (83'2) feet.
Supposing we take, as an extreme, one hundred feet of
descent as the distance requisite to give a rise of one
degree in temperature, and that the surface temperature
is zero, we should still have a temperature of 212
degrees — being the boiling point of water in ordinary
atmospheric pressure — at a depth of 21,200 feet, being
almost exactly four miles — which comparatively is little

198 THE MAGNETISM OF THE EARTH.

more than a mere pin-prick on the terrestrial surface.
It cannot be questioned, therefore, that at a distance of,
say, one hundred miles from the surface the tempera-
ture must be excessive.

Now we know that if a magnetized needle is heated
to a bright-red heat and is then allowed to cool, its
magnetization is lost. It is then simply an ordinary
unmagnetized piece of metal. It has, in fact, been
known for long that iron, if raised to a certain " critical
temperature," corresponding to a dull-red heat, loses it»
susceptibility and becomes magnetically indifferent.
This " critical temperature " for various samples of iron
and steel ranges from 1274 degrees to 1598 degrees
Fahrenheit.

It may, therefore, be concluded that the Earth
although acted upon as indicated, is not converted
structurally into a magnet, but that the magnetization
is entirely confined to the crust, and even in the crust
does not extend to any great depth. In fact it is not
improbable, particularly in view of the substance of the
Earth being mainly non-magnetic, that the magnetism
of the Earth is, to a very large extent, a purely atmos-
pheric phenomenon. In any case it is, evidently, merely
superficial. But although, this may be so, the effect of
the superficial magnetization is, of course, practically the
same to the surface-dweller as if the Earth as a whole
were really converted into a great magnet.

We see then, that the facts suggest that the Earth is
superficially magnetized through solar-electric influence
in association with the Earth's rotation, the effect being
to convert the Earth superficially into an electro-
magnet.

THE MAGNETISM OF THE EARTH. 199

In the present state of our knowledge of this difficult
and elusive subject, it would seem, therefore, that we
have fair grounds for believing, at least tentatively,
(1) that anelectrical stream or current is received by the
Earth on the hemisphere for the time exposed to the
Sun, which, through the Earth's rotation, is coiled
around the Earth daily, causing the Earth to become
superficially an electro-magnet ; (2) that the conversion
of the superficial crust of the Earth, or, at least,
the conversion of the enveloping atmosphere, into an
electro-magnet in this manner is the cause of the
directive tendency of the magnetic needle, both in de-
clination and dip ; while the diversities and peculiarities
connected with the method of magnetization are the
cause of the constant changes and anomalies in the
movements of the needle; and (3) that the method
of magnetization, the geographical variation in the
exposure of the Earth to the Sun, and the irregularities
of the Earth itself both in form and composition, are the
causes which decide the position of the magnetic poles
and equator, and the changes, whether daily, seasonal,
or secular, which occur in relation thereto.

It might very reasonably be supposed that the
attraction exercised by the Earth on the magnetic needle
would have some effect on the needle's weight. If an
object is attracted or drawn in any particular direction
— as for instance towards the Earth — it is evident that
greater force must be required to retain it in its position
than would be necessary were it not so drawn. The idea
that magnetization must affect weight is a very old one.

This subject was investigated very carefully by
Robert Norman, a seaman and ingenious artificer —

200 THE MAGNETISM OF THE EARTH.

who lived in the second half of the sixteenth century,
and who was the first to observe the magnetic dip by a
sound method. Norman, who was a contemporary of
Gilbert's and who forestalled the latter in some of his
magnetic investigations, proved by various experiments
tliat weight was unaffected by magnetism. Having
weighed some small pieces of steel in a very delicate
gold balance, he magnetized and reweighed them. He
found the weight quite unaffected, "though," as he
writes, " everyone of them had received vertue sufficient
to lift up his fellow." He then pushed a steel wire
through a small spherical piece of cork, and pared the
cork so that it sank to a certain depth in water. He
noted the depth and then magnetized the wire. He
found that the cork floated at exactly the same level,
although the wire now took up a certain definite
position in relation to the cardinal points. Norman,
afterwards arranged a magnetized needle on a small
piece of cork and floated the latter on water with the
view of noting any movement. He found that though
the needle as in the preceding experiment, took up a
definite direction, there was no indication of any move-
ment of translation.

Norman thus convinced himself that the action of
the Earth on magnets is purely directive. In fact the
power exerted by the Earth on the magnetic needle is
what is known as a "couple," that is, a pair of equal
but oppositely-directed parallel forces acting on the two
ends of the needle. As these two forces thus counter-
balance each other, being effective only as regards
directive tendency, gravity is quite unaffected by
magnetization.

THE MAGNETISM OF THE EARTH. 201

The careful study which has been given to the
subject of terrestrial magnetism during the past few
centuries has undoubtedly resulted in gain to science.
It cannot be said, however, that this perplexing problem
has yet been completely solved. Slowly and with
difficulty progress is certainly being made, if only through
the accumulation of painstaking observations in all parts
of the Globe, and the deductions made therefrom.

Slowly, but, we may hope, surely, we are advancing
towards a more perfect comprehension of this strange
phenomenon — a phenomenon which, though regarded in
the past as pertaining solely to the Earth, we now know
forms, or at least is associated with, a strange con-
necting link between the Earth and the great centre
of our system, the true nature of which still awaits
elucidation.

THE MOVEMENTS OF THE SUN AND
OF THE EARTH IN SPACE.

SYNOPSIS.

THE MOVEMENTS OF THE SUN AND
OF THE EARTH IN SPACE.

In early times it was universally believed that the
Earth was fixed in position, and that the Sun, Moon,
and stars revolved around it. The Pythagorean
Philolaus, who lived in the fifth century B.C., introduced
for the first time the motion of the Earth, although in a
somewhat fanciful and crude form. He appears to have
regarded the Earth, the Sun, the Moon, and the then
known planets as revolving around some central fire, the
Earth rotating on its own axis as it revolved. Three
other Pythagorean astronomers, belonging to the end of
the sixth and to the fifth century B.C., believed in the
rotation of the Elarth. These were Hicetas (of Syracuse),
Heraclitus, and Ecphantus. Plato (428-347 B.C.) is sup-
posed to have believed in the rotation of the Earth.
Aristarchus, of Samos, who lived in the first half of the
third century B.C., distinctly recognised that the Earth
not only rotated on its axis, but revolved around the
Sun, and that the latter, like the "fixed stars," was
motionless.

The belief in the motion of the Elarth, although thus
appearing some centuries before our era, was restricted

206 THE MOVEMENTS OF THE SUN

to a very few individuals — scientists far in advance of
the times. It seems subsequently to have quite dis-
appeared, and, as in the most ancient times, the im-
mobility of the Earth was universally recognized. The
Earth, it was agreed, was definitely fixed in position,
and was the centre of the revolutions of the celestial
bodies.

Copernicus, in the first half of the sixteenth century,
made public his belief that the Earth revolves around
the Sun annually, and rotates on its axis daily — the Sun
being immovable. Notwithstanding opposition and per-
secution, the conception of Copernicus gradually secured
acceptance.

More than two centuries later the idea seems to have
occurred to various astronomers that the Sun may not
be fixed, but may itself be moving like the Earth, and it
is only within the last century or slightly more, that this
belief has secured scientific acceptance.

Thus we commence with the idea of a fixed Earth
and revolving Sun. We advance to the idea of a fixed
Sun and revolving Earth, and this idea fluctuates and
completely disappears before it is at last resuscitated
and slowly secures acceptance. Finally we advance
still further to the conception that neither the Sun
nor the Earth is fixed — that, though the latter is re-
volving about the former, the Sun itself is also speed-
ing onward in space, moving in a mighty orbit of
whose character and dimensions we are as yet in
complete ignorance. Notwithstanding this we believe
that the movement of the Sun, vast as it is, is never-
theless governed by the great natural power which
holds the Moon to the Earth, and which retains the

AND OF THE EARTH IN SPACE. 207

planets themselves under the control of the Sim
whithersoever its course may lie.

The terrestrial orbit is generally described as being
an ellipse which deviates only slightly from a circle.
If the orbit be considered merely in relation to the Sun
as the centre of our system, this description is certainly
true. The Earth moves around the Sun every year in
a path which, in relation to the Sun, is nearly circular,
the eccentricity, as it is called, of the orbit— or its
deviation from a circle — being only one-sixtieth of the
mean radius, that is to say of the average distance of
the Sun from the Earth.

Thus the distance between the centre of the ellipse
and the centre of the Sun — which is situated in one of
the foci of the ellipse — is just about one-sixtieth part of
the mean distance of the Sun from the Earth. In an
ordinary illustrative diagram in which the major axis
of the orbit is represented by a line of, say, six inches
in length this eccentricity would be shown by the dis-
placement of the Sun's centre to the extent of one-
twentieth of an inch from the centre of the ellipse. In
fact on such a ratio, the eccentricity of the orbit, or its
variation from a circle, and the eccentric position of the
Sun would be quite indistinguishable to the eye of the
ordinary observer. The orbit would appear to be a
circle with the Sun centrally situated.

Our distance from the Sun, however, is so great that
this eccentricity results in our being about three millions
of miles nearer to the Sun when the Earth is in
" perihelion," or at its nearest to the Sun, than we are
when the Earth is at the opposite part of its orbit — at
the position known as " aphelion." The mean distance of

208 THE MOVEMENTS OF THE SUN

the Earth from the Sun is about 92,897,000 miles, while
the perihelion distance is about 91,355,000 miles, and tho
aphelion distance about 94,439,000 miles. The ratio of the
mean distance to the perihelion and aphelion distances
is as 10,000 to 9834 and 10,166 respectively, these ratios
being based on the variation in the apparent or angular
diameter of the Sun, as seen from the Earth.

The Earth is at perihelion about the 1st of January
and at aphelion about the 4th of July, while it is at
its mean distance from the Sun about the 2nd of April
and the 4th of October.

But although, when we consider the Earth's orbit
merely in relation to the Sun, we can, with sufficient
accuracy, describe it as an ellipse of comparatively small
eccentricity, to do so conveys a somewhat false impres-
sion of the real character of the Earth's movement.
This arises from the fact that while the Earth is moving^
along in its orbital course and tracing out its elliptical
path the Sun itself is not at rest but is moving rapidly
through space. Supposing, to use a very simple illustra-
tion, that a cab were driving smartly along the street
and that a boy amused himself by running around it as
it passed along, although we might possibly say that the
boy " circled " around the cab or that he " encircled " it^
we could scarcely with accuracy describe his path as a
circle nor yet as an ellipse. His course is, in fact, of a
more complex character. Yet we constantly describe
the Earth's orbital movement, which is of an analogous
kind, as being simply elliptical. Is not the true char-
acter of the orbital movement quite disguised by such a
description ?

That the Sun, accompanied of course by the planets^

AND OF THE EARTH IN SPACE. 209

is moving through space is no recent discovery. Halley,
in 1718, first drew attention to the fact that three pro-
minent stars (Sirius, Procyon, and Arctv/rus) had quite
appreciably changed their positions since Greek times.
Astronomers thereafter gradually came to realize that
the Sun, although it happens to be the centre of our
system, really falls to be classed among the stars. If,
then, certain of the so-called " fixed stars " were known
to be changing their place in the heavens was it not at
least possible that the Sun also was not definitely fixed
in position ? Thomas Wright (1711-1786), author of a
Theory of the Universe, and other astronomers, speculated
on this idea, and Tobias Mayer (1723-1762), a German
astronomer of merit, went to the length of suofsfestinsf
how such a motion might be looked for.

It was reserved for the elder Herschel, the gentle
and popular music teacher who became the most famous
astronomer of his time, to put the matter to the test.
He reflected that if the Sun were really moving through
space its motion ought to be rendered evident by an
apparent change of position of the stars. It is obvious
that, if one is moving in a certain definite direction,
objects situated in that direction will appear to separate
or open out as they are approached, while, on the other
hand, objects in the reverse direction will appear to
draw closer together. This is an evident effect of
perspective and can be readily tested by anyone walking
at night along a street of some length, lighted up by
lamps on each side. Again, a forward movement in
any definite direction is accompanied by an apparent
drift in the opposite direction of objects in comparative
rest situated on each side. Such an effect of rapid

210 THE MOVEMENTS OF THE SUN

motion is noticeable from the windows of a railway
carriage. Was it not possible, thought Herschel, to test
by these matters of common life the question whether
the Sun was moving in space ?

It will be seen that under the special circumstances
the problem which Sir William Herschel set himself to
try to solve was one of extreme nicety. The distance
of the stars is in every case so excessively great that
any apparent change of position had in actual distance
to be enormous in order to be distinguishable in angular
measurement. Besides, as regards the drift in the
opposite direction of the stars at right angles to the
line of movement, it is evident that we might actually
be moving along in space and that no drift might
appear on account of the observed stars moving at a
corresponding rate in the same direction. The move-
ment of the Sun might, in fact, be apparently reversed
in direction through the observed stars moving Tnore
rapidly in the same general direction as the Sun
itself.

It was, therefore, necessary to assume that there was
no reason why a star should be moving in any one
direction rather than in another. If this were so the
observation of a sufficiently large number of stars
would reveal whether there was such a preponderence
of apparent movement in any special direction as to
suggest that the true explanation lay in solar movement;
while a comprehensive survey would also show whether
the perspective effects we have described were such as
might be expected from the existence of proper motion
on the part of the Sun.

Sir William Herschel faced this difficult problem in

AND OF THE EARTH IN SPACE. 211

1783, and he arrived at the conclusion that the Sun
was certainly moving in space, and that the direction of
its movement — or, as it is now called, the " solar apex" —
was approximately in right ascension 260° 34' and
declination 26° 17' N., both these measurements being
applicable to the epoch 1790.

In the following years this matter was investigated
by several eminent astronomers, and they were fully
agreed that the Sun had a movement in space. These
investigators came to the following conclusions as to
the direction of the solar apex, the position in each
case being specified as for the epoch 1790 : —

Position of Solar Apex.
Right Ascension. Declination.
M. Argelander, from observations of

21 Stars having proper motions

exceeding 1" per annum ... 256° 25' -38° 37' N.

M. Argelander, from observations of

50 Stars having proper motions

between i" and 1" per annum... 255° 10' 38° 34' „
M. Argelander, from observations of

319 Stars having proper motions

between ^" and i" per annum 261" 11' 30° 58' „
M. Luhndahl, from observations of

147 Stars 252° 53' 14° 26' „

M. Otto Struve, from observations of

392 Stars 261° 22' 27° 36' „

From observations at different places

and times in the Southern

Hemisphere by M. Lacaille, Mr.

Johnson, and Mr. HendersonJ... 260° 1' 34° 23' „
Mean 257° 50' 30° 46' N.

Considering the extreme complexity of the problem
these results are so harmonious as to suggest that there
is little doubt as to at least the approximate direction of
the solar apex.

Of late years the determination more exactly of the

212 THE MOVEMENTS OF THE SUN

position of the solar apex has engaged the attention of
several eminent astronomers, and the following results
have been arrived at : —

Position of Solar Apex.
Right Ascension. Declination.

Prof. Lewis Boss, from 279 Stars of

large proper motion

289-3°

44-1° N.

Prof. Lewis Boss, excluding 26 Stars

of largest motion

288-7°

51-5° „

Prof. Porter, Cincinnati, from 576

Stars moving less than 30" per

century

281-9°

53-7' „

Prof. Porter, Cincinnati, from 533

Stars moving between 30" and

60" per century

280-7"

40-1° „

Prof. Porter, Cincinnati, from 142

Stars moving between 60" and

120" per century

285-2°

34-0° „

Prof. Porter, Cincinnati, from 70

Stars having a movement ex-

ceeding 120" per century

277-0°

34-9° „

Dr. Stumpe, Berh'n, from 551 Stars

having motions of from 16" to

32" per century ...

287-4°

45-0° „

Dr. Stumpe, Berlin, from 339 Stars

having motions of from 32" to

64" per century ...

287-2°

43-5° „

Dr. Stumpe, Berlin, from 106 Stars

having motions of from 64" to

128" per century

280-2°

33-5° „'

Mean

284-18"

42-26° N.

It will be noticed that there is an appreciable difference
between the result of the later calculations as a whole
and that of the earlier. This, however, is somewhat
exaggerated through the difference of date. As we have
mentioned the earlier calculations apply to the epoch
1790. The later apply to an epoch which we may
take to be on the mean one hundred years later. Owing
to the gyratory movement of the Earth's axis, the point
at which the celestial equator intersects the ecliptic —

AND OF THE EARTH IN SPACE. 213

which is known as the First point of Aries," — has a
westward movement of 50*1" per annum, being 1° 23' 30"
in a century. As it is from this point right ascension
is reckoned, and as the reckoning is in an easterly
direction around the celestial equator, we have to add
about 1° 23' 30" to the right ascension specified in the
earlier series in order to render the calculations compar-
able with those of the later series, while some alteration
would also have to be made in the declination.

Notwithstanding such correction, it is evident that
there is considerable divergence between the two series
of calculations. This is rendered the more noticeable
from the fact that each series is in itself so wonderfully
consistent. After all, however, perhaps the most sur-
prising feature is that in a matter requiring such
extreme nicety of observation and calculation the
divergence should not have been even greater. It will
be found that the difference in right ascension between
the means of the two series when corrected for epoch
is less than twenty-five degrees.

The tendency of scientific opinion for some years
has been to accept the result which lies nearest to the
mean between the two series as indicating with greatest '
accuracy the true direction of the Sun's movement in
space at the present time. The direction accepted by
many astronomers is, therefore, right ascension 277°,
declination 34*9° N., being the results arrived at by
Prof. Porter from the movements of stars having a
proper motion exceeding 120" per century.

Latterly, however, a further complication has
emerged.

Professor Kapteyn, of Groningen, on going into the

214 THE MOVEMENTS OF THE SUN

subject of stellar motion with special reference to the
movement of the Sun in space, found that the apparent
proper motions of the stars show drifts in two directions
and not in one only, although the latter might reason-
ably be expected were the stellar motion to any great
extent only an apparent one, resulting from the actual
motion of the solar system. Professor Kapteyn's con-
clusions were arrived at from an examination of no less
than 2500 stars whose positions had been recorded
about 1755 by Bradley, then Astronomer Royal. These
conclusions are confirmed by Mr. Eddington in a paper
contributed to the Monthly Notices of the Royal
Astronomical Society, from an examination of the
proper motions of over 4000 stars within 52° of the
north pole. The positions of the stars on which Mr.
Eddington based his calculations had been noted early
last century by Groombridge, a private astronomer of
eminence, who, from observations made at his own
observatory at Blackheath, compiled a catalogue of
4243 stars, the catalogue being published in 1838. The
stars utilised by Mr. Eddington were re-observed at
Greenwich about 1890,

Subsequently Professor Dyson, Astronomer-Royal
for Scotland, investigated the same matter. He confined
his attention to stars with large proper motions the
limits being from 20" to 80" a century. The number of
stars dealt with was 1100, and these were well dis-
tributed over the whole sky. He arrived at the same
conclusions as had previously been come to by Professor
Kapteyn and Mr. Eddington, viz. : — that there are two
well-marked streams of stars moving in difierent
directions.

AND OF THE EARTH IN SPACE. 215

It is very interesting to compare the findings arrived
at by these distinguished astronomers as to the direc-
tions of these two stellar streams. The following is a
comparative statement : —

Apex of Stream I. Apex of Stream II.

Right Ascension. Declin. Right Ascension. Declin.

Prof. Kapteyn ... 85° 11° S. 260° 48° S.

Mr. Eddington ... 90° 19° „ 292° 58° „

Prof. Dyson ... 94° 7° „ 240° 74° „

Mean ... 89-7-' 12-3° S. 264° 60° S.

The close approach to agreement in the results is
wonderful considering the difficulties involved, and this
in itself is strong proof of the approximate accuracy of
the general conclusions.

The result of the recent investigations has been to
suggest a doubt as to the conclusions previously come
to in regard to the solar apex. At the same time, it is
clear that the actual occurrence of such star drifts is a
further and very convincing argument in favour of the
belief that the Sun and its attendant planets are also
moving rapidly through space. It is scarcely conceiv-
able that in this respect the Star which happens to be
the centre of our system should differ from the otlier
stars. On the contrary, the more careful and prolonged
the observation of the stars the more confident do we
become that every star is moving rapidly in its
unknown course, that in the whole universe there is
really no such thing as a " fixed star."

It is interesting to observe that the directions of the
two stellar streams — if we disregard declination, which
is clearly of subordinate importance — are almost
diametrically opposite to each other. Taking the mean

216 THE MOVEMENTS OF THE SUN

of the three calculations it appears that one stream is
flowing towards right ascension 897°, while the other is
flowing towards right ascension 264°. This evidently
suggests exactly converse courses — say 90° and 270°.

Another remarkable fact is that the apex of the
second of these stellar streams closely approximates to
the position fixed upon as that of the solar apex. Thus
if the more recent discoveries tend in one way to throw
doubt upon the determination of the solar apex as being
based to some extent on the observation of a stellar
movement in one direction only, they seem in themselves
to suggest that if the solar apex is not in the direction
indicated by the earlier investigators, it must be in
exactly the opposite direction. If there is a well defined
stellar drift towards right ascension 89*7° and also
towards right ascension 264°, as observed from our
system, and if our system also is moving in space, as it
generally agreed, is it not likely that we are sharing in
one or other of these drifts? The conclusion seems
inevitable that the doubt, which has been hinted at
rather than expressed, as to the reliability of the
earlier observations is not warranted by the later dis-
coveries and is not requisite to their acceptance.

Supposing now that in the solar system there were
a great number of planets, readily observable, all moving
around the Sun, as the centre of the system, in the same
general direction but at different distances, with diverse
velocities, and at somewhat varying angles, many
having their orbits within that of the Earth and many
having their orbits outside that of the Earth, would not
these planets present to our view an appearance, as
regards streams moving in opposite directions, not

AND OF THE EARTH IN SPACE. 217

dissimilar to that which is actually presented by the
stars ?

Let us suppose neither Sun nor stars to be visible —
to the effect, as regards the Sun, of observation being
unobscured by its radiance, and, as regards the stars, of
obviating complication. Let us suppose our observa-
tions as to our movement in the terrestrial orbit and the
movements of the other planets in their orbits to be
limited to, say, two weeks. Is it not the case that in
such circumstances we should find that there were two
streams of the planets moving in almost diametrically
opposite directions ? It may be said that even allowing
this to be so we should have all the planets which were
on one side of the centre of the system drifting in one
direction, while those on the other side would all be
drifting in the opposite direction. This, no doubt, is
true, but it is not what, in the given circumstances,
would appear to us to be true. Our standard of
measurement and observation would necessarily be
based on our own position. What, after all, is the
ecliptic but simply an extension of the plane of the
path followed by the Earth in its orbital course, and
what is the celestial equator but simply an extension of
the terrestrial equator? We judge of the movements
of the stars by their change of position in relation to
each other and to these two great circles ; and the latter
depend entirely on the Earth's own position. Thus, in
the circumstances we have imagined, there would be
planets whose slow progress relative to others in the
same region of the sky and in relation to our own
changing position would appear to indicate their proper
motion in the opposite direction to that of their true

218 THE MOVEMENTS OF THE SUN

motion. In this way the slowest moving planets might
appear relatively to have large proper motion though in
an opposite direction. Thus on both sides of the centre
of the system there would be two streams of planets
moving in diverse directions, and each stream would
have its apex, or vanishing point, approximately at the
respective apices of our own line of progress for the
time being — that is to say at the vanishing points of a
tangent to the terrestrial orbit at the position then
occupied by the Earth.

Thus it would seem that the existence of two stellar
drifts in opposite directions is explicable on the hypo-
thesis that a circular or revolving movement is in progress,
in which the solar system is sharing ; and that, in con-
sequence of this state of matters, the apices of the two
stellar streams appear to us towards the respective
terminations of a line indicative of the direction of the
Sun's movement in space at the present time.

It has to be kept in mind, however, that there is no
reason to suppose that all the stars have a place in one
or other of these great stellar streams. This is, of
course, quite improbable.

It may be thought that if this hypothetical stellar
movement of revolution actually exists there must be a
central source of supreme attraction inducing it, and
which retains the stars in their stupendous orbits. It
can scarcely be doubted that there is no material body
capable of exercising such an attraction as would be
requisite for this purpose. Did such a body exist, it
would scarcely have to be sought out and identified. It
would make its existence and its superiority evident.

Such a central source of attraction is, however, not

AND OF THE EARTH IN SPACE. 219

called for to bring about a movement of revolution of
the character suggested. This is more easily explicable
by the members of the stellar stream being held together
simply by mutual attraction. Thus the constituent mem-
bers of the opposite sides of this hypothetical stellar ring
may really be revolving around each other just as if the
ring constituted in itself a system not unlike that of
binary stars. There is no reason to doubt, however,
that very many stars which are sharing in the move-
ment of the stellar streams are, at the same time, as
members of subordinate stellar systems, taking part
in minor revolutions, and that their movement is
thereby disguised. In like manner, in the solar system,
the Earth and the Moon revolve around each other,
although both at the same time are revolving around
the Sun.

The discovery that the stars show drifts in two
directions is one of the most important scientific
achievements of the new century. As Professor Dyson
remarks, " the hypothesis that the stars are moving in
two streams is of a revolutionary character and calls
for further investigation." We have suggested, as
apparently a feasible surmise, that the existence of
these two converse drifts may infer a stellar movement
of revolution common to both drifts. The discovery,
however, is so new and so unlooked for, that scientific
opinion is uncertain. Evidently in this department of
astronomy, continued investigation should prove of
fascinating interest and profitable in discovery.

In any case it is now beyond reasonable doubt that
the Sun with its attendant planets is moving rapidly
onward in its unknown course, and there is fair evidence

220 THE MOVEMENTS OF THE SUN

justifying the belief that its movement is at present in
a direction in right ascension lying between 256° and
289°, these being practically the limits fixed on by
skilled investigators. We may, therefore, reasonably
enough accept, with many of the best authorities, right
ascension 277° as being approximately the direction
of the solar apex in right ascension, and we may leave
the matter of declination — which practically represents
merely the angle of the movement, and the conclusions
in regard to which are far less certain — out of account
as of very subordinate importance and not specially
affecting our subject.

The question now arises as to the velocity of the
Sun's movement in its onward progress. This is, of
course, a subject of much uncertainty, in view of the
difficulties by which its determination is surrounded. It
has, nevertheless, been gone into by Professor Kapteyn
and others. It has been found that there are valid
reasons for believing that the Sun is moving at a speed
of about 19'89 kilometres per second, this velocity being
subject to a correction not exceeding 1'92 kilometres
vnore or less. We may, therefore, for the purposes of
our inquiry, accept 19*89 kilometres per second as being
a fair approximation to the actual speed with which the
Sun is moving onward in space.

A kilometre is '621 of a mile, or, more exactly,
39,37079 inches, so that the solar velocity, according
to British standards, is nearly 12'36 miles per second.
From this it follows that the distance travelled by the
Sun in our solar year is about 390,021,400 miles. The
mean distance of the Earth from the Sun is, as we have
seen, about 92,897,000 miles, so that the Sun's annual

AND OF THE EARTH IN SPACE. 221

journey is practically four times the mean radius of
the Earth's orbit.

We are now, therefore — on the basis of these cal-
culations as to the direction of the solar apex and the
velocity of the Sun's movement — in a position to judge
as to the true character of the Earth's revolution when
regarded as a movement in space.

We know that when the Sun is seen from the Earth in
right ascension 277°, we must, in so far as right ascension
is concerned, be approximately directly "behind" the Sun
as regards the solar movement in space — we must (dis-
regarding declination) be in the region through which
the Sun itself passed shortly before. Six months later,
when we have completed one half of our orbital journey
around the Sun, this state of matters must be exactly
reversed. We are then, as we may express it, in " front "
of the Sun. We are in the region which the Sun in its
onward progress has not yet arrived at, but which it will
attain shortly afterwards. At the stages intermediate
between these positions the Earth is necessarily away
from the Sun's path, its position being on either side of
the Sun.

Let us now endeavour to represent graphically the
character of the movement on the part of the Earth
thus indicated, and let us consider its peculiarities and
seasonal variations.

We find that the movement is represented by a
curve resembling an arc of a large circle followed by a
comparatively small loop. There is a very prolonged
and regular curve during the six months when the
Earth is passing from " behind " the Sun to the " front "
of the Sun, and consequently gradually gaining upon

222 THE MOVEMENTS OF THE SUN

the latter in its onward progress, then a loop and an
intersection as the Earth again returns to its position
" behind " the Sun. The complete movement is certainly
very different from either a circle or an ellipse.

Turning now to the seasonal change of position, we
find a rather interesting state of matters. The Sun is
in right ascension 277° about the 29th of December
annually. Consequently it is practically at the winter
solstice of the northern hemisphere that the Earth
occupies the position "behind" the Sun. It follows
that it is about our summer solstice that we are once
again crossing the Sun's course, this time in " front " of
the Sun. At the equinoxes we are most distant from
the Sun's path, being, at each equinox, at what we may
describe as the " side " of the Sun as it moves onward in
its course. At the March equinox our direction of
movement coincides with the Sun's own, while at the
September equinox the opposite is the case.

According to Sir J. Herschel* the Earth is on the
plane of the Sun's equator on or about the 11th of June
and the 12th of December, so that when the Earth is
cutting across the path of the Sun it is practically on
the plane of the Sun's equator, the change of plane on
the part of the Earth in its orbital movement in the
intervening period — being about seventeen or eighteen
days — being insignificant. It follows that about the
end of December annually the Earth is virtually in the
same region in space as was occupied by the Sun about
three months earlier, and that in the end of June we are
in the same region in space as will be occupied by the
Sun about three months later.

* "Outlines of Astronomy," p. 230.

AND OF THE EARTH IN SPACE. 223

At the intermediate dates, which are approximately
the times of the equinoxes, we are not on the plane of
the Sun's equator, as the latter is inclined to the ecliptic
at an angle of about 7° 15'. From the 12th of December
until about the 10th of March the Earth is moving
"southward" in relation to the Sun, and about the
latter date our plane in relation to the solar surface is
about 7° 15' to the " south " of the Sun's equator. We
then pass "northward" again, regaining the plane of
the solar equator about 11th June and continuing the
*' northward" movement until about the 12th of Septem-
ber when the movement again commences to change.
It follows from these movements that the south pole
of the Sun is included in the visible hemisphere from
12th December to 11th June, while the north pole of
the Sun is presented to the Earth to a correspond-
ing extent during the remaining six months of the
year.

It will be noticed that it is approximately the case
that when the Earth is on the plane of the Sun's
equator, being in June and December, the Sun is most
widely separated from the plane of the Earth's equator ;
and that when the Sun is on the terrestrial equator, at
the equinoxes, the Earth is almost at its greatest
angular distance from the plane of the solar equator.
We may almost say that whenever the south pole of the
Sun comes into the visible hemisphere — which is im-
mediately after the 12th of December — the Earth com-
mences to change its movement and to withdraw its
south pole from the terrestrial hemisphere presented to
the Sun. As the Sun's south pole is increasingly
directed to the Earth, this movement is accelerated.

224 THE MOVEMENTS OF THE SUN

attaining its maximum about the March equinox, or
shortly after the time when the south pole of the Sun is
most fully visible. The rapidity of the movement then
lessens as the south pole of the Sun is withdrawn from
the visible hemisphere, and it terminates at the June
solstice, or shortly after the complete disappearance of
the south pole of the Sun and the commencement of the
visibility of the solar north pole. The movement is the
converse during the ensuing six months.

The terrestrial equator is inclined to the ecliptic at
an angle of 23° 27', and the inclination of the terrestrial
axis resulting therefrom is practically unchanged in
direction during the orbital revolution. Notwithstand-
ing the Earth's change of position in its orbit, the
terrestrial axis remains constantly parallel to itself.
This also is true of the solar axis. Thus the seasons,
and the changed relations of the Sun and the Earth on
which they depend, result not from any physical turning
of the Earth to the Sun or of the Sun to the Earth, but
entirely from the angle which the Earth's orbit makes
with the terrestrial equator.

In following the plane of the ecliptic, the Earth, in
its orbital progress, moves "southward," as we may
express it, from the December solstice to the June
solstice, and this " descent," or " southward " movement
on the part of the Earth exposes the arctic or " upper "
regions of the Earth to the Sun, and brings into our
view the southern regions of the Sun. Again, from
June to December the Earth, in following the plane of
the ecliptic in its orbital course, "ascends" or moves
"northward," and this movement of "ascent" exposes
the antarctic or " lower " regions of the Earth to the Sun,

AND OF THE EARTH IN SPACE. 225

and brings into our view the "upper" or northern
regions of the Sun.

The plane of the ecliptic is subject to a very slight
variation through the actions of the planets, the
variation amounting, it is believed, to about 48" per
century. The effect at present is to lessen the angle
between the ecliptic and the terrestrial equator by
the amount mentioned. It is believed that the devia-
tion to one side and the other is rather less than
1* 21', making the complete oscillation less than twice
that amount.

It is a necessary consequence of the ratio which the
Sun's movement in space is supposed to bear to the
radius of the Earth's orbit — being virtually as 4*2 to 1
— that the distance which the Earth moves in its orbit
in corresponding periods at different seasons is exceed-
ingly variable. A glance at the diagram illustrating
the movement shows this to be so. We judge of the
distance travelled by the Earth in any given time by (1)
the length of the radius of the orbit — that is to say
by the distance between the Earth and the Sun; and (2)
the angular displacement of the Sun. But with the
Sun itself moving in space, these data fail us as proofs
of length of terrestrial movement. The Sun's angular
displacement, if the Sun itself is moving onward as well
as the Earth, necessarily arises from compound causes,
and is affected according as the movement of the Earth
varies in relation to the direction of the solar movement.
The diagram agrees with the known facts as to the
distance of the Sun and the angular displacement of the
Sun, and with the changing direction of the Sun in the
heavens. Yet it makes evident that our daily or
Q

226 THE MOVEMENTS OF THE SUN

monthly orbital progress fluctuates between wide
limits.

We may take it that our distance from the Sun in
the end of December, being about the time when the
Earth is in perihelion, is about 91,355,000 miles ; while
our distance from the Sun at the end of June, being
about the time the Earth is in aphelion, is about
94,439,000 miles. In the intervening period the dis-
tance travelled by the Sun is, we may suppose, about
195,010,700 miles. As the Earth during the period
passes from being "behind" the Sun to being in advance
of the Sun, it is evident that the Earth, must, between
the end of December and the end of June, have changed
its position in space by these three sums added together,
that is to say by no less than about 380,804,700 miles.
This, therefore, is the length of the chord of the arc
described by the Earth during the six months — assuming
that the curve is part of a circle, which for our purpose
is sufficiently near the truth. The angular length of
this arc can be got from the proportion which the
separation of the Sun and the Earth in March — say
92,897,000 miles— bears to half the length of the chord.
The latter is the sine and the former the versed sine of
half the arc. From the trigonometrical ratios of these
parts we find that the angular length of the arc passed
over by the Earth in the given circumstances, between
about 29th December and 29th June, is 104 degrees.
This being so, we find, from the relative ratios of the
parts of a circle, that the distance travelled by the
Earth during these six months is about 438,582,000
miles, being nearly 27*8 miles per second.*

* The working of this calculation is shown in the Appendix.

AND OF THE EARTH IN SPACE. 227

On the other hand, the distance travelled in the
ensuing six months is comparatively small, and at the
end of the period the Earth, from an astronomical point
of view, is almost at the same position in space as it
was at the beginning of the half year. We are assuming
that during the six months the movement of the Sun
is about 195,010,700 miles. At the commencement of
this period the Earth is at the point where its orbit
intersects the Sun's line of progress, and its position is
about 94,439,000 miles in advance of the Sun. At the
end of the period the Earth is again at a point where
its Orbit intersects the Sun's line of progress, but its
position now is about 91,355,000 in rear of the Sun.
Adding together the distance of the Earth from the Sun
on both these occasions we get 185,794,000 miles. This
is only 9,216,700 miles less than the estimated distance
travelled by the Sun in the period, and, in view of the
changed position of the Earth in relation to the Sun at
the end as compared with the commencement of the
half year, this also is approximately the distance
between the Earth's position in space at the beginning
of the half year and its position in space at the end of
the half year. This is, of course, subject to the
divergence in plane between the Sun's equator and the
line of the Sun's progress, in the portion of the Sun's
course lying between the positions of the Earth at the
beginning and the end of the half year.

The great variation in the displacement of the Earth
in space in the two half years is very striking, and it
has necessarily an important bearing on the measure-
ment of stellar distance ; in which the movement of the
Earth in its orbit is employed as a fundamental. Our

228 THE MOVEMENTS OF THE SUN

reasoning, however, is of course dependent on the ratio
between the velocity of the Siin in space and the
velocity of the Earth in its orbit; the former being
taken as practically 4*2 radii of the orbit.

The velocity of the Sun's movement is, however, at
present very uncertain, and the estimated velocity of the
Sun TTiay be utterly erroneous. Sir William Herschel,
who appears to have been the first to attempt in any
way to form an idea as to the velocity of the Sun,
expressed the opinion that we might, generally speaking,
estimate that the solar motion could certainly not be
less than that of the Earth in its annual orbit, being,
say, 3'1416 diameters of the orbit or about 583,690,000
miles, annually. His son. Sir John Herschel, concluded,
from calculations made by other astronomers, that the
Sun, was moving with a velocity of 1'623 radii of the
Earth's orbit yearly, being — according to his own calcu-
lation — about 154,185,000 miles per annum. Professor
Kapteyn, at one time, estimated the velocity of the Sun
to be 16 kilometres, or rather under 10 miles per second,
being about 313,743,000 miles per annum, or very slightly
less than 3*4 radii of the Earth's orbit. As we have seen
the present estimate is rather more than 4 radii of the
Earth's orbit. The estimates, therefore, have varied
considerably, but the variation is not so great as might
be expected in such an abstruse matter.

It might very naturally be anticipated that the
character of the Earth's orbital movement, with the
Sun progressing rapidly through space, would be similar
to the movement of the Moon in its monthly journey
around the moving Earth. The relations between the
Moon and the Earth, are evidently analogous to the

AND OF THE EARTH IN SPACE. 229

relations between the Earth and the Sun. Yet the
analogy between a planet and its satellite and the Sun
and the Earth fails to give any assistance in the way of
enabling a hypothetical judgment to be formed as to
the probable relation between the velocity of the Sun
in its journey through space and that of the Elarth in
its orbital progress. Were the ratios at all similar in
the case of the Earth and Moon on the one hand and
the Sun and Earth on the other, we might take it that
the Earth's path could be represented by a series of
undulations on either side of the Sun's course. To
allow of this, however, the Sun's progress in the time of
the Elarth's revolution would require to bear a much
higher ratio to the radius of the Earth's orbit than it is
believed actually to bear. The ratio would require to
be not dissimilar to that which the Earth's orbital
progress in a lunar month bears to the radius of the
lunar orbit.

The Moon makes its sidereal revolution in 27d. 7h.
43m., at a mean distance of 238,840 miles. During the
time the Moon is making this revolution the Earth
moves forward in its orbit about 43,661,287 miles, or
nearly 183 radii of the lunar orbit, as contrasted,
according to present belief, with a movement on the
part of the Sun of only 4"2 radii of the terrestrial
orbit during the time of the Earth's revolution. Before
deciding, however, that this discrepancy furnishes
grounds for believing that the accepted velocity of the
Sun is ludicrously inadequate, one must investigate
further as to the relative velocities of the planets in
their orbits, and of their satellites in their revolutions
about their primaries.

230 THE MOVEMENTS OF THE SUN

Jupiter has eight known satellites, no less than three
having been discovered since the twentieth centurj^
began. Let us take the one believed to be nearest to
the planet — that known as number V. The distance of
this satellite from Jupiter is about 111,800 miles, and
its time of revolution is llh. 57m. The mean distance
of Jupiter from the Sun is about 483,288,000 miles,
which makes the length of its orbit about 3,037,800,000
miles. The planet performs its orbital revolution in
4332 days. It follows that during the time of the
satellite's revolution Jupiter advances in its orbital
progress about 349,000 miles, or only about 3 radii bf
the satellite's orbit. This is actually a less ratio than
that ascribed to the Sun in relation to the Earth. This
shows that no opinion can be formed from the analogy
which exists between the Sun and the Earth and a
planet and its satellite, and our knowledge regarding
the orbits and the times of revolution of the planets
and their satellites, as to the probable ratio between the
Sun's progress during the time of the Earth's revolution
and the length of the radius of the Earth's orbit.

Thus, although the determination of the velocity of
the Sun is very uncertain, there appears to be no reason,
in the present state of astronomical knowledge, to dis-
credit the accepted figures; and, if the Sun is really
moving in space with the velocity stated, then the
terrestrial orbit must be as represented.

In any case, whatever be the velocity of the Sun's
movement, it is evident that, with a moving Sun, the
character of the Earth's orbital revolution — when con-
fiddered as a movement in space — cannot be accurately
described as an ellipse. While this is so, it must be

AND OF THE EARTH IN SPACE. 231

admitted that for practical purposes it is most con-
venient to overlook the Sun's movement and to regard
the movements of the Earth and the other planets as
having reference to a Sun of unchanging position.
When attention is restricted to the members of the solar
system only such an attitude has obvious advantages,
and as there is no relative displacement — seeing that
the Sun and the planets go together in this movement
in space — it is to some extent quite justifiable. When,
however, we deal with the members of our system in
their relation to the sidereal universe this position
would seem to be indefensible.

LOGAKITHMS AND THEIR INVENTOR.

STIfOPSIS.

LOGARITHMS AND THEIR INVENTOR.

It is somewhat paradoxical that logarithms, the very
purpose of whose invention was to abbreviate and
simplify complicated and difficult calculations — a pur-
pose which they are so admirably adapted to fulfil —
have themselves come to be regarded by many as almost
a symbol of complication. As, in these papers, loga-
rithms are frequently referred to or made use of in
connection with various computations, it is desirable
that we should give some consideration to their aim and
history. In doing so, we shall, in accordance with our
design, consider the subject chiefly from a simple and
popular standpoint. We shall, therefore, when we come
to deal with the finding of logarithms, confine ourselves
entirely to ordinary arithmetical methods.

It is, indeed, the case that in the early history of'
logarithms the method of calculation was purely arith-
metical, the method adopted being one which, although
somewhat tedious in the working, can readily be carried
out by any schoolboy whose arithmetical studies have
advanced so far as to enable him to work out the square
root of a number.

The closeness of the connection between logarithms
and arithmetic, in the mind of the inventor of loga-
rithms, is evidenced by the name which he gave to his

235

236 LOGARITHMS AND THEIR INVENTOR.

invention. The word logarithm is derived from the two
Greek words — logos, the ratio, the calculation; and
arithmos, a number. The word, therefore, signifies
practically "the calculation of numbers." As the word
arithmetic is itself derived from this Greek word
arithTnos, we may, in fact, take the meaning of the
word logarithms to be simply "arithmetical ratios," or
"arithmetical calculations."

John Napier, the inventor of logarithms, was bom at
Merchiston, near Edinburgh, in 1550, the year in which
the Scottish 'Reformation is considered as having com-
menced. His father, Alexander Napier, was the seventh,
in lineal descent from father to son, to succeed to, or to
become laird of, the estate of Merchiston. The first
Napier who owned the estate — " Alexander Napare " as
he called himself — was provost of Edinburgh in the
reign of King James I. of Scotland. His eldest son not
only succeeded him in the estate but also in the provost-
ship, and this was again repeated in the next generation.
It thus came about that three of the direct ancestors of
the inventor of logarithms held the provostship of
Edinburgh at different times in the fifteenth century,
a very important era in the history of that city. The
father of John Napier was the great-great-grandson of
the third of these provosts.

The grandfather of the inventor of logarithms was
killed at the battle of Pinkie in 1547, and he was suc-
ceeded by his eldest son, Archibald, then a boy of
thirteen. Having come into the estate at such an
early age, and thus being rendered wealthy and in-
dependent, Archibald Napier married about two years
after his father's death — ^being in 1549 — his marriage

LOGARITHMS AND THEIR INVENTOR. 237

being celebrated at what appears to us the extra-
ordinarily early age of fifteen. John Napier, the in-
ventor of logarithms, who was born in the following
year, was the first child of this youthful marriage.

In Napier's early days the Reformation was greatly
exciting the Scottish people, and Napier became in his
teens an enthusiast in the Protestant cause. An
incident of this period is related in an address "To
the Godly and Christian Reader," at the commencement
of a volume by Napier which was published in 1593.
This volume is entitled A plaine Discovery of the whole
Revelation of St. John. The anecdote and the subject-
matter of the book both reveal the placid calculator of
"the ratios of numbers" as a keen religious contro-
versialist. Napier writes : — •

" In my tender years and bairneage (i.e. childhood) in
Sanct-Androis (i.e. St. Andrews) at the Schooles, having,
on the one parte, contracted a loving familiaritie with a
certaine Gentleman, a Papist, and on the other part
being attentive to the sermons of that worthy man of
God, Maister Christopher Goodman, teaching upon the
Apocalyps, I was so mooved in admiration against the
blindness of Papists, that could not most evidently see
their seven-hilled citie Rome, pointed out there so lively '
by Saint John, as the mother of all spiritual whoredome,
that not onely bursted I out in continual reasoning
against my said familiar, but also from thenceforth, I
determined with myselfe (by the assistance of God's
spirit) to employ my study and diligence to search out
the remanent mysteries of that holy Booke; as to this
houre (praised be the Lorde) I have bin doing at al such
times as conveniently I might have occasion."

Napier succeeded his father as proprietor, or "baron,"
of Merchiston in 1608, he being then about fifty-eight

238 LOGARITHMS AND THEIR INVENTOR.

years of age. He had married in 1572 a daughter of
Sir John Stirling of Keir, who survived only until 1579.
A few years later he married again, the name of his
second wife being Agnes Chisholme. By his first mar-
riage he had two children, a son and a daughter ; and
by his second marriage ten children, five sons and five
daughters.

Although frequently referred to as "Lord Napier"
or " Lord of Merchiston," Napier was not really a peer.
Under the old Scottish law landowners who held their
estates immediately of the Crown were termed "barons,"
being described in formal documents as, for instance,
"John Napier, lord of the barony of Merchiston." When
titles of nobility were introduced into Scotland, the
" barons," upon whom such titles were conferred, came
to be known as " the greater barons," while the Crown
vassals not ennobled were distinguished as "the lesser
barons." Both the "lesser" and the "greater" barons
however had seats in the Scottish Parliament until 1427
In that year an Act was passed dispensing with the
attendance of " the lesser barons" on condition of repre-
sentatives being sent from each county, these representa-
tives being called " Commissioners of the Schire." The
" lesser barons " could not sign legal documents by their
surname alone, or by the name of their lands, as a peer
signs his title only. They signed like other commoners,
but — particularly in ancient times — they frequently
added to the signature the name of their lands with the
prefix "baron of " (for instance) Merchiston.

Napier was thus not a peer although he was one of
the " lesser barons." He was, however, the last head of
the family who ranked as a "lesser baron." His eldest

LOGARITHMS AND THEIR INVENTOR. 239

son, Archibald, who succeeded him as proprietor of
Merchiston, was raised to the peerage by King Charles
I. in 1627, with the title of Napier of Merchiston. This
title, which is still held by Napier's descendants, is now
combined with that of Ettrick, in the person of Lord
Napier and Ettrick.

Napier's first literary venture — A plaine Discovery
of the whole Revelation of St. John — is believed to be
the primary original work relating to Biblical inter-
pretation ever published. It is described by a recent
writer as "a serious and laborious work" of much merit.

After the issue of this book Napier appears to have
greatly occupied himself with the invention of secret
instruments of war. In a document dated 7th June
1596, which bears his signature and which is now pre-
served in the Bacon collection at Lambeth Palace, a list
is given of inventions made by Napier for the defence
of the country against a Spanish invasion. The inven-
tions specified consist of (1) a mirror for burning the
ships of the enemy ; (2) a piece of artillery capable of
destroying everything around an arc of a circle ; and (3)
a round metal chariot so constructed that its occupants
can move it rapidly and easily while firing out through
small holes in it. According to Sir Thomas Urquhart,
of Cromarty, the piece of artillery was actually tested
upon a plain in Scotland, and was found to fulfil the
claims of its inventor.

Among Napier's other practical ideas was the sug-
gestion that salt might with benefit be applied to the
land, a suggestion which at the time was looked upon
as chimerical, but which in more recent times has been
put to the test with advantageous results.

240 LOGARITHMS AND THEIR INVENTOR.

Whatever Napier's other claims to fame may have
been — and they are by no means insignificant — they
were all eclipsed by his brilliant invention of logarithms.
So much is this the case, that Napier is now universally
known simply as " the inventor of logarithms."

The volume in which Napier made his immortal
invention public appeared in 1614. In accordance with
the custom of scientists of the day it was written in
Latin. The title is ponderous : —

Mirijlci Logarithmorum Canonis descriptio, Ejusque
fUbsVyS, in utraque, Trigonometria ; ut etiam in omni
Logistica Mathematica, ATnplissimi, Facillimi, & ex-
peditissivii explicatio. Authore ac Inventore, loanne
Nepero, Barone Merchistonii, &c. " The description of
a Wonderful Law of the Ratios of Numbers, and Its
use in both forms of Trigonometry; as also in all
Sexagesimal Mathematics, a Very Full, Easy, and un-
encumbered explanation. Author and Inventor, John
Napier, Baron of Merchiston, &c."

Napier was fortunate in the reception of his great
work. It was everywhere received with admiration by
men of science. In fact it astonished the scientific
world. Kepler, a leading astronomer of the time, voiced
scientific opinion when he said that Napier was the
greatest man of his age in the department to which he
had applied himself.

A copy of the work came into the hands of Henry
Briggs (1556-1630), one of the greatest mathematicians
of the day, who was then professor of geometry at
Gresham College, London. Briggs, in a letter to Usher,
afterwards Archbishop of Armagh, gives expression to
his feelings :—

LOGARITHMS AND THEIR INVENTOR. 241

" Napier, lord of Markinston, hath set my head and
hands at work with his new and admirable logarithms.
I hope to -see him this summer, if it please God, for I
never saw a book which pleased me better, and made me
more wonder."

Briggs visited Napier in 1615 and these two names
are thereafter intimately associated in the development
of logarithms. An interesting account of the first
meeting between these two distinguished men, each of
whom had an extreme admiration for the other, is given
by William Lilly (1602-1681) in his Life and Times
which was published in 1721. Lilly is described as an
Astrologer, but in those days astrology and astronomy
were curiously intermingled. He writes : —

"I will acquaint you with one memorable story related
unto me by Mr John Marr, an excellent mathematician
and geometrician whom I conceive you remember : he
was servant to King James and Charles I. At first
when the Lord Napier, or Marchiston, first made publick
his logarithms, Mr Briggs, then reader of the astronomy
lecture at Gresham College in London, was so surprised
with admiration of them that he could have no quietness
in himself, until he had seen that noble person the Lord
Marchiston, whose only invention (i.e., the invention of
whom alone) they were : he acquaints John Marr here-
with, who went into Scotland before Mr Briggs, purposely
to be there when these two so learned persons should
meet. Mr Briggs appoints a certain day when to meet
at Edinburgh : but failing thereof, the Lord Napier was
doubtful he would not come. It happened one day as
John Marr and the Lord Napier were speaking of Mr
Briggs; *Ah, John,' saith Marchiston, 'Mr Briggs will
not now come ; ' at the very instant one knocks at the
gate ; John Marr hasted down and it proved Mr Briggs
to his great contentment. Ho brings Mr Briggs into
my Lord's chamber, where almost one quarter of an hour

242 LOGARITHMS AND THEIR INVENTOR.

was spent, each beholding the other almost with admira-
tion, before one word was spoke. At last Mr Briggs
began — ' My Lord, I have undertaken this long journey
purposely to see your person, and to know by what
engine of wit or ingenuity you came first to think of this
most excellent help unto Astronomy, viz., the Logarithms;
but, my Lord, being by you found out, I wonder nobody
else found it out before, when now known it is so easy.'
He was nobly entertained by the Lord Napier, and every
summer after that, during the Lord's being alive, this
venerable man, Mr Briggs, went purposely into Scotland
to visit him."

The system of logarithms made public by Napier is
not that now in general use and distinguished as the
" common " system of logarithms. Neither is it that in
more restricted scientific use, although the latter system
is known as the " Napierian." The latter is, however,
very closely related to the system first promulgated by
Napier. Napier's book contains an account of the
nature of logarithms and a table giving the logarithms
for arcs and angles for every minute of the quadrant, to
seven or eight decimal figures. His application of the
system was thus mainly trigonometrical.

The merit of the inventor of logarithms cannot,
however, be restricted to any special system or to any
particular application. It necessarily followed from the
actual invention that the most convenient system would
soon become apparent, and that the application of the
invention to a wider purpose than had originally Ijeen
contemplated would gradually become evident. This
was certainly recognized by Briggs, to whom belongs
the credit of havinor first suggested the formation of a
decimal system of logarithms, a suggestion with which
Napier heartily concurred. This decimal system it is

LOGARITHMS AND THEIR INVENTOR. 243

which forms the base of the " common " form of loga-
rithma

What, now, are logarithms ? Instead of attempting
to define the word let us endeavour to explain practi-
cally. We may suppose, for the sake of clearness, that
the system now in general use was that first made
public by Napier, although, as we have noticed, this
was not really the case. Still, whatever the base of the
system, the underlying principle is one and the same;
and this equally applies whatever the special application
of the system, whether trigonometrical or otherwise.

We may suppose then that Napier in studying
figures was struck with the idea that a system might
be arranged whereby one number might be acceptxjd as
symbolical of another. In algebraic calculations we
accept certain letters — say x and y and z — as symbolical
of numbers. Why should we not do the same with
figures ? If then, we accept figures as symbolical, we
may plan a system whereby numbers of moderate dimen-
sions, and easily handled shall represent the great and
complicated numbers which have to be dealt with in
various scientfic investigations.

Suppose now that we set down 1 as representing 10;'
2 as representing 100; 3 as representing 1000, and so on.
This being done, let us observe what are the relations of
the three higher numbers — 10, 100, and 1000 — to each
other, and what are the corresponding relations to each
other of the three lower numbers — 1, 2, and 3 — which are
accepted as representing the higher numbers respectively.

It is evident at a glance that 10, 100, and 1000 are
the successive "powers" of 10 — that is to say, they are
the successive products of the number when multiplied

244 LOGARITHMS AND THEIR INVENTOR

by itself. They are the first power, the second power
(or square), and the third power (or cube), respectively.
Thus these successive powers of 10, the first, the second,
and the third, are represented respectively by 1, 2, and
3; the cardinal numbers 1, 2, and 3 corresponding ex-
actly with the ordinal numbers indicating the power —
1st, 2nd, and 3rd. We see, therefore, that when numbers
are set down in this peculiar relation to each other the
squaring of the natural number corresponds with the
doubling of the symbolic number; the cubing of the
former with the trebling of the latter, and so on.
Conversely, the halving of the symbolic number corres-
ponds with the extraction of the square root of the
number which it represents, the division of the former
by three with the extraction of the cube root of the
number which it , represents, and so on. This is all
evident by merely studying the simple statement : —

—

10 X 10 (100)

10 X 10 X 10 (1000)

To Napier belongs the honour of having discovered
this peculiar relation of numbers. May we not well
exclaim with Briggs — " I wonder nobody else found it
out before, when now known it is so easy ? "

In the foregoing statement the numbers 1, 2, and
3, are the logarithms of 10, 100, and 1000 respectively
to the " base " 10. This indicates that 10 is the number
whose first, second, and third powers are represented
by the numbers 1, 2, and 3 respectively.

Logarithms may, of course, be formed to any desired
base. Thus if we desired to form a table of logarithms
to the base 5, we should commence by noting 1=5;

LOGARITHMS AND THEIR INVENTOR 245

2 = (5x5) 25; 3 = (5x6x5) 125, &c. Similarly, if the
base selected were 7, we should have, 1 = 7; 2 = (7 x 7) 49;

3 = (7x7x7) 343, &c. The number 10 as a base haa,
however, certain advantages over any other number
as a base, as we shall see later on.

If, now, 1 is accepted as the logarithm of 10, 2 as the
logarithm of 100, 3 as the logarithm of 1000, and the
logarithm thus rises or falls by 1 for every correspond-
ing increase or decrease respectively in the power of 10,
it is clear that the logarithm of 1 must be 0. Similarly
this must be the case whatever the base of the system,
as the same reasoning is applicable to any other base as
we have applied to the base 10. Thus, taking 10 as the
base, we can at once construct the following : —

Natural Numbers 1, 10, 100, 1000, 10,000, 100,000, 1,000,000, &c.
Logarithms thereof 0, 1, 2, 3, 4, 5, 6, &c.

It is clear at a glance that as is the logarithm of 1,
and 1 is the logarithm of 10, the logarithms of all
numbers between 1 and 10 must be more than and
less than 1, that is to say they must be fractional only.
In like manner, the logarithms of all numbers between
10 and 100 must be more than 1 and less than 2, being
1 and a fraction. So the logarithms of all numbers
between 100 and 1000 must be 2 and a fraction; between
1000 and 10,000, 3 and a fraction; between 10,000 and
100,000, 4 and a fraction, and so on. This shows us that
so long as the number of figures in the natural number
is unchanged, the integral part of the logarithm is un-
changed, only the fractional part varying with the
variation in value.

We see, therefore, that all numbers above 1, and less
than 10, have no integral part in their logarithms, but a

246 LOGARITHMS AND THEIR INVENTOR.

fractional part only. If the number is 10, but less than
100, the integral part is 1; if the number is 100, but less
than 1000, the integral part is 2; the integral part in-
creasing by 1 for each figure annexed to the number.
Thus the integral part is invariably one less than the
number of integral figures in the natural number.

We have noticed that we can readily conclude from
the fact that the logarithm of 10 is 1, while that of
100 is 2, and that of 10,000 is 4, that the halving of
the logarithm corresponds with the extraction of the
square root of the number represented. Thus 10 is
the square root of 100, and its logarithm is half the
logarithm of 100. Again 100 is the square root of
10,000, and the logarithm of the former (2) is one-
half the logarithm of the latter, which is 4.

Now we have noted that the logarithm of 1000 is
3, while that of 100,000 is 5, but we have not found
the number whose logarithm is 1|, or the number
whose logarithm is 2|, or, as these logarithms are ex-
pressed decimally, 1"5 and 2'5. It is clear that the
numbers whose logarithms are 1*5 and 2'5 must be
the square roots of 1000 and 10,000 respectively. It
is only in every alternate case in the series that the
square root is itself an exact power of 10.

What then is the square root of 1000, or the square
root of 100,000 ?

As the mode of extraction of the square root of a
number is a matter the recollection of which many
people lay aside with their schoolbooks, and as the
extraction of the square root is almost a fundamental
in connection with logarithms, we may give the means
of extraction in some detail.

LOGARITHMS AND THEIR INVENTOR. 247

To extract the square root of a number we divide
it off from the right into pairs of figures, or, if the
number is partly or wholly a decimal fraction, then
the division of the figures into pairs is made from the
decimal point. We then divide the first, or left-hand
pair of these figures by the nearest perfect square, not
greater in value than the pair of figures thus divided.
The divisor forms, as quotient, the first figure of the
square root. We annex the next pair of figures to
the remainder; and we then double the preceding
quotient, and accept this as the first fraction of the
next divisor. We find, by inspection, how many times
this first portion is contained in the new dividend
excluding the last figure of the latter. We then annex
to the divisor the figure expressing the number of
times it is so contained, at the same time annexing
the same figure to the quotient, and completing the
operation as in ordinary division. The same process
is repeated for every succeeding number of the square
root, and may, if (as is generally the case) the square
root is non-terminating, be prolonged to any desired
extent of decimal places.

In dealing with 1000, we mark off the last two
ciphers, so that the number consists of two pairs of
figures, 10 and 00. The nearest perfect square to 10 is
3. We therefore divide 10 by 3, and mark 3 as the first
figure of the square root. Multiplying 3 by 3 we get 9.
We therefore mark 9 below, and subtract it from the 10,
as in ordinary division, the remainder being 1. We now
annex the two ciphers, converting the 1 into 100. Multi-
plying now the first figure of the quotient (3) by 2, we
get 6 as the first part of the new divisor. We see, by

248 LOGARITHMS AND THEIR INVENTOR.

inspection, that 6 will go into 10 only once. We there-
fore note 1 after the 6 making it 61, and at the same
time we note 1 as the second figure of the square root.
As 61 multiplied by 1 is 61, we mark 61 below, and
subtract it from 100, the remainder being 39. To 39 we
now annex two ciphers, and at the same time mark a
decimal point after the second figure of the square root,
showing that the integral portion of the root is now
ended. Doubling 31, being the first portion of the root,
we get 62 as the first part of the next divisor. This is
contained in 390, six times. We therefore annex 6 to
62, making it 626, and at the same time note 6 as the
next figure (or first decimal figure) of the square root.
Multiplying 626 by 6 we get 3756, which, subtracted
from 3900, leaves 144 as remainder. Adding two ciphers
to this remainder we continue as before.

3) lO'OO (31 0227766

9
61)100
61
626)3900
3 756
6322)14400
12644
63242)175600
126484

632447)4911600
4427129

6324547)48447100
44271829

63245546)417527100
379473276

632455526)3805382400

. 3794733156

10649244

LOGARITHMS AND THEIR INVENTOR. 249

The portion of the root already obtained is always
doubled to obtain the first part of the next divisor. It
comes, of course, to the same thing if the preceding
divisor is accepted as the first portion of the new divisor,
provided that the last figure of the former is doubled,
and, if the product obtained by the doubling of the last
figure amounts to 10 or more, the preceding figure
increased by 1.

We find in this way that the square root of 1000 is
31-6227766. ... As the logarithm of 1000 is 3, and as
the logarithm of the square root of a number is half the
logarithm of the number, it follows that the logarithm
of 31-6227766 ... is 1*5.

We now come to one of the great advantages of 10
as the base in a system of logarithms. We have gone
to some trouble in extracting the square root of 1000.
We now find that in doing so we have, except for the
shifting of the decimal point, also extracted the square
root of 10, of 100,000, of 10,000,000, and of every other
power of 10 having an odd number of ciphers after the
unit. Thus we may make the following series : —

Natural Noa. 10 100 1000 10,000 100,000 1,000,000

Square roots. .316227766 10 31-6227766 100 316-227766 1000

Natural Nob. 10,000,000 100,000,000 1,000,000,000 10,000,000,000, Ac
Square roots. 3162-27766 10,000 31622-7766 100,000, &c.

We find therefore that the logarithm 3-16227766 . . .
(the square root of 10) is '5, being one-half of the loga-
rithm of 10; that the logarithm of 31-6227766 . . . (the
square root of 1000) is 1*5, being one-half of the loga-
rithm of 1000 ; that the logarithm of 316227766 . . . (the
square root of 100,000) is 2*5, being one-half the logarithm

250 LOGARITHMS AND THEIR INVENTOR.

of 100,000 and so on. In fact, we see that as long as the
figures of the number represented are the same, and
unchanged in their order, only the integral part of the
logarithm changes, the fractional part being unaffected.

Thus we establish the two important facts that so
long as the number of integral figures in the natural
number represented remains the same, the integral 'part
of the logarithm is unchanged ; and that so long as the
actual figures of the natural number represented remain
the same, and in the same order, the fractional part of
the logarithm is unchanged.

These results follow from our system of enumeration
being really itself on what may be called a decimal
basis. There can be no doubt that this speciality of our
method of enumeration arises from the fact that the
number of fingers on both hands is ten, so that in the
counting of the fingers it is necessary on reaching ten
to begin over again. The fingers were, in all pro-
bability, the basis of the first series of numbers as,
indeed, to this day, they very often furnish an important
aid to enumeration in early childhood. This is the
theme of the nursery rhyme : —

" When th' Earth was new, and very few
Could count from one to ten,
Men reckoning up their^fingers were
Accounted learned men.

The Earth's now old and growing cold,

But still the custom lingers ;
And most of us do " counting " learn

By telling-off our fingers."

In all probability, if we had commonly six fingers on
each hand instead of five, as occurs in rare instances, our

LOGARITHMS AND THEIR INVENTOR 251

system of enumeration would have been duodecimal
instead of decimal.

It comes about then, from the fact that we have just
five fingers on each hand, that in ordinary numbers
there is, in a measure, a repetition after every ten, and
that in logarithms to the base ten the fractional part of
the logarithm is unaffected by any change in the value
of a number, so long as the number consists of the same
figures, with their order of progression unaltered.

In the early days of logarithms some difficulty must
certainly have been experienced in the maintenance in
relation to fractional numbers of the arrangement that,
80 long as the number represented remained unchanged
in the figures and their order, the fractional part of the
logarithm should be unaltered, only the integral part
varying with the variation in value of the number
represented.

It is, of course, evident that, as the logarithm of 1 is 0,
the logarithm of a fraction such as (for instance) '1 or
•25 or 75, must be a minus quantity. The difficulty
referred to does not arise when the number represented,
although fractional, falls into the natural order of
powers of 10, that is to say when the logarithm consists
merely of an integral part — or, as it is called, an index
or characteristic — although in this case such integral
part must necessarily be a minus quantity. Thus just
as the logarithm of 10 is 1 and the logarithm of 1 is 0,
so the logarithm of '1 is —1, and the logarithm of "01
is —2. In such a case there is no difficulty. The
difficulty occurs when the logarithm has a fractional
part. Thus the logarithm of 200 is 2-30103; the
logarithm of 20 is 1 30103, and the logarithm of 2 is

252 LOGARITHMS AND THEIR INVENTOR.

•30103. It is evident that as the number is divided by
10, the logarithm is lessened by 1. Clearly then the
logarithm of '2 (being 2 divided by 10) must be - -69897,
as that is the number obtained if we subtract 1 from
•30103. Similarly the logarithm of "02 must be
— 1*69897, that being the number which is one less
than —'69897. The series might be continued in-
definitely. It is obvious that in every case the loga-
rithm of the fractional number would differ from the
logarithm of the number consisting of the same figures
(in unchanged order) as an integral quantity. The
fractional part of the logarithm of the former would, in
fact, be the complement of the corresponding part of
the logarithm of the latter — that is to say it would be
the difference between it and 1. Although we have
dealt only with the logarithm of 2, as increased and
diminished by the powers of 10, it is clear that our
remarks apply equally to any other number in its
integral and fractional relations respectively. Such a
difference between the logarithms of integral numbers
and fractions would certainly have impaired the com-
pleteness of a system of logarithms to the base of 10,
and would have militated against its general acceptance.
This difficulty was met by a most ingenious and
beautiful device which forms the crowning feature in
the perfecting of Napier's grand invention.

The solution was found by deciding that when the
number represented was fractional the index of the
logarithm should become negative while the fractional
part of the logarithm still remained positive. The
simplicity and absolute novelty of this arrangement are
most admirable.

LOGARITHMS AND THEIR INVENTOR. 253

We have noticed that the integral part of the loga-
rithm is known as the index or characteristic. The
fractional part is named the mantissa, which is the
Latin word for an addition or an increment — for some-
thing added to something else. Thus the mantissa of
the logarithm is the part which is axided or appended to
the index or characteristic.

The distinction between the negative index and the
positive mantissa in the logarithm of a fraction is in-
dicated by the minus sign being placed over the index,
instead of being prefixed as in ordinary negative numbers.

We can now, therefore, trace out a descending series
of numbers and show the corresponding logarithms,
just as we have done with an ascending series.

Natural numbers 1 -1 '01 -001 -0001 -00001
Logarithms thereof I 2 3 4 5

It will be noticed that if we reckon the decimal
point as being itself a figure — and include the first
effective figure, being, in the present series, the figure 1 —
the series follows exactly the same rule as in the ascend-
ing series, the negative index of the logarithm being,
like the positive index, one less than the figures in the
number represented. Thus the number 1 with its loga-
rithm forms the connecting link in the ascending and
descending series : —

Ifatnntl No*. Logarittinit.

^ 10000 = 4

I 1000 = 3

I 100 = «

I 10 = 1

la 1 = ?

I l = 1

I o^

■001 = 2

•0001 = I

254 LOGARITHMS AND THEIR INVENTOR.

Taking now the square root of 1000, as already
worked out, we can follow its logarithm as the number
lessens decimally. The square root is, as we have seen,
31-6227766 ... the logarithm of which is 1*5. We see
now that the logarithm changes as follows : —

Number 31-6227766.. 3-16227766.. -316227766.. -0316227766..
Logarith. 1-5 -5 1-5 2'5

Number -00316227766.. 000316227766.. -0000316227766..
Logarithm 3-5 4-6 6-5

The working out of the logarithms of the numbers
coming between exact powers of ten is a somewhat
tedious process whatever be the method by which it is
done. For many years after their invention logarithms
were, as has been mentioned, calculated arithmetically,
but latterly they have been calculated chiefly by
algebraic and other methods. It is said that in the
construction of a table of logarithms the arithmetical
method is the better, but that for the calculation of
any individual logarithm other methods have rather
the advantage. We shall, as already stated, confine
ourselves to arithmetical methods.

The arithmetical calculation of the logarithm of any
iriven number is based on the workinof out of means, or

to o '

averages, from two known logarithms and the numbers
which they represent. Thus we know that 3 is the
logarithm of 1000, and that 4 is the logarithm of 10,000;
and we know that as (for instance) 657.9 lies between
these numbers its logarithm must lie between these
logarithms. The diflSculty arises in the determination
of its exact position logarithmically between the
numbers bounding it, whose logarithms are known.

LOGARITHMS AND THEIR INVENTOR. 255

In order to obtain the logarithm of any given
number whose logarithm cannot at once be found (the
logarithm of any number being, of course, very easily
found, if the number is clearly a definite power or
root of a number whose logarithm is known), we have to
proceed by taking the arithmetical mean of the bounding
logarithms and what we may call the logarithmic mean
of the bounding numbers, the latter being the square
root of the product of the two numbers. By repeating
this process as often as may be requisite we gradually
narrow the bounds of the number whose logarithm is
desired, until at last we are able to determine the
logarithm by common proportion.

Taking the number just mentioned, 6579, let us
endeavour to find its logarithm by this method. As the
number lies between 1000 and 10,000 its logarithm
evidently, as we have seen, lies between the logarithms
of these numbers, which are respectively 3 and 4. The
logarithm must, therefore, be 3 and a fraction.

Working on the averages of the numbers and of
their logarithms we have first to get the logarithmic
mean of 1000 and 10,000. This is the square root of
the product of those numbers. The product is 10,000,000-
and its square root is 3162"27766 . . ., which consequently
is the number represented by the logarithm 35, being
the mean of the logarithms 3 and 4.

We notice that the number I whose logarithm is
desired (6579) now lies between 3162-27766.., and
10,000, and that its logarithm is, therefore, between
3'5 and 4. We have thus to find the logarithmic mean
of these numbers (316227766 . . . and 10,000), and the
arithmetical mean of these logarithms. We continue hi

256 LOGARITHMS AND THEIR INVENTOR.

this manner, always enclosing the given number in
closer and closer limits, until at last we work up to very
narrow bounds, when we can proceed by proportion.

The following are the steps which by this process we
have to follow to find the logarithm of 6579 to seven
decimal places : —

Bounding Kos.

fiooo

\ 10,000

/3162-27766 . .
\ 10,000

/5623-4132 . . .
\ 10,000

/5623-4132 . . .
\7498-9421 . . .

_ /6493-8163 . . .
• \ 7498-9421 . . .

f, /6493-8163 . . .
^- \6978-3058 . . .

^ /6493-8163 . . .
'• (6731-7038...

f. /6493-8163...
^- \661 1-6902 . . .

„ /6552-4882 . . .
''•teei 1-6902...

/6552-4882 . . .
\6582-0226 . . .

/6567-2388 . . .
\6682-0226 . . .
/6574-6265 . . .
\6582-0226 . . .

f 6578-3235 . . .
\6582-0226 . . .

f 6578-3235 . . .
16580-1728 . . .

10.

11.

12.

13.

14.

Logarithmic

mean, or square

root of product, of

bounding Nos.

r^ogarithm of
bounding Kos.

3162-2'

■766... {3

5623-4132... (f^

7498-9421

6493-8163

/3-75

\3-875

6978-3058

6731-7038

6611-6902

6552-4882

6582-0226

6567-2388

6574-6265

6578-3235

6580-1728

6579-2481

■{I
■ {tl

■ 13-^

r3j

• 13-6

Logarithm of
mean.

/3-8125
(3-875

f3-8125
\3-84375

f3-8125
\ 3-828 125

g r3-8125

•'••• \3-8203125

f3-8l640625

• \3-8203125

/3-81640625

• \ 3 -8 18359375

3-8173828125
3-818359375

81787109375
818359375

818115234375
818359375

818115234375
8182373046875

}:,.5

1 3-75
1 3-875

}

1
/

3-8125

3-84375

828125
[^3-8203125
|3-81640625
1 3-818359375
13-8173828125
|3-81787109375
13-818115234375

3-8182373046875
3-81817626953125

We now proceed by proportion on the basis of the
twelfth and fourteenth steps. The former gives us the

LOGARITHMS AND THEIU INVENTOR. 257

logarithm of 6578"3235 . . ., which is slightly less
than 6579, and the latter gives us the logarithm of
6579*2481 . . . which is slightly more than the given
number. The difference between these two numbers,
whose logarithms we have found (6578-3235 . . . and
6579-2481 . . .) is '9246 and the difference, to seven
decimal places, between their logarithms is -0000610.
As the difference between the numbers (-9246) is to the
difference (-2481) between the given number (6579) and
the number approaching it most closely (65792481), so
is '0000610 to the sum to be subtracted from the
logarithm of 65792481 in order to convert it into the
logarithm of 6579. The calculation is sufficiently
detailed to secure accuracy to seven decimal places.
•9246 : -2481 :: '0000610 : -0000164.

The fourth proportional is '0000164 which, con-
sequently, is the sum to be subtracted from the
logarithm of 65792481 (being 3'8181763) to change it
into that of 6579. The required logarithm is therefore
3-8181599.

It is very clear that this process, although not
difficult, is somewhat prolonged and rather wearisome,
yet it is really more abbreviated than that generally
adopted in determining definitely the logarithms of
important numbers in the early days of the invention.
Thus Briggs, in working out the logarithm of 5, went
through no less than twenty-two sepai^ate processes
exactly similar to the fourteen we have gone through in
determining the logarithm of 6579.

It is possible, however, in some cases to abbreviate
the double process of multiplying and extracting the
square root — or determining what we have called the

258 LOGARITHMS AND THEIR INVENTOR.

logarithmic mean. Thus the following rules might be
accepted as conveniently applicable in certain cases : —

1. The logarithmic mean of any two numbers is the
square root of one of the numbers multiplied by the
square root of the other. Example : — To get the loga-
rithmic mean of 81 and 64 we might multiply these two
numbers together and extract the square root of the
product. It is evident, however, by mere inspection
that the square root of 81 is 9, and that that of 64 is 8.
Multipljdng 9 by 8 we get 72, which is the logarithmic
mean, or the square root of the product.

2. The logarithmic mean of any two numbers may also
be found by dividing the lower number into the higher,
extracting the square root of the quotient and multiplying
it by the lower number. Example : — Find the loga-
rithmic mean of 4 and 256. Dividing 256 by 4 we get
64, the square root of which is 8. Multiplying 8 by 4 we
get 32, "which is the required mean. This really infers
that the logarithm of 32 is midway between — that is to
say is the mean of — that of 4 and 256. The logarithm
of 4 is -6020600, while that of 256 is 2-4082400. Adding
these logarithms we get 3*0103000 and dividing by 2
(to obtain the mean) we get 1-5051500 which is the
logarithm of 32.

3. If one number be divided into another, one half of
the quotient — plus one — is to the square root of the
quotient as the arithmetical mean is to the logarithmic
mean. This gives the relation between the arithmetical
and the logarithmic means of any two numbers.
Example: — Given numbers, 90 and 10. Dividing 10
into 90 we get 9 as quotient. Adding 1 and then
dividing by 2 we get 5. The square root of the quotient
(9) is 3, while the mean of the two numbers (90 and 10)
is 50. Thus 5 is to 3 as 50 is to the logarithmic mean.
The latter is therefore 30.

As we have seen the addition of logarithms

LOGARITHMS AND THEIR INVENTOR. 259

corresponds to the mvltiplication of the numbers
represented, that is to say, by adding the logarithms of
any two numbers we get the logarithm of the product
of the numbers represented. Similarly the subtraction
of logarithms corresponds with the divisioTi of the
numbers represented. Thus, if we subtract the loga-
rithm of 50 (1-69897) from that of 100 (200000) we get
the logarithm of 2 ('30103) being the quotient obtained
if 100 is divided by 50.

Again, if we divide the logarithm of any number by
2 we get the logarithm of the square (or second) root of
the number ; if we divide the logarithm by 3 we get the
logarithm of the cube (or third) root; if by 4 we get
that of the fourth root, and so on. Thus, by the aid of
logarithms, the extraction of roots of numbers is ex-
tremely simplified. Conversely, if we multiply the
logarithm of a number by 2, we get the logarithm of
the square (or second power) of the number; if we
multiply the logarithm by 3 we get the logarithm of
the cube (or third power) of the number ; if by 4 the
logarithm of the fourth power, and so on.

Supposing now that instead of squaring a number,
and thus doubling its logarithm, we multiply a number
by its square root, how is the logarithm affected ? It
requires some thought to realise that in this case the
logarithm is increased by one-hxdf. Thus if we multiply
100 by the square root, which is 10, we get 1000, and
the logarithm of 100, which is 2, is increased by one-
half, and becomes 3. The same argument necessarily
from the principle of logarithms, applies to other similar
ratios of change, whether of an increasing or decreasing
character. We may therefore conclude ae follows : —

260 LOGARITHMS AND THEIR INVENTOR.

By multiplying a number by : —

(1) Its 2nd root, we add ^ to its logarithm.

(2) „ 3rd „ „ I „

(3) „ 4th „

>j 4 >»

»f

(4) „ 5th „

(5) „ 6th „

&c.

(fee.

By dividing a number by : —

(1) Its 2nd root,

we subtract ^

from its logarithm

(2) „ 3rd „

U »l

(3) „ 4th „

»1 l»

(4) „ 5th „

II II

(5) „ 6th „

>» Jf

1! II

&c.

&C.

It is evident that these facts lend themselves to the
calculation of any desired logarithm. The chief difficulty
arises in connection with the determination of the
fractional part between any two exact roots.

To put the matter to the test let us suppose that we
desire to find by this means the logarithm of 5. It is
convenient to treat the number as 50 and to correct the
index of the logarithm subsequently. Dividing 50 into
100 we get 2. We have to find what relation, as a root,
2 bears to 100. We therefore multipy 2 by itself until
it amounts to 100 noting each power of the number 2,
which is thus obtained : —

2 = 1st power.

4 (2x2) = 2nd

8 (4x2) = 3rd

16 (8x2) = 4th

32 (16 X 2) = 5th

64 (32 X 2) = 6th

128(64x2) = 7th

LOGARITHMS AND THEIR INVENTOR. 261

We find that 2 lies between the sixth and seventh
root of 100, and we conclude from this that, in order to
convert the logarithm of 100 (which we know to be
2000000) into that of 50, we have to subtract from it
between one-sixth and one-seventh. Can we now ascer-
tain the precise point between one-sixth and one-seventh
at which the amount to be subtracted actually lies ?

We may obtain this information by successive
determinations of the logarithmic mean of the numbers
bounding 100, these numbers being, to commence with
64 and 128 — being respectively the sixth and seventh
powers of 2.

If we divide 64 into 128 the quotient is 2, and if we
extract the square root of 2 and then multiply the
square root by 64 we shall get the logarithmic mean
of 64 and 128. By then extracting the square root
of the squAire root already found (this new root being
the fourth root of 2) and multiplying this new root by
the lower of the two numbers whose mean is then
desired, we shall obtain the second mean. The same
process can be repeated again and again until we secure
two numbers bounding 100 very closely on each side.
We can then, by proportion, fix the ratio which may be •
accepted as applicable to 100.

In working by this method the best course is to
extract all the requisite roots to begin with. Let us
assume that the roots we shall require, to enable us to
determine the desired logarithm to seven decimal places,
are eight in number. We shall accordingly extract
eight square roots, the first being the square root of 2,
the second the square root of the root first found, the
third the square root of the root second found, and so

262 LOGARITHMS AND THEIR INVENTOR.

on. Each root after the first will therefore be the
square root of the immediately preceding root, while
the first will be the square root of 2, which is the
quotient obtained if 64 (being the lower of the bounding
numbers already known) is divided into 128, (being the
higher of these numbers).

The following are the eight required square roots,
those first extracted being taken to some extreme, so as
to form a proper basis for those at and near the end of
the series : —

(1) 1-4142135623731 (5) 1-021897148654

(2) 1-1892071150027 (6) 1-0108892861

(3) 1-0905077326653 (7) 1-0054299

(4) 1-044273782427 (8) 1-0027113

As the first bounding numbers are 64 and 128, we
multiply the first of the roots by 64. It is, of course,
unnecessary to do this to the extreme of the decimals
given in the root.

In order, however, to ensure subsequent accuracy, it
is advisable to include in the multiplication, say, 10
decimals. We accordingly multiply 1-4142135624 (the
last decimal being increased by 1 in respect that the
following figure is not below 5) by 64. The product,
which is the logarithmic mean of 64 and 128, is
90*5096680, the last three figures of the product being
omitted as unnecessary.

The numbers bounding 100 now become 90*5096680
on the lower side, and 128 on the higher side. We
therefore multiply the second root by 90*5096680, as
the lower of the second pair of bounding numbers, and
we thereby get the logarithmic mean of 90*5096680 and
128. We thus obtain a third pair of bounding numbers

LOGARITHMS AND THEIR INVENTOR. 2G3

and we then multiply the third root (to the extent of,
say, nine decimal figures) by the lower number of this
pair. This process is repeated until the given roots are
exhausted.

The following are the results obtained : —

T r>.,<...itiir»;n Power of 2 Power of 2

Bounding Nos. mean represented by represented

■ bounding Nos. by mean.

(l){j|g ( 90-5096680 |^ } 6-6

(2j|M-609ee80 J ,07-6347412 {^l^ } 6-75

Wll^SS} 9«-">"»^M6;?5 f8-«^«

c=){X;?1^1?h«':»''^™^ {^:g?5 I^o^^^

(6){l^S?!l} 99-"82817 \t^ll^, (6-640625

(«){l^lISh«'04e805. {«|S5h««3'2»

When (with the view of abbreviating the process of
multiplication) decimal figures are omitted from both
the root and the lower of the bounding numbers, prior-
to the multiplication of the one by the other, it is
necessary to exercise care in order to obviate subsequent
error. In general, when decimal numbers are omitted
we increase the last retained figure by 1 if the first
of the omitted figures is 5 or greater than 5; while we
ignore the omitted figures if the first of them is less
than 5. If such a course were adopted in this case
serious error might result. We have to be guided not
by the value of the first of the omitted decimals in the

264 LOGARITHMS AND THEIR INVENTOR.

root and lower bounding number separately, but by the
aggregate value of these decimals in the root and
bounding number together. Thus supposing the root to
be 2574635 and the lower bounding number to be
27*658616 and that we decide to omit the last three
decimal numbers in each case, in only one of the
numbers should we increase the preceding decimal by
one. Although the first of the omitted decimals is in
each instance 6, their aggregate value is only 12 and
this, it is clear, justifies the single increase only. If the
aggregate value of the first of the omitted decimals is
less than 5, in neither the root nor the bounding number
should the preceding figure be altered. If the aggregate
value of the first omitted decimals is 5, but less than 15,
the preceding figure should be increased by one in
either the root or the bounding number, but not in
both. If the aggregate value is 15 or more, then an
increase of the preceding figure by one in both the root
and the bounding number is justified. Usually the
circumstances bring the operation into the second class,
and thus justify an increase of one in the last retained
decimal in either the root or the lower bounding
number. This matter is of importance, as the multi-
plication of one number by the other multiplies a
preliminary error correspondingly, so that what may
previously have been insignificant may become very con-
siderable. Of course greater accuracy may be secured
by inserting an additional decimal, beyond what had
been contemplated, in either the root or the bounding
number, giving effect to the aggregate value of the
required correction, in so far as it does not justify the
alteration of a preceding figure.

LOGARITHMS AND THEIR INVENTOR. 265

Proceeding now by common proportion on the sixth
and eighth steps of the foregoing series, being 99*7762817
and 1000468051 and the powers represented by these
numbers respectively as above (viz. : — 6640625 and
'664453125) we find that the amount which has to be
subtracted from the power represented by 1000468051
(being 664453125) in order to convert it into the power
represented by 100, is '00067585. This makes the
power represented by 100 (in relation to 2) 6'6438554.

We see, therefore, that we have to subtract » „,,o>k.

10 OOn nnO 6438554

(which is, of course, ' ' ) from the logarithm of 100,
in order to convert that logarithm into the logarithm of
50. We ascertain what this fraction represents by
adding seven ciphers to the logarithm of 100 (being 2),
and then dividing it by 66438554. The quotient is
•3010300, and this we subtract from 2 as the logarithm
of 100. The remainder is 1-6989700. Deleting now
the index 1 in order to convert the logarithm from that
of 50 to that of 5, we get the logarithm of 5 to seven
•decimal places as '6989700.

It will be seen that on the whole this method of
finding the logarithm of a number is distinctly less
prolonged than the method previously described, al-
though it also is certainly somewhat tedious.

It is evident that this process like the preceding one
consists essentially in connecting the number whose
logarithm is known with the number whose logarithm
is desired, in accordance with the principles upon which
logarithms are founded.

It is of no importance whether the number whose
logarithm is desired is divisor or dividend. If the
number whose logarithm is known is the greater, then

266 LOGARITHMS AND THEIR INVENTOR.

the conversion is made by subtraction from its logarithm ;
if it is the less, the requisite correction is made by
addition to its logarithm.

We may illustrate very briefly the remark that it is
unimportant whether the number whose logarithm is
known is treated as dividend or as divisor. Let ua
suppose that we know the logarithm of 16, and that
we desire to find therefrom, by the method just described,
the logarithm of 64. We divide 16 into 64 and get 4 as
a quotient. It is evident that 4 has simply to be
squared or raised to the second power in order to equal
16. Thus we see that — 16 being the lower number —
we have simply to add to the logarithm of 16 one-half oi
its value in order to convert it into the logarithm of 64.
Let us suppose now that we are dealing with the same
two numbers, but that the number whose logarithm we
know is 64, and that we wish to find therefrom the
logarithm of 16. We divide 16 into 64 as previously.
We have now to convert the quotient (4) into 64. In
order to do this we have to cube it or raise it to the
third power. We see therefore that — 64 being the
higher number — we have to subtract from the logarithm
of 64 one-third of its value in order to convert it inta
the logarithm of 16.

The relation of the required correction to the power
to which the quotient has to be raised, in order to equal
the number whose logarithm is known, is very clearly
shown when the former is expressed as a vulgar fraction.
The denominator of such a fraction exactly corresponds
with the power to which the quotient has to be raised
provided that the numerator of the fraction is expressed
as 1. Thus, representing the 2nd power, we have a

LOGARITHMS AND THEIR INVENTOR. 267

correction of ^ ; representing the 3rd power, a correction
of \, and so on.

Of course the logarithms of all numbers have long
ago been found with much accuracy and many excellent
tables of logarithms are in existence. The calculation
of logarithms is therefore now-a-days more a matter of
mental training and exercise, than a matter of either
public or private benefit in any other respect.

No method has yet been devised by which any
desired logarithm can be rapidly calculated so as to
secure the advantage of logarithmic aid in the absence
of a lengthy table of logarithms. By the following
brief table, however, the logarithm of any number can
be found to five decimal places with comparatively
little trouble and also the number which is represented
by any given logarithm — the latter being generally
distinguished as the anti-logarithm : —

268

LOGARITHMS AND THEIR INVENTOR

Tables for Finding the Logarithms of Numbers and the Numbers correspond-
ing to Logarithms by simple arithmetical n;iethods.

/ FIRST TABLE.

Column A.

Column B.

Column C.

Column D.

Numbers Between

10 and 20

1023052

19872480

698970000

20 „ 30

354574

14063390

522878700

30 „ 40

179094

10702930

397940000

4d „ 50

107900

8612000

301030000

60 ,, 60

72082

7197310

221848700

66 „ 70

51550

6179170

154902000

70 „ 80

:38692

5412280

096910000

80 „ 90

30106

4814190

045757500

96 „ 100

24094

4334810

SECOND TABLE (Corrections).

10000 +0
9986 -I
9900 -6
9800 -8
9779 -8
9700 -7
9600 -4
9300 +0
9400 +4
9300 +7
9206 +8
9200 +8
9100 +6
9012 +1
9000 +0
8990 - 1
8900 -8
8800 -11
8779 -11
8700 -10
8600 -6
8500 +0
8400 +6
8300 +10
8204 +12
8200 +12
8100 + 9
8009 +1

8000 +0
7994 -1
7900 -12
7800 -IG
7778 -16
7700 -14
7600 -8
7500 +0
7400 +8
7300 +15
7206 +17
7200 +17
7100 +13
7006 +1
7000 +0
6996 -1
6900 -18
6800 -24
6779 -25
6700 -22
6600 -13
6500 +0
6400 +13
6300 +23
6206 +26
6200 +26
6100 +20
6004 +1

6000 +0
5997 - 1
5900 -30
5800 -40
5780 -41
5700 -36
5600 -21
5500 +0
5400 +21
5300 +38
6206 +44
5200 +44
5100 +33
5002 +1
5000 +0
4998 -1
4950 -32
4900 -54
4850 -68
4800 -73
4779 -73
4750 -72
4700 -65
4650 -53
4600 -38
4550 -20
4500 +0
4450 +20

4400 +39
4350 +56
4300 +70
4250 +78
4206 +81
4200 +81
4150 +76
4100 +62
4050 +37
4001 +1
4000 +0
3999 - 2
3950 -67
3900 -113
3850 -142
3800 -154
3780 -155
3750 -152
.3700 -138
3650 -113
3600 -80

3550
3500

-37
+

3450 +43
3400 +84
3350 +121
3300 +150
3250 +169

3204 +176

3200 +175

3150 +165

3100 +135

3050 +81

3001 +2

3000 +0

2999 -4

2975 -96

2950 -179

2925 -248

2900 -305

2875 -350

2850 -384

2825 -406

2800 -418

2779 -421

2775 -421

2750 -414

2725 -399

2700 -376

2675 -346

2650 -310

2625 -268

2600 -221

2575 -170

2550 -116

2525 -59

2501
2500
2499
2475
2450
2425
2400
2375
2350
2325
2300
2275
2250
2225
2200
2175
2150
2125
2100
2075
2050
2025
2001
2000
1999
1975
1950
1925

-2

+ 60

+119

+178

+235

+ 289

+ 339

+385

+ 424

+ 457

+ 481

+ 497

+ 501

+ 494

+ 474

+ 439

+ 389

+ 321

+ 235

+ 128

+ 6

-18

-431

-802

-1119

1900
1875
1850
1825
1800
1777
1775
1750
1725
1700
1675
1650
1625
1600
1575
1550
1525
1501
1500
1499
1475
1450
1425
1400
1375
1350
1325
1300

-1380

-1590

-1748

-1857

-1920

-1939

-1917

-1855

-1756

-1624

-1460

-1268

-1051

-812

-555

-283

-11

+ 11

+ 290

+ 584

+876

+ 1163

+ 1439

+ 1699

+ 1938

+2150

1275
1250
1225
1200
1196
1175
1150
1125
1100
1075
1050
1025
1001
1000

+2329

+ 2478

+2565

+2606

+ 2<)08

+2587

+ 2499

+ 2334

+ 2082

+ 1734

+ 1278

+ 704

+ 31

Sfl'

I ® g a

63 J3 ■— ■""

EXPLANATION OF THE TABLES.

L To Find the Log-arithm of a Given Number.

The logarithm can be found to five decimal places. Tlie nximbers in column A of the First Table may
be considered as units, tens, hundreds, or thousands, &c. If the given number consists of less than four
figures, aflSx a cipher or ciphers making it up to 4 figures. N.B. — In every case where figures are deleted and
the first of these is 6, or greater than 6, the immediately preceding figure should be increased by 1.
Procedure: — (1) Select in column A of the First Table the numbers bounding the given number, and
subtract the given number from the greater of these bounding numbers and note the remainder ; (2) Take
from column B the number directly adjoining the bounding numbers and multiply it by the remainder ; (3)
Delete the last two figures of the product and an additional figure for every numeral more than four in the
given number ; (4) Add the nnmber in column C directly adjoining that taken from column B ; (5) Multiply
again by the remainder first obtained, and tlien repeat the ojieration detailed in No. 3 ; (6) Add the number
appearing in column D o|)posite tlie bounding nuniljers, and accept the sum obtained as a decimal fraction of
9 ignnm— prefixing a cipher or ciphers if necessary, so as to make up 9 figures — and subtract it from 1 ; (7)

LOGARITHMS AND THEIR INVENTOR.

269

16-26

Bemainders

K<-(laee (he number to 6 figures bj the deletion of the final figures and then apply the correction or pro-
fwrtional correction shown or indicated in the Second Table as applicable to the given number, treating any
9xceM orer 4 figures in the given number as a decimal fraction. Then delete the sixth figure. The remaining
I > figures form the decimal part of the required logarithm, the index of which is, of course, one letia than the
integral figures of the given number.

The logarithm of any single numeral, or of a numeral followed merely by a I'HlftP TABU.
cipher or ciphers, such as 3 or 300, 8 or 80,000 can be obtained by simply sub-
tracting the number in column D (opposite the given number as the hightr number
in column A), as a decimal fraction from I.

n. To Find the Number corresponding to a Given Logarithm.

The number can be found to four figures. Procedure (1) Treat the logarithm
* (decimal {v^rtion) as a 7-pIace decimal fraction, and subtract it from 1. (2) Sub-
'■ tract from the remainder the nearest lower number, if any, in column D of the
' Kirat Table as a decimal fraction. In column A, on the same line as the number
t.aken from column D (or if there is no lower number in column D, then on the
l:ue on which no number appears in that column), will be found the bounding
numbers of the required number, the lower of these being the first of the required
tiiiares. (S) In the Third Table, in the column headed "Remainders," find the
luuubera nearest to the final remainder obtained, and on the same lines in the
.oluiuns of the Third Table headed by the bounding numbers will be found the
iiiuubera nearest to the required number, which, if not appearing in the Table,
can readily be obtained by simple proportion. If, on subtracting the number in
ot>lumn I), as detailed in No. 2, there is no remainder, the required number is the
/- 'Ohrr of the bounding numbers in column A.

36-46

EXAMPLES.

Find the Log. of 6Sio7. Bounding Nos., in column A, 60,000 and
70,000. Difi^erence from the higher of those 6743. Multiply 51550 (from
column B) by 6743. Amount 347t)01650. Deleting the last 3 figures, this
is 347602. From column C we get 6179170 which plus 847602 = 6526772.
Multiplying by 6743 we get 44010023596. Deleting the last 3 figures, we
have 44010024, which, plus 154902000 (from column D) = 198912024.
Accepting this number as a decimal fraction and subtracting it from 1
we get -801087976. Deleting the last three figures, we have -801088. The
Second Table gives correction for 6300 as + 23, and for 6400 as + IS.
Proportionally for 6325-7 it is + 20. Adding this number,
(bersfore, we have '801108. Deleting the last figure, we have
*80111 as the decimal part of the log. As the given number
oonsists of 5 figures the index is 4, so that the required log.
i* 4-80111.

Find the number correiponding to Log '9756311. Affix
so as to make up 7 figures. Then 1 - -9756320 = 0243680.
There is no lower number in column D to subtract, so
that the first figure of the required number must, as
shown by column A, be 9. We find by the Third Table
that the number -0243680 lies between the
" Remainders " -020S721 and -0222764, and
that in the column in the Third Table
headed by the same bounding numbers as
we found in column A (90 - 100) these
"Remainders" correspond to the
numbers 9400 and 9500,

By simple
proportion the remainder
0243680 corresponds to the
number 9454, the
required number be-
ing therefore 9454.

8d-9d

96-106

9000
9100
9200
9300
9400
9500
9600
9700
9800
9900
10000

8000
8010
8100
8190
8280
8370
8460
8550
8640
8730
8820
8910
9000

76-80

7000

7040

7111 -i

7120

7200

7280

7360

7440

7520

7600

7680

7760

7840

7920

8000

66-76

6000

6020

6090

6125

G160

6222-2

6230

6.300

6370

6440

(iolO

6580

6()50

6720

6790

6860

6930

7000

66-<

5000

5040

5100

5142?

5160

5220

5250

5280

5333-S

5340

5400

5460

5520

5580

6640

5700

6760

5820

5880

5940

6000

46-66

4000

4050

4100

41,50

4166-61

4200

4250

42a')f

4300

4.350

4375

4400

4444-4

4450

4500

4550

4600

4650

4700

4750

4800

4850

4900

4950

5000

3000

3040

3080

3120

3160

3200

3240

3280

3320

3333-3

3360

3400

3428f

3440

3480

3500

3520.

3555-5

3560

3600

3640

3680

3720

3760

3800

3840

3880

3920

3960

4000

26-36

2000

2010

2040

2070

2100

2180

2160

2190

2220

2250

2280

2310

2340

2370

2400

2430

2460

2490

2500

2520

2550

2571?

2580

2610

2625

2640

2666-6

2670

2700

2730

2760

2790

2820

285C

2880

2910

29-10

2970

3000

1000

1020

1040

1060

1080

1100

1120

1140

1160

1180

1200

1220

1240

1260

1280

1300

1320

1333-3

1340

1.360

1380

1400

1420

1440

1460

1480

1500

1520

1540

1560

1580

1600

1620

1640

1660 ,

1666-6

1680

1700

17141

1720

1740

1750

1760,

1777-/

1780

1800

1820

1840

1860

1880

1900

1920

1940

1960

1980

2000

•3010300
•2924298
•2839967
•2757241
•2676062
•2596373
•2518120
•2441251
•2365720
•2291480
•2218488
•2146702
•2076083
•2006595
•1938200
•1870866
•1804561
•176091.3
•1739252
•1674911
•1611509
•1549020
•1487417
•1426675
•1366771
•1307683
•1249387
•1191864
•1135093
•1079054
•1023729
•0969100
•0915150
, ^0861861
•0809219
-0791813
•0757207
•0705811
•0669467
•0655015
•0604807
•0579920
•0555173
•0511525
•0506100
-04.57575
-0409586
•0362122
•0315171
•0268721
•0222764
•0177288
•0132-283
•0087739
•004.3648
•0000000

270 LOGARITHMS AND THEIR INVENTOR.

Napier, to whose genius we are indebted for log-
arithms, did not very long survive his brilliant invention.
His discovery was made public in 1614, and he died on
the 4th of April 1617. There is some uncertainty as to
his place of burial. This was generally understood to
have been St Giles' Cathedral, Edinburgh, but there is
authority for believing that he was really buried in St
Cuthbert's Church, Edinburgh. In any case the place
of burial is in Edinburgh.

Among the minor improvements in connection with
figures for which we are indebted to Napier, seems to
be the use of the decimal point. The former custom
was apparently the insertion of a bar or line under the
decimal portion of a number, a practice which evidently
could not have been nearly so convenient as the dot
which it is believed Napier originated.

In the year 1617, very shortly before his death,
Napier published a small duodecimo volume of 144
pages in which he gave an account of a method of
performing the operations of multiplication and division
by means of a number of small rods. These materials
for calculation maintained for many years a place in
science, and became known as " Napier's Bones." This
little book is known (from the first word of the title) as
the Rabdologia, It attracted very general attention,
and several editions were published on the Continent,
an Italian translation appearing in 1623, and a Dutch
translation in 1626. There is some reason to suppose
that this book, although published later, was really
written some time before the volume in which Napier
made public his invention of logarithms. The Rabdo-
logia is now merely of antiquarian interest. It is

LOGARITHMS AND THEIR INVENTOR. 271

pointed out in connection with this work by a recent
writer that " Nothing shows more clearly the rude state
of arithmetical knowledge at the beginning of the
seventeenth century than the universal satisfaction with
which Napier's invention was welcomed by all classes,
and regarded as a real aid to calculation."

After Napier's death, his son, Robert Napier, pub-
lished in 1619 an explanation of the mode of construc-
tion of logarithms. The Constructio, as it is called — as
an abbreviation of its lengthy title — was written almost
entirely by Napier himself, and this, as his son points
out, several years before the name "logarithm" had
been invented. In the Gonstinictio logarithms are re-
ferred to as " artificial numbers."

There is good reason for believing that the invention
of logarithms — although their first inception may have
come as "a happy thought" — was really the result of
many years of labour and study carried out with the
set purpose of aiding in the progress of science. Napier
must have been well aware that the advance of science
in many ways, and astronomical progress in particular,
were excessively impeded by the drudgery and labour
involved in the long and irksome calculations which
had to be gone through.

Napier has done more than any other single in-
dividual to assist the continued progress of astronomical
knowledge, by clearing away the difficulties and hind-
rances which formerly obstructed the practical worker
in the solution of the various problems with which he is
confronted.

When Napier published his work on logarithms, no
part of Great Britain had, in that generation, taken any

272 LOGARITHMS AND THEIK INVENTOR.

part whatever in the advance of science. With the
single exception of Napier himself, there is no British
author of the time — Newton being of a later generation
— whose name can be placed in the same rank as those
of Copernicus, Kepler, Galileo, and other great Con-
tinental scientists. Napier, too, was what we may
describe as sui generis. Amongst this brilliant galaxy
of genius he stood alone in his own sphere. "The
invention of logarithms," says Dr. James W. L. Glaisher,
F.R.S,, " has been accorded to Napier with a unanimity
which is rare with regard to important scientific dis-
coveries." Although Briggs is entitled to great credit,
and he and Napier together may be said to have
perfected logarithms, to Napier alone belongs the merit
of this great invention.

Sir George Mackenzie, a distinguished Scottish
jurist of the seventeenth century, when lecturing
upon the laws of his country, used to state that
some customs flourished in one country some in an-
other. Nature allowing no universal excellence, but the
Almighty designing to gratify every country of his
creation. With professional enthusiasm he went on
to suggest that in this way Scotland stood out pre-
eminent for its admirable system of registration of
titles. May we not, with far greater reason, say that
Scotland stands pre-eminent, at least in science, during
the seventeenth century, as the birth-place of the in-
ventor of logarithms.

In Napier's time, before the issue of his great work,
Scotland had no place in science. "It was," says Dr.
Glaisher, " perhaps the last country in Europe from
which a great mathematical discovery would have been

LOGARITHMS AND THEIR INVENTOR. 273

expected." Napier, by his great invention, carried his
country directly to the front rank. Scotland not only
ceased to be a negligible quantity in so far as the
scientific world was concerned, but became prominent
in science, and this simply as being the country of
Napier.

One would naturally expect that the memory of
such a genius as Napier would be specially honoured in
the country of his birth and in the city with which he
and his family were for many generations so closely
and so honourably connected. In the vestibule of St.
Cuthbert's Church in Edinburgh, a tablet has been
placed to the memory of Napier, and his Q^gj appears
on the Keith Medal of the Royal Society of Edinburgh,
but the Scottish capital contains no monument of a
public character to this most illustrious Scotsman.
Amongst the numerous statues set up in "Modern
Athens" to genius in many forms, one will search in
vain for such a tribute to John Napier, of Merchiston.

But perhaps, after all, Napier's best and most lasting
monument is his general and universal recognition as
" the Inventor of Logarithms." May we not truly say
that "his sound has gone into all the Earth and his
words unto the ends of the World ? "

GRAVITATION THE CHIEF CAUSE

OF THE
GENERAL OCEANIC CIRCULATION.

SYJIfOPSIS.

Gravitation in relation to ocean movements
usually associated chiefly with those of a sub-
ordinate character — More important connection
suggested — Figure of the Earth — Gravitation in
its terrestrial aspect — Terrestrial, solar, and lunar
gravity, and centrifugal force — Eflects on the
waters of the ocean — Superficial poleward move-
ment from equatorial regions — Course of the flow
— The gradient from equator to poles — Eastward
drift of poleward movement — Other causes aflect-
ing the flow of the water polewards — Temperature
of ocean in polar regions^ Counter-flow of water
from polar areas to equatorial regions — Solar and
lunar gravity the inducing causes — The influence
of the Moon — The gradient from the poles to the
equator in relation to Moon — Influence of Sun —
Gradient in relation to Sun — Solar and lunar
attraction on the waters the converse of terres-
trial attraction — Movement of water from the
depths in the polar oceans towards the surface in
tropics — Westward deflection, &c. — Balancing of
converse movements — Efiects of Earth's rotation —
Ocean ridges — Temperature — Relative salinity —
Centrifugal force — Ocean temperature and salinity
in tropics — Relative density — ^Diurnal, periodic
and seasonal variations in the movements of the
waters — Ascensional and descensional circulation
— Prevailing winds in relation to ocean circulation
— Relative temperature and salinity contributing
causes to the general movements primarily in-
duced by gravitation.

GRAVITATION THE CHIEF CAUSE
OF THE GENERAL OCEANIC
CIRCULATION.

It does not appear that the effects of gravitation have
ever been specially considered in relation to the general
movements of the waters of the ocean, except, indeed,
in the subordinate character associated with the move-
ments occasioned by the relative specific gravity of
neighbouring bodies of water. Yet, there are reasons
for thinking that in the wider aspect suggested by the
word itself gravitation has a material bearing on the
problem of oceanic circulation.

In considering the causes of the circulation of the
waters of the ocean it is important to keep in mind thcj
figure of the Earth. The Earth is an irregular spheroid,
its polar diameter being about 7,899 miles, and its
equatorial diameter about 7,925^ miles. The polar or
axial diameter is the least, and the equatorial diameter
the greatest, of the diameters of the spheroid ; and the
diameter, generally speaking, lessens with separation
from the equator and approach to either of the poles.
The difference between the polar and equatorial diameters
is about 26J miles, that being about the excess of the

277

278 GRAVITATION THE CHIEF CAUSE OF

mean equatorial over the polar diameter. It follows
that, at the equator, the waters of the ocean are, on the
mean, piled up to a height of about 13| miles more than
they are at the poles ; that is to say the surface of the
ocean at the equator is about 13| miles farther distant
from the centre of the Earth than is the surface at each
of the poles.

It may be noticed also that even in this respect there
is irregularity, as the equator is not by any means a
perfect circle but is somewhat elliptical — the greatest
equatorial diameter exceeding the least by nearly two
miles. Thus in two regions of the equator — being at
the extremities of the longest diameter — the sea surface
must be nearly a mile farther from the centre than
it is at the extremities of the least equatorial diameter.
The greatest equatorial diameter is (according to Sir
Archibald Geikie) in longitude 14° 23' E., and 194° 23'
E. (the latter being 165° 37' W.), whilst the least
equatorial diameter is at right angles thereto.

Setting aside, however, the variation in the diameter
at the equator itself, the mean diameter at the equator
exceeds the polar diameter by about 26^ miles; and,
consequently, the surface at the equator is, on the mean,
about 13] miles more distant from the centre of the
Earth than is the surface at each of the poles.

Gravitation, in its terrestrial aspect, certainly acts
as an attractive power towards the Earth's centre, and
it is equally certain that water always finds its own
level. The explanation of the apparent exception to
these rules which occurs in the case we are considering
is, as is well known, to be found in the Earth's rotatory
motion and the attraction of the Sun and the Moon.

THE GENERAL OCEANIC CIRCULATION. 279

Through the former, we have centrifugal force opposing
terrestrial gravity, while the attractive influence of the
Sun and of the Moon is, through the relative positions
of these bodies and the Elarth, more powerful in the
equatorial regions than in other parts.

It is clear that the elevation of the waters at the
equator and their gradually lessening elevation with
separation from the equator, as compared in both cases
with their polar level, is comparable to a great mountain
extending around the Earth having its summit as a
ridge stretching along the equator wherever the ocean
exists and having its base in the arctic and antarctic
regions. This ocean -elevation has, indeed, many similar-
ities to land elevations although in other respects it is
absolutely dissimilar.

As the elevation of the waters at the equator is
decided by the balancing of terrestrial, solar and lunar
gravity and of centrifugal force, these counteracting
forces must necessarily regulate the precise amount of
the equilibrious elevation of the waters in each particular
equatorial position. The waters of the ocean, however,
are never in a state of rest. They are, through their
excessive mobility — not only as a whole but molecularly;
— in a state of constant agitation. Such agitation may
arise from atmospheric causes, from changes induced by
varying specific gravity, or from the var^nng relations
of the Sun and Moon to the locality. Whatever the
cause or causes may be, the ocean is in a state of con-
stant restlessness and from this it must necessarily come
about that portions of the superficial waters in the
tropics will constantly be thrown up to a slight extent
beyond the height at which centrifugal force and gravity

280 GRAVITATION THE CHIEF CAUSE OF

are exactly balanced. Whenever this occurs it would
seem that terrestrial gravitation must have due effect,
and that a movement of the superficial water will arise
from the equatorial regions in the direction of one or
other pole. The equatorial elevation will in these
circumstances become practically a watershed and a
movement "down-hill" (as we may express it) of the
surface water must originate, the water affected by this
movement being, at its origin, most probably a mere
superficial film.

In this way, then, we can reasonably account for the
beginning at the equatorial regions of a superficial
movement of the waters of the ocean towards each of
the geographical poles. In fact it is difiicult to see how
the concurring circumstances can exist without such a
movement being originated. If the equatorial elevation
of the waters results from the balancing of the diverse
effects of terrestrial and solar and lunar gravity and of
centrifugal force, it would seem evident that, whenever
a temporary elevation of the water occurs beyond the
exact position of equilibrium, terrestrial gravity must
predominate, and a poleward movement of the surface-
water occur — that is, of the water which is temporarily
forced upward beyond the position of equilibrium. In
the unstable condition of the ocean, such a state of
matters must be of constant occurrence, and such a pole-
ward motion of the surface-water is exactly analogous
to the ordinary flow of water on a land surface from a
higher to a lower level.

If, then, a poleward flow of water from the equatorial
regions has its origin in this way, what will be the
course of its progress ?

THE GENERAL OCEANIC CIRCULATION. 281

When water starts to flow from the summit of a
mouatain or hill its destination is necessarily the base.
In the matter of ocean-circulation the surface at the
equator corresponds with the summit of the mountain,
and the base is to be found at each of the poles. It has,
however, to be noticed that it is not at the ocean
surface, at each pole that the base of this " mountain "
exists. It is, on the contrary, at the bottom of the
polar oceans. This is almost self-evident. The summit
of this water " mountain " is the part which is farthest
distant from the Earth's centre, and the base is, of
course, the parts which are nearest to the Earth's centre.
The surface at the equator is the part of the ocean
farthest removed from the centre of the Earth and the
floor of the ocean at the poles respectively is the part
nearest to the centre of the Earth. Thus the tendency
of a surface movement of the waters originating in
equatorial regions, through the influence of terrestrial
gravity, must be towards the bottom of the ocean in the
polar regions.

On the land surface the progress of water from a
higher to a lower level is intimately connected with the
gradient. If the gradient is steep the flow is rapid, if"
the gradient is very slight the flow is languid. Similarly
the poleward flow of water from the equatorial regions,
the origin of which we have considered, must be affected
by the gradient.

What then is the gradient between the surface of
the sea at the equator and the bottom at the poles ?

On this point we can judge only by averages. We
have noticed that, on the mean, the polar diameter of
the Earth is about 26| miles less than the equatorial

282 GRAVITATION THE CHIEF CAUSE OF

diameter. It has been found by polar explorers that
the depth of water in the polar regions is not materially
different from the mean depth in other parts of the
ocean, although at one time it was supposed that the
polar oceans would prove to be exceptionally shallow.
Let us take it that in the neighbourhood of each of the
poles the depth of the ocean is two miles. This would
make the polar diameter of the Earth — from the bottom
of the ocean at the north pole to the bottom of the
ocean at the south pole — about 30^ miles less than the
mean equatorial diameter from surface to surface of the
ocean.

Thus the bottom of the ocean at the pole will be
about 15| miles nearer to the centre of the Earth
than is the surface of the ocean at the equator. The
meridional circumference of the Earth is about 24,816
miles, so that the distance from the equator to either
pole (being one-fourth of the meridional circumference)
is about 6,204 miles. Thus the gradient between the
equator and the pole is about 15 1 miles in about 6,204
miles. This is about 1 in 407 or, say, one inch in
thirty-four feet.

It is evident that a flow of water commencing in the
equatorial regions in a poleward direction cannot take a
direct course. The rotatory velocity induced by the
Earth's diurnal motion is about 25,000 miles a day at
the equator — being more than a thousand miles an
hour — while at the poles it is non-existent; and the
velocity continuously decreases with separation from
the equator and approach to either pole. At the 60th
parallel, for instance, it is about 12,000 miles a day, or
rather less than one-half of what it is at the equator.

THE GENERAL OCEANIC CIRCULATION. 283

As velocity of motion is lost only gradually, water
leaving the equator in a poleward direction must also
have a motion in the direction of the Earth's rotatory
movement of greater velocity than the regions through
which it passes. The Earth's rotation is towards the
east, so that water flowing from the equator northward
will throughout its course — when its natural bias is
not otherwise counteracted — have a north-easterly move-
ment ; while, similarly, water leaving the equator on a
southward course will flow in a south-easterly direction.
Thus the former will, if observed as a current, be
correctly described as a south-westerly current, or a
current flowing from the south-west; while, in like
manner, the latter, in its progress towards the antarctic
regions, will be a north-westerly current.

In our observation of streams on the land-surface
we usually find that streams flowing any considerable
distance receive as they proceed additions to their volume
in the way of tributaries. The circumstances which
decide the course of the primary stream similarly decide
the course of the flow of other waters, which possibly
may have their rise in the regions through which the
original stream is passing. It is to be expected that a
corresponding state of matters will exist in relation to
the flow towards the poles of water from the ocean-
surface at the equator. The circumstances which
originate such a flow will, in like manner, apply in
other parts of the higher latitudes and in the regions
through which in its progress polewards such a flow
must pass. It has also to be kept in mind that gravity
is an increasing force. The flow of a stream downhill,
if unobstructed, and if the gradient is constant, tends to

284 GRAVITATION THE CHIEF CAUSE OF

become more rapid. This ocean-flow must doubtless be
influenced in this way, and thus a flow which in its
origin may be imperceptible may develop into a notice-
able current.

Again, it is clear that, under any circumstances, a
stream or flow of water will take the course of least
resistance. We have seen that the water flowing from
the equatorial regions at the surface is drawn towards
the polar regions at the bottom, its course being de-
flected eastward through the Earth's rotation. There is
no reason, however, to suppose that the flow of the
water will, in the northern hemisphere, be directly north-
eastward from the surface at the equator to the bottom
at the north pole; or that it will pursue an analogous
course in the southern hemisphere. On the contrary,
there can be no question that the course of the flow of
the waters will be influenced by many circumstances.
It may be either obstructed or hastened by atmospheric
currents, or its natural course may by this means be
entirely diverted. It may come in contact with counter
or divergent movements of the waters and its course
may thereby be changed or modified or its movement
may be entirely dissipated.

The natural tendency of the poleward flow, as
influenced solely by the terrestrial gravity which is its
originating cause, is to sink gradually from the surface
to the bottom, as it passes from the equatorial to the
polar regions. This tendency, however, will be aflected
by its state as regards temperature and salinity, in
comparison with the waters through which it passes^
If warmer, it will tend to rise ; if colder, to sink ; if
Salter, it will tend to sink ; if less saline, to rise ; while

THE GENERAL OCEANIC CIRCULAFION. 285

if it differs both in temperature and salinity its position
will be influenced by each of these qualities.

Similarly, its position in the waters and its course
will be aftected by material obstructions to its progress.
Decreasing depths or elevations or ridges in the ocean
floor, may throw the flow upward or divert its course ;
while a long coast-line across the course may stop the flow
entirely, and a coast-line partially obstructive may cause
a moderate flow to develop into a considerable current.
Very probably we have an example of a long coast-line
stopping the flow, in the fact which (according to Dr.
H. R. Mill) was revealed by the survey of India — that
the sea-surface is three hundred feet further away from
the centre of the Earth at the head of the Arabian Sea
than it is at Ceylon ; while, in the Gulf Stream, as it
leaves the Gulf of Mexico, we probably have an illustra-
tion of partial obstruction through the direction of the
coast-line.

We have seen that water flowing from the surface
at the equator towards the ocean floor at the pole by a
direct course would have a gradient of descent of about
1 in 407. As the water cannot follow a direct course,
but must in its poleward journey be constantly deflected
eastward by the rotation of the Earth, the length of its
course must be greatly increased and the gradient
consequently lessened.

Thus there is reason to suppose that through the
influence primarily of terrestrial gravitation there must
be a constant flow of water from the equatorial to the
polar regions, the flow having its origin at the surface
in the former position and its goal being the ocean floor
in the latter position, a goal which its relative warmth.

286 GRAVITATION THE CHIEF CAUSE OF

and consequent relative specific gravity, will militate
against it ever actually reaching.

It has been found by explorers that in the Arctic
Ocean the temperature of the surface vkrater is generally
about 29'2° F. At about 110 fathoms the temperature
suddenly increases to about 33° F. to 33*5° F. ; while
both temperature and salinity range highest at a depth
of between 120 and 350 fathoms. The temperature at
these depths varies from 35° F. to 39*9° F., or, on the
mean, about 8'25° F., warmer than the water at the
surface. Underneath this warm layer there comes a
second cold layer, the middle of which lies at about 500
fathoms; the temperature found by Nansen at that
depth being about 31-9° F. From 1000 fathoms to the
bottom the temperature has been found to be fairly
uniform at from about 331° F. to 33'4° F., being, on the
mean, over four degrees warmer than the water at the
surface.

In the Antarctic Ocean it has been found that a
somewhat analogous state of matters occurs. The
surface temperament down to about 50 fathoms is
generally between 29° F. and 30° F. At 50 fathoms, or
thereabout, the temperature begins to increase, and it
rises until the depth is about 165 fathoms, when it is
found to be about 35° F. This temperature, which is
from five to six degrees warmer than that of the surface,
is maintained down to about 800 or 825 fathoms.
The temperature then again falls gradually to about
31° F., at which it continues fairly constant to the
bottom. These Antarctic soundings were obtained by
the German deep-sea expedition in the Valdivia, in
1898-9.

THE GENERAL OCEANIC CIRCULATION. 287

As regards the variation in the temperature at
different levels in the Arctic Ocean, Dr. Nansen says : —

" Our observation showing that under the cold surface
there was warmer water — sometimes at a temperature as
high as I'O" C. (33 "8° F.) — was surprising. Again, this
water was more briny than the water of the polar basin
has been assumed to be. This warmer and more strongly
saline water must clearly originate from the warmer
current of the Atlantic Ocean (Gulf Stream)." "Farthest
North," XL, 634.

It seems clear that there cannot be a constant flow
of water from the equatorial regions to the polar area
without a counter-flow existing in the opposite direc-
tion. We have suggested that a flow of water pole-
wards from the tropics is chiefly induced by the
influence of terrestrial gravity, and we have seen that
there are valid reasons for such a suggestion. It is
certain, however, that a flow of water in the opposite
direction — that is, from the polar to the equatorial
regions — cannot be induced by this cause. Terrestrial
gravity could never induce water to flow uphill, and a
flow of water from the poles towards the equator would
clearly be an uphill flow. Yet, unless there exists a
return flow towards the tropics, the poleward flow would
result in upsetting the ocean equilibrium, or else in
bringing about its own termination notwithstanding the
poleward attraction.

There are, nevertheless, reasons for believing that
just as there exists a constant flow of water from the
equator polewards, so there exists a constant flow of
water from the poles towards the equator; and that,
like the poleward flow, the flow towards the tropics

288 GRAVITATION THE CHIEF CAUSE OF

may be due chiefly to the action of gravity. In the
latter case, however, the source of attraction is not the
Earth but the Sun and the Moon.

At first sight it seems incredible that the Sun and
the Moon can, at the Earth's surface, exercise a gravita-
tional influence on the circulation of the waters of the
ocean at all comparable with that exercised by the
Earth itself. Let us see how it can possibly come
about that, chiefly through their influence, a flow of
water is induced from the poles towards the equator
compensating the flow in the opposite direction and
preserving the equilibrium of the waters of the ocean.

We shall confine our attention in the first place to
the Moon, although there is reason to suppose that its
influence is insignificant as compared with the influence
of the Sun. The mass of the Moon is, of course, very
small compared with the mass of the Earth. If we
take the figure 1 to represent the mass of the Earth,
the mass of the Moon will be represented by "01228.
At the surface of the Earth at the equator a body falls
from a state of rest about 16'095 feet in a second ; while
at the surface of the Moon it falls only 2Q5 feet.* The
semi-diameter or radius of the Moon is about 1,081
miles, and the mean distance of the Moon from the
Earth is about 238,840 miles. As the force of gravity
varies inversely as the square of the distance, and as
the Moon's radius is contained 221 times in the mean
distance which separates the Moon from the Earth, the
force exerted by the Moon at the mean distance of the
Earth will be to that exerted at the lunar surface as 1

* The force of gravity on the surface of a sphere is found by dividing
the mass by the square of the radius.

THE GENERAL OCEANIC CIRCULATION. 289

is to 48,841, the latter being the square of 221. Thus,
at the mean distance of the Earth, the force exercised
by the Moon would be sufficient to make an unsupported
body fall towards the Moon to the extent of '00065 of
an inch in one second of time, the attraction being thus
almost inappreciable in comparison with that of the
Earth.

We have now to notice a very remarkable fact. We
have seen that the gradient of the poleward flow of
water is very slight. We have found that the gradient
on the direct course would be about 1 in 407. We find,
on the other hand, that the gradient of a flow of water
moving under the influence of lunar attraction from the
polar regions towards the tropics must be exceedingly
great. There is, indeed, reason to think that the excessive
gradient makes up to some extent for the exceedingly
weak attraction.

The mean distance of the Moon from the Earth is,
as we have just noticed, about 238,840 miles. This is
from centre to centre. The equatorial radius of the
Earth is about 3,963 miles, so that the surface at the
equator, when turned towards the Moon, is nearer to
the Moon by 3,963 miles than is the centre of the Earth;,
its distance consequently being 234,877 miles. The
geographical poles, on the other hand, are — at any rate
when the Moon is above the equator — slightly farther
away from the Moon than is the centre of the Earth.
Taking the axial radius of the Earth to be 3949*5 miles,
it can readily be found that, under the circumstances
specified, the distance of the geographical poles from
the Moon is about 238,873 miles. Thus the poles are
about 3996 miles farther away from the Moon than is

290

GRAVITATION THE CHIEF CAUSE OF

the surface at the equator when the latter is turned
towards the Moon — 3996 miles being the difference
between 238,873 miles and 234,877 miles. Thus, we
may take it that water passing from the pole to the
equator will thereby approach nearer to the Moon, or,
otherwise, will fall towards the Moon, to the extent of

A B =238,840 miles.
C B= 3,963 „
AC=AB-CB=234,877 miles.

AN =238,873 miles.
AN2=AB'^+NB2
NB =3949-5 miles.

Diagram illustrating method of finding the difference between the
Moon's mean distance from (1) the geographical poles and (2) the
geographical position at the equator turned towards the Moon for the
time being.

about 3996 miles ; and it will do so in passing over a dis-
tance on the Earth's surface (taking the course to be a
direct one) of about 6204 miles. The gradient on the
direct course, is, therefore, 3,996 miles in about 6204
miles, or about 1 in 1*553 — say, practically, 1 in 1^.

Coming now to the Sun. The mass of the Sun is to
that of the Earth as about 331,000 is to 1. At the Sun's
surface a body would, from a state of rest, fall in one
second about 444 feet; and — taking the mean solar
diameter to about 866,400 miles — the surface of the Sun is
about 433,200 miles distant from the centre. The mean
distance of the Sun from the Earth (centre to centre) is
about 92,897,000 miles. J3ividing the radius of the
solar orb (433,200 miles) into the Sun's mean distance

THE GENERAL OCEANIC CIRCULATION. 291

from the Earth (92,897,000 miles) and squaring the
result (214*44) we find that the force of gravity exerted
by the Sun at the centre of the Earth is about l/45,985th
part of that which the Sun exerts at the solar surface.
Dividing, now, 444 feet by 45,985 we find that, at the
distance of the centre of the Earth, the Sun exerts
sufficient attraction to cause an unsupported body, from
a state of rest, to fall towards it with the velocity of
about '11586 of an inch in a second of time. In relation
to the Sun the geographical poles may be considered as
at the distance of the centre of the Earth. As the
corresponding force exerted by the Moon is '00065 of an
inch, it would appear that the force exerted by the Sun
is more than 178 times as great as that exerted by the
Moon.

The gradient in relation to water drawn from the
polar to the equatorial regions by the influence of the
Sun is practically the same as we have found to apply
to the corresponding attractive power of the Moon.
Water passing from the pole to the equator will —
assuming the Sun to be in the zenith at the part of
the equator towards which the water is drawn — pass
about 3,963 miles nearer to the Sun by accomplishing a
direct journey of about 6204 miles. This is equivalent to
a gradient of about 1 in 1'565 — say, practically 1 in 1|.

It will be noticed that in relation to the Sun and the
Moon what we regard as the surface and the bottom of
the ocean respectively are reversed — assuming that by
the surface we mean the part farthest from the centre
and by the bottom the part nearest to the centre,
situated on the same radius. Again, just as we have
found that in relation to the Earth, the bottom of the

292 GRAVITATION THE CHIEF CAUSE OF

ocean at the pole is absolutely the nearest, and the
surface at the equator absolutely the most distant, from
the centre, so we find in relation to the Sun or Moon
(when above the equator) that the ocean at the pole is
the most distant and the surface at the equator the
nearest to the centre, and that it is actually the bottom

Tropt'eal

^C»r\n^»9

^ttareticRegianS

Diagram showing the diversity according to geographical situation in the
relation of the waters of the Ocean to the Sun or the Moon.

of the ocean at the pole which has the steepest gradient
in relation to solar and lunar attraction. The eflfects of
gravity as exercised by the Earth, on the one hand, and
by the Sun or the Moon on the other, are thus diametri-
cally reversed.

This is made clear by considering for a moment,
how the Earth will appear as viewed from the Sun at,
say, the equinox. The equator is in the centre, and the
poles at diametrically opposite parts of the circumfer-
ence, the separation between the poles and the equator
on the terrestrial disc being almost negligible. The
equator, however, is bulged out towards the Sun while
the poles are continuously at a distance corresponding
to the centre of the terrestrial ball. In fact, allowing
for the Earth's rotation, the polar waters would practi-
cally, in relation to the Sun, form the part most distant
continuously from the solar centre; while the super-

THE GENERAL OCEANIC CIRCULATION. 293

ficial water at the part of the equator turned towards
the Sun for the time being would be the nearest to the
solar centre. A similar state of matters must exist in
relation to the Moon and the terrestrial ocean.

The waters being extremely mobile, and molecularly
under the influence of gravitation, are quite different
from a solid body which is attracted as a whole. We
have noticed that the ocean is in a state of constant
unrest and of more or less unstable equilibrium, and we
have seen that there is reason to believe that, through
the influence of terrestrial gravity, a constant flow is
set up from the tropics polewards. It is evident that
this poleward flow must upset the balance which has to
be maintained between the polar and equatorial oceans.
The water at the tropics must thereby be reduced
below, while the water at the poles must be raised
above, the due proportions fixed by gravitation and
centrifugal force. We suggest that the primary cause
of the transference of the superfluous waters from
the polar to the equatorial regions is the gravitational
influence of the Sun and the Moon — the excess water
"running down," as we may express it, towards the
attracting bodies. Probably this movement will origin-
ate at or near the bottom in the polar oceans, being the
position from which the gradient is steepest, and the
destination of the flow will be the surface in the tropics.

As we have seen, the poleward movement of the
waters is deflected eastward in its course by the Earth's
rotation. The converse movement of the waters must,
of course, be deflected in the opposite direction — that is^
towards the west. It will, when observed as a current
in the northern hemisphere, be found to move from the

294 GRAVITATION THE CHIEF CAUSE OF

north-east. This movement towards the equator will
be affected by obstructions, specific gravity, temperature,
etc., similarly to the movement in the opposite direction.

We may take it as certain that these converse
movements of the waters — towards the poles in the
one case and towards the equator in the other — balance
each other, so that as much water is drawn towards the
equator as is drawn away from it. Were this not so,
the equilibrium of the ocean would necessarily be upset.
We have, therefore, to consider how the counter forces
are equalised.

This is a very difficult matter. We can best deal
with it by noticing the various influences which will
affect the diverse movements, and how they will affect
them.

Both movements will have the steepness of their
gradients lessened through the deflection resulting from
the Earth's rotation. The lessening of the gradient
will, however, not operate equally. The Sun and the
Moon have an apparent daily movement towards the
west, and this is the direction of the deflection caused
by the Earth's rotation to the flow of water equator-
ward, induced primarily by their influence. Thus even
in its rotatory deflection, the flow towards the tropica
is, in a sense, responding to the source of its attraction.
The lessening of the gradient by this cause in the flow
towards the equator is thus not at all so great as the
lessening of the gradient which must be occasioned by
the same cause in the poleward flow, since, in con-
nection with the latter, no such mitigating circumstances
arise.

Again, the influence of ocean ridges, or elavations of

THE GENERAL OCEANIC CIRCULATION. 295

the ocean bottom, will operate unequally. The upward
deflection of the poleward flow caused in this way to an
ocean movement whose attraction is continuously down-
ward must have an exceedingly great eflect in the
lessening of the attractive influence. On the other
liand, such an upward deflection in the flow towards
the tropics, instead of being obstructive is stimulative.
The attractive force is itself drawing the flow upward
in any case, so that its deflection upward by natural
obstacles must further strengthen the attractive effect.

The influence of temperature on the flow towards
the bottom of the ocean in the polar regions of the
heated surface water of the tropics will clearly be
obstructive, as the warmth of the tropical water will
tend to make it maintain a higher level than it would
otherwise follow in its poleward progress. Temperature,
therefore, will also result in a lessening of the gradient
in the poleward flow. Similarly the coldness of the
water flowing from the poles to the tropics will be
obstructive. Its tendency under solar and lunar attrac-
tion would be to pass to the surface at the equator, but
its relative coldness will tend to keep it far below the
surface. It is noticeable, however, that though relative
temperature must operate as an obstruction to both the
poleward and the equator-ward flow, it will not operate
equally. In the case of the flow towards the tropics the
progress of the water towards the source of attraction is
not materially different at a great depth from what it is
at a high level, while the opposite is the case in the
poleward flow. Thus, the influence of temperature on
the poleward flow is such as greatly to lessen the steep-
ness of the gradient, while in the equator-ward flow its

296 GRAVITATION THE CHIEF CAUSE OF

influence can only slightly lessen the steepness of the
gradient.

Relative salinity will very probably assist attraction
in each case. The water of the tropics being more saline
than that of the arctic and antarctic basins will tend to
sink in its poleward progress, while the water of the
polar regions, being less saline than that of the tropics,
will tend to rise in its equator-ward progress.

Centrifugal force will be obstructive to the pole-
ward flow but will aid the flow towards the equator.

Thus the poleward flow arising primarily from
terrestrial gravity will be adversely affected by (1)
deflection arising from the Earth's rotation: (2) ocean
ridges : (3) temperature, and (4) centrifugal force ; while
it will be assisted by relative salinity. The flow towards
the equator arising primarily from solar and lunar
gravity wnll be adversely affected by (1) deflection
arising from the Earth's rotation, and (2) temperature
(although both these causes of obstruction will be less
strongly adverse than in the case of the counter flow) :
while it will be assisted by (1) ocean ridges, (2) relative
salinity, and (3) centrifugal force.

It would seem, therefore, that on the whole, these
causes operate greatly more in opposition to the pole-
ward flow occasioned by terrestrial gravity than to the
flow in the opposite direction resulting from solar and
lunar gravity.

We have seen that on the direct course, if un-
obstructed, the flow towards the poles arising from the
Earth's attraction would follow a gradient of 1 in 407 ;
while the flow towards the equator arising from solar
and lunar gravity would follow a gradient of about

THE GENERAL OCEANIC CIRCULATION. 297

1 in 1|. There is reason to think that the effect of the
various circumstances relative to the converse flows in
the manner we have indicated is such as to make the
resulting influences practically the same as if the effective
gradient of the poleward flow were 1 in 4973, and that
of the flow towards the equator 1 in 3. With these
gradients, or somewhat proportional gradients, the power
of terrestrial gravity on the poleward flow would
balance the power of solar and lunar gravity on the
equator-ward flow, and such balancing of the relative
causes of the converse flows is evidently an essential to
their very existence.*

It has been found that, at and near the equator, the
temperature and, in a less degree, the salinity are
noticeably lower, at a comparatively short distance
below the surface, than they are at a corresponding
depth at a greater distance from the equator. This is

♦Let V=the mean effective velocity of flow in feet per second;

G= force of gravity in feet per second (being the velocity

attained at end of one second of a body falling from a

state of rest — i.e., twice the distance fallen in first

second ; g= gradient. Formula — V= »J2 G g.

(1) Flow from tropical to polar regions, with effective gradient

under terrestrial attraction of 1 in 4973.
V = V2 X 32-19 xT = V2 X 32-19 X -0002010859 = VOl 2945910242

4973
.-. Mean effective velocity of flow poleward = -113780 feet per second.

(2) Flow from polar to tropical regions with effective gradient under

solar and lunar attraction of 1 in 3.
V = V2x 2 (-11586+^00065) xl= V2x -0194183 x •§
12 3

= VOl 2945555556.
.*. Mean effective velocity of flow equator-ward = -113779 teet per
second.

298 GRAVITATION THE CHIEF CAUSE OF

reasonably explained by the flow towards the equator
of the cold and less saline waters of the polar oceans.
Mr. J. J. Wild, a member of the Civilian Scientific
Staff of the Challenger expedition, makes some allusion
to this fact. Writing in reference to observations of
temperature in the section of the Atlantic Ocean lying
between latitude 30" N., and latitude 37" S. — the central
portion of the section being thus very near the equator —
he says : —

"The most remarkable feature of the central belt is
the rapid decrease of temperature in the surface-stratum
of the ocean amounting to from 15° to 19° C. (27° F. to
34*2° F.) within less than 200 fathoms, in comparison with
the much slower decrease in the northern and southern
belt, in the lower strata of which we observe a gradual
increase of temperature as we recede from the equator,
much more rapid, however, in the former than in the
latter." {Thalassa, p. 74).

Mr. Wild, again, in referring to the conditions of
temperature near the equator in the Atlantic and Pacific
Oceans states that in the Atlantic the surface tempera-
ture is 25° C. (77° F.) down to a depth of 30 fathoms,
where there is a steep gradient to an intermediate
current which extends from 100 fathoms to 400
fathoms. The cold bottom-stratum is reached at a
depth of 500 fathoms with a temperature of 4° C.
(39 2° F.). In the Pacific a surface-stratum at a tem-
perature of 28-8° C. (83-84° F.) falls to 11-3° C. (52-34° F.)
at 150 fathoms, and to 3° C. (37-4° F.) at 800 fathoms,
the bottom-stratum commencing at 900 fathoms with a
temperature of 2-5° C. (365° F.). Mr. Wild adds :--

" How soon the cold bottom-stratum is reached in the
equatorial belt is one of the unexpected discoveries due

THE GENERAL OCEANIC CIRCULATION. 299

to recent deep-sea exploration. In the warm seas which
bathe the British Islands, a temperature of 4° C. is not
registered until we arrive at a depth of 900 fathoms;
and, at 1500 fathoms the temperature is still 2-5° C."
(Thalassa, p. 45).

Mr. Wild observes : —

" From the principal storehouse of heat in the tropics,
warm currents proceed towards the temperate and frigid
zones, and return thence in the character of cold currents
towards the regions of the equator. That this is so is
found by the results of all observations made up to the
present day, and it is in perfect agreement with the
well-known agency of water as a storer up and carrier of
heat." {Thalassa, p. 41).

With reference to salinity Dr. H. R. Mill says : —

**In the far north and the far south, bordering the
polar ice, the surface water is comparatively fresh.
Two zones of maximum salinity occupy the trade-wind
regions and are crossed by the tropics, while a belt of
low salinity lies a little north of the equator."

It is known that from the equator to the tropics the
surface water gradually increases in density, the density
at the equator being under 1*027, while at the tropics it
is appreciably over 1"027. From the tropics, however,
to the polar areas the density gradually diminishes
towards the poles, being less than 1'025 within the
polar circles.

It is clear that if the Sun and, in a minor degree,
the Moon have an appreciable influence in relation to
oceanic circulation, the circulation must be affected by
the varying relation of these luminaries to the terres-
trial oceans. Probably this variation would more
directly affect the movement of the waters from the

300 GRAVITATION THE CHIEF CAUSE OF

higher to the lower latitudes. But it cannot be doubted
that, as the converse movements of the waters are to a
great extent dependent on each other, the movement
towards the higher latitudes must also be influenced by
this cause. It is probable, therefore, that, in oceanic
circulation generally, the diurnal and periodic and
seasonal variations are fully reproduced. Mr. Wild
makes some reference to this matter : —

" As a necessary consequence (of the apparent motion
of the Sun) the volume, rate, and direction of the diiferent
currents are found to vary with the seasons; cold currents
preponderating at one time of the year, and warm currents
at another." {Thalassa, p. 54).

It is unnecessary to refer at any length to the
partial circulation of a more or less vertical character
which must be induced in the ocean waters by diversity
of temperature and variation of specific gravity. It
will be noticed that in this connection, there is a certain
symmetry associated with a general circulation of the
nature suggested. The movement of the water from
the equatorial towards the polar regions originates, we
have supposed, at the surface in the tropics. In this
situation the water is more exposed to solar radiation
than it is in any other part of the Earth. This being
so, the terminus of the movement set up in these waters
is the floor, or, at any rate, the lower portions, of the
polar oceans. These are the parts most completely cut
away from solar radiation. This movement thus carries
into these areas water of a higher temperature than
they normally contain. The natural tendency of the
warmer water is to rise to the surface ; so that this
warmer water carried into the polar regions induces a

THE GENERAL OCEANIC CIRCULATION. 301

vertical or ascensional and decensional circulation, and, to
some extent, distributes warmth in the cold polar seas.

On the other hand, the movement from the polar
oceans will probably originate chiefly at or near the
bottom in the cold dense water of these parts, and this
movement will draw this water slowly towards the
surface in the tropics. In the equatorial regions the
vertical or ascensional circulation induced by excessive
heat is naturally most rapid, so that the cold water is
soon drawn upward and speedily warmed.

It seems to be commonly believed that the primary
agents concerned in oceanic circulation are the pre-
vailing winds and atmospheric conditions, subordinate
causes being relative temperature and salinity and con-
sequent specific gravity. There would appear to be
reason for maintaining that the prevailing winds and
the general circulation of the ocean are both products of
analogous causes rather than that they are respectively
cause and effect. Our hypothesis is that the primary
cause of the general circulation of the waters of the
ocean is to be found in the law of gravitation in its
relation to such a mobile fluid as water, in association
with the spheroidal shape of the Earth. The relative
temperature and salinity of associated bodies of water
and their consequent relative specific gravity have,
however, undoubtedly great weight as subordinate and
contributary causes to the extensive movements which
are known to characterize the general circulation of the
waters of the ocean.

TWILIGHT AND DAWN.

SYIfOFSIS.

TWILIGHT AND DAWN.

Strictly speaking, the term " twilight " applies both to
the light which precedes sunrise and the light which
remains after sunset. Colloquially, however, as in
poetic usage, the former is distinctively known as
"dawn," and the term "twilight" is applied only to
the latter.

Thus Tennyson speaks of

" Twilight and evening bell
and after tJmt the dark ; "

while Milton writes

" Fairest of stars, last in the train of night,
If better thou belong not to the dawn."

But, although we can correctly refer to the morning
twilight, just as to the evening twilight, the word
" dawn " has not the same flexibility. The latter term,
when used in this connection, applies exclusively to the
first appearance of light in the morning, to the light
which heralds the uprising of the Sun.

There is some reason to suppose that originally the
word " twilight " had a more restricted application, that,
as it still does colloquially, it formerly in correct and
scientific use applied exclusively to the light which

X 305

30G TWILIGHT AND DAWN.

follows the setting of the Sun. The word is derived
from the Anglo-Saxon, and is really a compound of the
words two and light. In English the word two has
entirely lost — if, indeed, it ever possessed — the effective
pronunciation of the second letter, although this is still
preserved in the partially obsolete alternative form of
the word — twain. Probably in the Scottish form — twa
— we have a near approach to the original pronunciation.

Thus twilight originally meant simply two-light or
twa-light, the prefix signifying not the cardinal number
two but the ordinal number second. Certainly such a
use of the word two is now-a-days very exceptional.
We have still, however, an example of the use of the
word as equivalent to second in the old phrase of
commendation which is often applied to the regiment
of the Royal Highlanders — the Black Watch — which
was formerly known as " The Forty Second," the phrase
being " The gallant Forty-Twa." Here the word twa is
obviously employed as identical in meaning with seamd,
and this appears to have also been the case in the use of
the prefix in the word twilight.

Thus twilight was evidently the second-light, the
light which followed after the setting of the Sun. The
popular meaning of the word twilight — the light
immediately following sunset — would thus seem to have
been originally the exclusive signification of the term.

In some ways it may be considered regrettable that
in scientific and accurate use twilight has now come to
include dawn, that the word has not retained what
apparently in former times was its definite and restricted
meaning. Although, from the physical point of view,
there is no appreciable difference between the character

TWILIGHT AND DAWN. 307

of the light preceding sunrise and that of the light
succeeding sunset, there is yet a certain distinction.

The dawn, or morning twilight, which ushers in the
day, iirst appears — at least in the lower temperate and
tropical latitudes — towards the East. It ascends from
the horizon and increases in brilliance until at last it is
lost in the glory of the direct light of the Sun. The
evening twilight, on the other hand, is the remnant of
departing brightness, the shade of a lustre which is
vanishing away. It is most noticeable towards the
western horizon and gradually gives place to the black-
ness of night. The twilight of morning is, as it were,
the springtime of the day, while the twilight of evening
is its autumn or fall. There would seem to be some
reason for the recognition of these distinctions by the
restriction of the use of the word twilight to the ligfht
which follows sunset, instead of employing it also as an
alternative to the word dawn.

After all, however, just as old age merges insensibly
into second childhood, so it happens — particularly in
the higher latitudes — that the twilight of evening
changes imperceptibly into the dawn of morning. The
Sun sinks below the horizon and twilight supervenes.
Twilight wanes, then waxes, and the Sun again appears.
Thus there is no break in