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Gift of
The Yale Divinity School
1909

GEOMETRY AND FAITH.

" No man, therefore, can doubt, but toward the atteining of knowledg
incomparable, and Heavenly wisdome, mathematical speculations, both
of Numbers and Magnitudes, are means, aids and guides ; ready, certain
and necessary. " — John Dee, of London, 1570.

GEOMETRY AND FAITH
A SUPPLEMENT
TO THE
NINTH BRIDGEWATER TREATISE
BY
THOMAS HILL

'The truths of Natural Beligiou are impressed iu indelible characters
on every fragment of the material world "

THIRD EDITION GREATLY ENLARGED

BOSTON
LEE AND SHEPARD PUBLISHERS
NEW YORK CHARLES T. DILLINGHAM
1882

" This is the chiefe Glory of Geometry, that it loyters not, or employes
it self about these inferiour Machines, from whence it had its Original,
but hath soared up into Heaven, and resetled humane minds, (groveling
before in the dust) in Ciwlestial Seats." — John Leeke and George Serle,
1661.

Copyright, 1882,
By LEE AND SHEPARD.

All rights reserved.

CONTENTS.

Page.
CHAP. I. SYMMETRY IN SPACE  I
II. SYMMETRY IN TIME  10
III. NUMBER  IS
IV. THE CALCULUS  20
V. APPLIED MATHEMATICS  24
VI. MOTION  27
VII. MUSCULAR ACTION  32
VIII. GEOMETRICAL INSTINCTS .... 37
IX. MOTION ETERNAL IN DURATION ... 43
X. MOTION OMNIPRESENT IN SPACE . . 46
XL THE SPHERE OF HUMAN INFLUENCE . . 50
XII. MAGNITUDE  54
XIII. CHANCE AND AVERAGE  64
XIV. PHYLLOTAXIS . ,  71
XV. NUMBER AND PROPORTION .... 90
XVI. THE DEVELOPMENT OF FORMS . . 102

* * * yempezQiav, * * o dq davfia ovx avOqwrtivov, alia, yayovbg
OsTov (favsqbv av yiyvoiro zcji dwausvco gvvvousiv' * * o 61 deiov r sari
xcu Oavuaarbv roig tyxadoywoi w xou Siavoovfilvotg' — The Epinomis.

PREFACE,

"The present work lays no claim to originality." A humble
gleaner in the fields of Mathesis and Theology, I offer only a few
of the common fruits, well known to those who have more thoroughly
surveyed the boundaries of these two domains. And I have ventured
to connect them with the name of the " Bridgewater Treatises," not
because I consider, myself worthy to appear in company with their
writers, but simply that I may thus the more earnestly express my
admiration for the treatise of Babbage.
Waltham, Mass., 1849.

PREFACE TO THE SECOND EDITION.
I have rewritten the greater part of this work, and altered so much
the expression and illustration of my thought, that I might have given
it a new title, but for the affection which twenty-five years' familiarity
has bred for the old one.
Portland, Me., 1874.

Evfioloyqrov, tcprr rov ycto dsi ovrog 17 ysmfierQix^ yvajaig kativ
'Olxov ago., a yevvais, xpv%ijg ngog dlrjdeiav ea] av, xal ansgyaarixdv
(piloaoyov diavoiag ngog to avca ayiiv a wv xenon 6v deov e%ppev'
— Plato's Republic, Book VII.

GEOMETRY AND FAITH.

I.
SYMMETRY IN SPACE.
The universe, actual, possible and impossible, is composed of
four elements, spirit, matter, space, and time, which are by no
alchemy transmutable into each other. Many alchemists continue,
even in this closing half of the nineteenth century, to make the
attempt, and some even flatter themselves that they are succeed
ing ; but the sturdy reply of human consciousness is, that the four
elements are diverse and not transmutable ; or, if any trans
mutation is possible, it must be confined to this, that matter may,
in some manner, be an effect of spirit. But to us, finite spirits,
nothing more is granted than the re-arrangement, the partial con
trol of matter, not its creation. Matter, as we know it, is dis
tinguished by its being the recipient and dispenser of force ; which
force, so far as we know it, is from spirit alone. This obedience
of matter to spirit gives justification to our suspicion' that it is the
creation of spirit.
Space and time are without parts, and are indivisible except by
a mental act. This division is suggested to us by manifest motion
in matter. Force shows itself in matter by moving it ; that motion
calls our attention to the space and time, within which the motion
(i)

Z GEOMETRY AND FAITH.
is taking place ; and we divide mentally this space and time, first
from the remainder of the boundless contiguities, secondly into
smaller parts. Thus geometry and algebra are generated, the
sciences which deal respectively with space and time, those pure
entities, the relation of which to the Infinite Spirit we cannot com
prehend, but which we become familiar with in the finite portion
embraced in our experience, in the universe and its history.
In geometry, the mind imposes upon indivisible space arbi
trary boundaries of division, according to arbitrarily selected laws
or conditions. These boundaries are of three kinds, surfaces, lines
and points. The point is a zero of magnitude in space, but never
theless is not nothing ; which is nowhere, while the point is some
where. This contradiction in terms, that a point should have no
extension, and yet have a position, is one of those instances, in
which geometry abounds, in which the mind is compelled, by the
necessity of direct vision, to admit each of two truths, which are to
logic mutual contradictories. The mathematician modifies the law
of non-contradiction by confining it to propositions concerning finite
quantities. A lower form of a zero of magnitude in space is the line, which
is extended, at each point, only in two opposite directions ; and the
lowest form is the surface ; for which there can, at each point, be
drawn a line, such that the surface extends, in every direction,
only perpendicular to that line. Geometers define these lower
forms of zeroes, or boundaries in space, by the further self-contra
diction of imagining the movement of a point ; a double contradic
tion, since space is itself incapable of motion, much more a zero
of magnitude.
A geometrical line is defined as the path of a point, moving ac
cording to certain conditions, which always limit its motion, in

SYMMETRY IN SPACE. 6
each of its positions, to one of two opposite directions. Or, it
may be defined as a continuous series of all the points which fulfill
certain conditions, among which must be the condition that each
point is contiguous only to two others, one on the opposite side to
the other. So also a surface may be defined as the space in which
a point moves, when, in each position which it assumes, a straight
line may be drawn through it, and its motion be permitted, in any
direction at right angles to that line, and in no other. Or, the
surface may be defined as a series of points, through any one of
which a straight line may be drawn, such that all the contiguous
points lie in a direction at right angles to that line. To either of
these definitions of a surface, we must add, in order to make a
geometrical surface, some other conditions which the points must
fulfill. When the geometer has selected these conditions and would in
vestigate the form which the points, so conditioned, would enclose,
he is not contented with the mere act of reason ; he endeavors to
bring imagination to his aid ; to make a sensible image of the form.
If he has been blind from his birth, he imagines his fingers feeling
out the form ; otherwise he embodies it visibly, as in a drawing,
or in a model. If he would convey a knowledge of it to others,
he calls matter to his aid, and forces atoms of chalk, black lead,
wood or thread, to fulfill approximately the conditions which his
geometric law imposes upon the series of points. This drawing,
or model, is an expression of his idea, an enunciation of his law.
A geometrical figure, whether upon the blackboard, or the printed
page, or in a block of wood, or a set of stretched threads, is incon
trovertible evidence that a geometer has been expressing, by this
means, a geometrical thought.
The laws which please the geometer most highly are those

4 GEOMETRY AND FAITH.
which give us symmetrical figures, figures in which part answers
to part ; either on opposite sides of one line or one surface, or
about more than one line or surface. This taste is not peculiar to
the geometer ; symmetry pleases the most savage, as it does the
civilized man ; and men whose whole ability lies in other direc
tions, as well as the mathematician. A striking proof of the uni
versality of this taste was shown in the sudden and universal pop
ularity attained by the kaleidoscope. In a few years that toy of
Brewster found its way to every parlor, and the heart of every
child, ay, and every man in Christendom. Yet its sole magic
consists in the symmetry which it imparts to a few fragments of
irregular form. But that magic is sufficient to enchant all who
come within its sway. We have never found any one uninterested
in an extempore kaleidoscope, made by throwing open the piano,
and placing brightly colored articles at one end of the folding lid.
All regularity of form is as truly an expression of thought as a
geometrical diagram can be. The particles of matter take the
form in obedience to a force which is acting according to an intel
lectual law, imposing conditions on its exercise. It does not alter
the reality of this ultimate dependence of symmetry upon thought,
simply to introduce a chain of secondary canses, between the origi
nal thinking and the final expression of the thought.
Many of the geometer's a priori laws were, indeed, first sug
gested by the forms of nature. Natural symmetry leads us to
investigate, first, the mathematical law which it embodies ; then,
the mechanical law which embodies it. Thus all the benefits which
have come to our race from the pursuit and discovery and use of
the keys of physical science, have been bestowed upon us through
these suggestions of geometrical thoughts in the outward creation.
But in the pursuit of mathematical knowledge men began, at an

SYMMETRY IN SPACE. 5
early age, to invent and investigate a priori laws, laws of which
they had not received any suggestion from nature. And the
intellectual origin of the forms of nature was made still more mani
fest when these a priori laws, of man's invention, were, in many
cases, afterwards discovered to have been truly embodied in the
universe from the beginning; as, for example, Plato's conic sec
tions in the forms and orbits of the heavenly bodies, and Euclid's
division in extreme and mean ratio.
This division in the extreme and mean ratio was invented by
the early geometers, without any known suggestion. It is evi
dent that this division might be illustrated in a great variety of
ways. A whole must be divided into two parts, such that the
first shall bear the same relation to the second that the second
does to the whole. No matter what the whole is, a division of it
approximately in this manner would be an expression of the idea
of extreme and mean ratio. If the whole were a quantity (dis
tance, angle, surface, volume, value, time, velocity, &c), and the
relation were that of magnitude, the whole would be to the smaller
part as unity is to half the difference between three and the square
root of five. If, on the other hand, the whole were a work of art
of any kind, or a system of thought, the relation would not be
one of mere magnitude ; and the division would be a work of more
ingenuity. But, whatever the whole, or the relation, the proper
division would be an expression of the idea.
Now we have, in nature, at least three embodiments of the law
of extreme and mean ratio, two of which are very striking. The
botanists find that two successive leaves, counting upward on the
stem, stand at an angle with each other that is either one-half,
one-third, two-fifths, three-eighths of the whole circle ; or some
higher approximation to this peculiar proportion. The seed ves-

6 GEOMETRY AND FAITH.
sels and buds on a spike of broad-leaved plantain afford one of the
most instructive examples. They are usually set on a high ap
proximation, so that the order is not apparent. Take a piece of
the spike, an inch or so in length, between your hands, and gently
twisting reduces it to three ; while a slight twist in the opposite
direction brings out five rows, which a harder twist reduces to
two. The efficient cause of this arrangement we do not know. It has
been ingeniously suggested that it might be produced by a simple
law of the genesis of cells. Let us suppose that each cell emits
a new cell at regularly recurring intervals of time, and that the
new cell begins to generate cells at the expiration of two intervals
after its birth. A cell developing on a plane, under this law,
would produce its cells in the phyllotactic order of the leaves, in
the terminal rosette of a plant. But it is difficult to see how this
hypothesis can be made to include and explain the whole phe
nomena of the arrangement.
The final causes, although the devout mind always recognizes
the impossibility of man's attaining a certainty concerning all the
final causes of the phenomenon, are more obvious. It has been
shown that this division of the circle insures in the only perfect
way to each leaf its chance at zenith light, its best chance at air ;
in short, that this phyllotactic law distributes the leaves most
evenly about the stem.
In the solar system, if we divide the periodic time of each
planet by that of the planet next farthest from the sun, we shall
have, beginning with the quotient of Uranus' year divided by
that of Neptune and ending with the quotient of Mercury's year
divided by that of Venus, a series of fractions agreeing very
closely with the approximations of the phyllotactic law. The

SYMMETRY IN SPACE. 7
problem was similar. The planets would not have remained in
proper subjection to the sun had they been allowed to group
themselves too frequently in one rebellious line, hanging upon
the golden chain of his attraction, dragging him and themselves
from their proper orbits. They must be kept evenly distributed
about the sun ; and since they are moving, the times of their
revolution, their angular velocities, must be divided by the same
law as that which divides the stationary angles of the leaves.
We have then in the plants a geometrical or angular illustra
tion, and in the planets an algebraical or temporal illustration, of
the mathematical idea of extreme and mean ratio. The infer
ence seems irresistible, — these two illustrations, which cannot be
imagined as having any causal or genetic connection, owe their
intellectual relation to having sprung from One Mind.
This is a striking illustration, but the same inference may be
drawn frop every form in nature, — planet, crystal, plant, and
animals. All natural forms conform more or less closely to geo
metrical ideals ; sufficiently near to suggest these ideals to men
fitted to receive the suggestion ; sufficiently near to show that the
whole of nature may, in one sense, be regarded as a series of draw
ings and models, by which to teach the mathematics to students in
the school of life.
The final causes may never, however, be considered as wholly
known. The perfection of the Divine workmanship is shown in
the adaptation of each object in nature to a great variety of ends.
The geometrical laws, on which the world is built, are adapted to
all the wants and all the needs of every creature. Our human
needs are innumerably various, and nature finds means to satisfy
them all. Our intellect craves symmetry, and through symmetry
is first led to the perception of geometric law. But we love the

8 GEOMETRY AND FAITH.
symmetry before we perceive the law. The sense of beauty is
satisfied, even in externals, most perfectly, and fills us with most
pleasure, in things that the understanding fails to analyze and
define. Much has been written concerning an analysis of the
beauty of outline ; one great painter thinking it consists in flexure,
others assigning it to a spiral, or a helix, or an ellipse ; while
Darwin refers it to early association, while yet a suckling, with
the form of the mother's breast. I venture with diffidence to
give my own opinion, that the perception of beauty in outline is
the unconscious perception of geometric law, — just as the per
ception of harmony has been demonstrated to be the unconscious
perception of arithmetical ratios in time, or algebraic law. The
beauty of outline, I would say of external form, independently of
expression, is in proportion to the simplicity of the geometric law,
and to the variety of the outline which embodies it. Nor is it
essential to the highest enjoyment of beauty that the conformity
to geometric ideals should be perfect, any more than it is essential
to the highest music to have the harmony perfect. On the con
trary, the higher degrees of beauty are apt to be found in forms
that suggest, rather than embody, the ideal ; and especially in
figures potentially, but not actually, symmetrical. The monotony,
which might result from unbroken regularity of form, is avoided,
and a new grace is given, for example, to the higher animals, by
their temporary disguise of symmetry, in their varied positions and
movements. In the sea shells, the same end is attained by the
spiral form, which so many of them take ; in which there is not an
actual symmetry, but only a law of symmetry, the perfect develop
ment of which would require an infinite number of convolutions.
In the forms of vegetative life, there is the widest departure
from actual symmetry, and yet a constant suggestion of its laws.

SYMMETRY IN SPACE. 9
The phyllotactic law secures to the tree a general regularity, and
equal growth upon every side ; and yet, by complication of detail,
combined with occasional failure or destruction of buds, secures
an endless variety of graceful forms, in each species. May we
not then name beauty as another final cause, another end secured
by the adoption of the division in extreme and mean ratio ? The
approximations are beautiful to us, and the pleasure given to us
was foreseen when the law was adopted. May it not also have
been felt ; and may not the forms of flowers be but approximations
toward the expression of an infinite beauty, hidden, from all finite
sense, in the incommensurable ratio of that surd ? That the ex
ternal symmetry of animals may have beauty as its final cause, is
rendered probable from the lack of symmetry in the viscera, which
are hidden from sight.
Whatever be our speculations upon such points, this at least is
manifest, that the sense and the presence of beauty are kindly
adapted to each other in the world. Even shapeless matter de
clares its Creator's power ; the perfect symmetry of crystalline
forms, the potential symmetry of all the organic worlds, show forth
His wisdom and His love.

II.

SYMMETRY IN TIME.
Time has but one dimension, and is divisible only into before
and after. In the zero of now, the future is becoming the past ;
and this suggests the division of the future, and of the past, by the
insertion of imagined presents, zero boundaries, dividing time into
periods ; these imagined future nows becoming actual, as we suc
cessively reach them, and those past having been actual, as we
passed them. As time flows only forward, the imagination runs
backward into the past with the greatest difficulty ; indeed I am
not certain whether it is possible for the imagination to run back ;
when I attempt to do so, I find that I leap backward, by longer or
shorter leaps, but never run continuously in imagination through
time, except forward, from the moment to which I have leaped.
By symmetry in time, therefore, we do not mean a similar ar
rangement of intervals before and after a certain moment. This
has occasionally been attempted in per recte et retro chanting ; but
it is a transference of geometrical symmetry to time, where it is
out of place, and tasteless. Symmetry in time is the arrange
ment of two or more similar series of intervals, to follow the same,
or successive movements. When the set of similar intervals fol
lows the same moment, it constitutes keeping of time ; when suc
cessive moments, rhythm ; unless the intervals are very short,
when rhythm becomes tone, or color, and keeping time becomes
harmony. (10)

SYMMETRY IN TIME. 11
The passion for harmony and rhythm is an essential element in
human nature. It is a passion which varies greatly in intensity in
different persons, but it is never wholly absent. Savage nations
have some rudiments of music and of song. The naked Fuegian,
when the stormy winds of his inhospitable straits pause for a while
in their wild uproar, chants his songs or hymns in a rude measure
and melody. The dullest ear for harmony has an ear for rhythm
sufficient to perceive the difference between prose and verse. It
is when the intervals of time in rhythm become so short, as to be
separately imperceptible, that the rhythm is called simply tone ;
and harmony is the simultaneous movement of tones. Man finds
pleasure in all forms of symmetry in time, whether the parts are
perceptible, or imperceptibly short ; and the world has been made
in exquisite adaptation to this taste of man.
Space has three dimensions, time but one. Yet, in some re
spects, time is richer in its contents for man, than is space. The
beauty of forms in space, is almost equalled by the beauty of
color, and color arises simply from symmetry of times ; it is a kind
of tone. Color, indeed, is more expressive, more directly produc
tive of pleasure to the eye, than form. The latter appeals more
to the intellect, and is more directly expressive of intellectual
ideas ; the former appeals more to the heart, and gives a sweeter
pledge of the Divine Love. But beside color, symmetry in time
gives us music in all its vast variety of forms and expressions. In
music there is a beauty as distinctly intellectual as that of geo
metrical figures, and a power of expression which geometric form
attains, scarcely even in the human figure and face. Nor can we
omit to mention heat, which although not giving direct pleasure
to the mind, and the heart, as beauty, color, and music do, is
still essential to the life of the body and to its comfort. Many

12 GEOMETRY AND FAITH.
chemical changes are also produced by minute symmetric mo
tions. A minute symmetrical division in space produces no sensation
for us, except as it may lead to symmetrical motion. Thus the
minute symmetry of the particles of a solid in a clear liquid solu
tion, is revealed to us only through the motion of light, and the
changes which that motion experiences in going through the solu
tion. In like manner the temporal symmetry of motion in the ray
of light coming from the stars, brings us more information than its
mere property of making visible can give ; for, on cross-examina
tion by the spectroscope, it confesses the chemical secrets, and the
degree of heat in the star at the time of its leaving, ages ago ;
and also the direction of the star's motion.
Man's organization, and his surroundings, are adapted to his
love of music. His voice is capable of being regulated to musical
tones of various pitch. The metals are sufficiently elastic to
render them sonorous ; and, as the foundation of all, the air itself,
by its elasticity, becomes the vehicle of sound and the instrument
of music.
Two gases, intermingled, remain to a certain extent indepen
dent of each other ; and, inasmuch as sound travels in each gas
with a velocity proportioned to the density and elasticity of that
gas, there will be, from a single source of sound, two sounds prop
agated in the mixture, with different velocities, interfering with
each other, and destroying the pure tone of a musical sound.
Now the atmosphere is a mixture of heavy oxygen, with lighter
nitrogen. The elasticities are, however, so nearly adjusted to the
densities, that sounds travel in either gas with nearly the same
volocity, so that the air sounds in an organ pipe, as if one gas.
Had sound travelled in these two gases at rates differing as much,

SYMMETRY IN TIME. 13
as the rate in them differs from that in most of the gases known to
us, the use of wind instruments of music would have been impos
sible ; probably all music, even the tones of the human voice,
would, in that case, have been discordant to an ear at any consid
erable distance from the source of sound. With the intense and
elevating character of the pleasure derived, first from the tones of
human speech, from the melody of birds, and other natural music,
and secondly from the art of music, in our minds, we cannot but
be grateful for this adaptation of the mingled atmosphere to the
needs of man, in his higher nature.
All undulatory motion produces a symmetrical division of time.
The beauty of color, like that of tone, arises from an implicit per
ception of rhythm. The harmony of tints in the landscape,
like that of the sounds in a strain of music, arises from the har
mony of times in which the vibrations of the mediums occur. The
pleasure, in either case, arises from an implicit, or unconscious,
perception of keeping time. Heat also has its colors, or tones, as
is known to all who have noticed that the sun's heat passes freely
through glass ; which is impervious to the heat of a fire. What
other advantages to man may hereafter be discovered, in this
coloration of heat, time alone can show ; but when we consider to
what an extent, through the providence of God, glass is employed,
it seems not irreverent to own our gratitude to Him that this sub
stance reflects back the warmth of our apartments, and keeps it
within, but allows the heat of the sun to pass through from with
out. Besides these hidden proofs of creative foresight, and benefi
cence, in the concealed, or minute symmetry of time, we shall
find open and abundant proofs in manifest rhythmic movements.
In the play of alternating muscles, we perceive an adaptation of the

14 GEOMETRY AND FAITH.
physical frame to the intellectual taste. In walking, for example,
there are few persons who do not feel the increase of pleasure and
of power gained by keeping step. A single drum-tap, regulating
the tramp of a large body of men, has sometimes an effect almost
equal to that of music. The rhythm of verse, and of music, de
lights many who are comparatively insensible to both melody and
harmony properly so called. Who that reflects upon the genius of
Bach, of Handel, of Haydn, and Beethoven, and considers the
effect which music, such as theirs, has upon the world, can doubt
the kindness of that superintending Power, who kindled the fire
in their hearts, and through that in ours ; who also adapted the
air, and the various materials, for man, by which he pours out his
musical conceptions ? Who that reflects upon the genius of a
Homer, and a Shakespeare, and remembers to how many millions
their verse has given delight and instruction, can doubt that the
same Beneficence gave the poet his power, and men the heart to
be touched with poetry ?

III.

NUMBER.

Number is not, like space and time, matter and spirit, an inte
gral part of the Universe, nor is it a necessary attribute of either
of these. Space and time are without parts or limits, and are, in
themselves, so diverse that they would not suggest even the idea
of duality. Thus also amorphous matter suggests no number.
Number is an impress of thought, it is a pure creation of Spirit ;
and its constant suggestion in the forms and periods of nature, is
a clear demonstration that nature is the work of an Intellect which
controls both space and time in thought. The human intellect
early learns number from the text-book of outward nature ; and
delights in tracing, further than nature goes, the laws of number.
From the great usefulness of this earliest abstract science, and
from the fascination of its pursuit, arithmetic has, in modern
schools, been allowed to usurp the place of geometry ; and the
pupil has been taught to reason upon abstract numbers before he
has learned to conceive clearly imaginary forms. From the same
fascinating power, number has sometimes, in the minds of great
men, like Pythagoras, been allowed to occupy a disproportionate
share of attention ; as though number included all proportion and
beauty. Even the Hebrews, with all their clearer light of truth,
appear to attach a mysterious power to number.
There is a power in number. When our human thought at
tempts the survey of space and time, and would subdue these
(iS)

16 GEOMETRY and faith.
realms to obedience under our intellect, we find ourselves com
pelled, before we can attain any precision in our forms, to intro
duce number. The reason can deal, to some extent, with con
tinuous quantity, moving under continuous law, and not in the
proportion of numbers. But the imagination cannot take a step
with any clearness, much less can the hand build with any satis
faction, without referring quantities to a unit of quantity, to which
the ratio shall be that of two numbers to each other. And of
course our finite intellects handle with most ease the smaller num
bers; so that these become to us the most important; and there is
not a number under ten which has not some strong associations
with it in the human mind, which give it a kind of sanctity.
These mystic charms cluster especially around the odd numbers
three, five, seven, and nine ; which seem to have an individuality ;
the first-named three being primes, while four is but two twos ;
and six, two threes ; and these charms were felt in the earliest
ages of human history.
But nature also loves these numbers ; and they are illustrated,
even to the untaught mind, by many phenomena ; organic beings
possess a unity, which is absolute ; the sexes, of both plants and
animals, give us duality ; the powers within, and those above, sug
gest the threefold division ; the points of the compass, the limbs of
mammals, give us the number four ; the fingers of the hand, five,
and so on. And the increasing knowledge of the physical world,
in our nineteenth century, brings us increasing proof that God,
who planned heaven and earth, was acquainted with numbers ;
made all things in number, weight, and measure ; and adopted
the smaller numbers, either out of preference for them, or in
condescension to the minds of his children, whom he has placed
here for their preparatory education.

NUMBER. 17
Chemistry is a science of this century, and it teaches us that,
from the beginning, the numbers two and three have been domi
nant powers in the Universe. Simple unites with simple to form
a couple, a compound. This couple rarely takes a third element
to form a triplet. The couplets and triphts unite again in com
pound couplets, and thus the innumerable variety of substances
is built up under the simplest possible combinations of number.
Follow these substances through all their various modes of mo
tion and action ; in their weights, in their attractions, their gas
eous condition, their volume, their specific heat, their color at a
high temperature ; and they are found still to be bound together
by simple laws of arithmetical proportion.
Consider also the law of extreme and mean ratio, as exhibited in
the leaves of plants. In itself the law transcends the power of
number, and had the plants fulfilled it with absolute accuracy, it
might have been, even yet, hidden from the mathematician's eye.
But the plants give it to us only by approximations ; approxima
tions which demonstrate that the exact law was known to the
Builder of the plant, and is by him revealed to the mathemati
cian ; but which give to the unlearned the simpler conception of
the first four prime numbers ; in the beautiful varieties of leaves
opposite, and leaves ternate, five pointed and seven pointed stars.
The laws of musical harmony are especially to be noted. When
the waves of the air are perceived only as continuous musical
tones, and the individual vibrations are not at all recognized, why
should the ratio of four to five give us pleasure, and that of eight
to eleven give us none ? What process of education in our ances
try, what association of ideas, renders the effect of the one com
bination harmonious, of the other discordant ? Any attempt to
explain it will but strengthen the conclusion, that to the Builder

18 GEOMETRY AND FAITH.
of the ear the laws of number were known, and that the ear was
constructed with reference to them.
The harmonies of light and heat are not sufficiently well under
stood to make the argument here so apparent. Yet there is,
doubtless, in these departments also, an adaptation of the human
sense to the perception of effects arising from simple numerical
proportions in the frequency of vibrations. In the matter of
geometric form, while the value of proportion has been felt by all
artists, and all architects, the value of numbers in the proportions
has not been universally conceded, nor its place assigned. Yet
I have by experiments, upon unprejudiced persons of good taste,
strengthened greatly my inclination to accept Hay's law, — that
angles, real or potential, are the essential elements of geometric
beauty ; and are beautiful in proportion to the numerical sim
plicity of their ratio to the right angle.
With these manifest indications that the divine thought, the
ideals of the creation, include number as an essential element, we
may well understand the enthusiasm of early thinkers over the
properties of the smaller numbers. The sacredness of the num
ber three has been made especially prominent in Christendom.
The four elements of the ancients, and Erigena's fourfold division
of nature, show the power of the points of the compass to impress
their number on the human mind. The five digits of the hand,
and the prevalence of fivefold divisions in the floral kingdom, give
us the five-pointed star with its symbolism ; point up, for manhood
and virtue; point down, for beastliness and sin. The lily tribe
gives us the six-pointed star ; and six, a perfect number, in which
the sum of the factors equals the product, is fitting as a symbol of
the descent of the divine into the human trinity, the indwelling of
God in man ; the Perfect perfecting his child. The seven notes

NUMBER. 19
of the diatonic scale, the seven distinct colors, and other natural
examples, fall in with the seven days of the week, the quartering
of the moon's period. Jew and Gentile alike have hallowed the
number seven, and no other number occurs so frequently with
sacred associations in Jewish and Christian literature. Higher
primes than seven do not enter much into our human thought, nor
appear to be embodied distinctly in any par.t of creation known to
us. The weeks in the year are four times thirteen ; that is, there
are about thirteen moons in the year ; the only example I re
member of a prime number above seven prominently suggested
by nature. The nine muses, the ten numerals, the twelve months,
and twelve apostles are numbers not prime.
Music, painting, the coloring of nature and art ; architecture,
sculpture, drawing, the beauty of proportion and form ; how large
a portion of our earthly pleasure and spiritual culture depends
on these ; and these draw their charm in some mysterious way
from the numbers two, three, five, seven. The number of prime
numbers is unlimited ; and since the first four give us, in the har
mony of tones and colors, and in the proportions of form, such
varied sources of high pleasure, such potent modes of spiritual ex
pression, we may reverently hope, that in the immortal life, the
same Beneficent Power which makes two, three, five, and seven,
thus minister to our joys below, will open to us more of the infinite
treasures which lie hidden in the boundless fields beyond.

IV.

THE CALCULUS.
Space and time are so entirely diverse in their nature, that
there is no connection or relation between them ; except through
the mind, as percipient of both; or through will, manifesting itself
in motion. In contemplating space we see it as external to the
mind ; our consciousness does not sharply locate its own where
abouts ; we fancy ourselves near the Eyegate or Eargate of the
town of Mansoul; but cannot say precisely where our council
chamber may be situated. Not so with time, our consciousness is
sharply defined ; we are neither in the past nor in the future ; our
conscious moment is the now, without duration. Hence we can
more readily imagine ourselves freed from limitations of space than
from those of time. We can imagine to ourselves, time in the flow
of our own thoughts ; the thought of space necessarily takes us
out of ourselves. But when we go out of ourselves and contem
plate space, we carry time with us in the very action of our
thought. In all closer contemplation of outlines, the attention is
transferred successively to different points of the figure, and time
is occupied by that transfer. Thus we come naturally, and almost
inevitably, to regard the line as the path of a moving point, the
surface as generated by a moving line.
Thus space and time, though heterogeneous, are united into one
science of mathematics by human thought ; and the laws of alge
bra, or time, are applied to geometry, or space. By this simple
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THE CALCULUS. 21
device, into which Descartes and Newton were led by nature's own
guidance, the human mind has extended almost indefinitely its
geometrical acquisitions ; it was by carrying, as it were, its native
element of time with it into the domain of space that it has con
quered so vast a field.
When we remember how intense the delight which man feels in
the discovery of mathematical truths ; how many of the noblest
thinkers of the race have owed their finest discipline to this pur
suit ; how rich the harvest of practical benefits which have flowed
from the application of mathematics to the arts and sciences ; how
magical their effect has been in banishing superstition, and elevat
ing the general tone of human thought and human endeavor, we
may surely own, with gratitude, the marks of divine wisdom and
love, in this gift to man, of the power to penetrate space, and ap
ply to it the laws of time. It is a peculiar gift, not a necessary
accompaniment of intellect, for sometimes the brightest intellects
possess it in only a very feeble degree. Thankfully, therefore, do
we acknowledge the presence of an Infinite Spirit, giving good
gifts to man in the inspiration of a Leibnitz and a Lagrange, as
well as of a Handel and a Shakespeare.
The main source of this power given by algebra to the geometer,
is the comprehensiveness of the language put into his hands. The
introduction of general and abstract terms is always a means of
enlarging the grasp of thought, and increasing the clearness of
reasoning. Space has its three dimensions, its elements of magni
tude and direction ; and although, in one aspect, the simplest of
all possible objects of thought, may yet, for purposes of reasoning
concerning it, be advantageously reduced, by algebraical language,
to the one term of quantity, capable only of flowing in one direc
tion, and being considered as greater or less than a given magni-

22 GEOMETRY AND FAITH.
tude. But the generality thus introduced is made vastly more
general by using symbols which shall combine, in one letter, vari
ous forms and relations in space, defined according to judiciously
selected and easily interpreted laws. Thus, for example, all possi
ble triangles, plane and spherical, and all their properties are im
plied in the single equation, r = pq ; and a similar condensation
of meaning is attained in mechanical science. Another source of
the peculiar power of the calculus arises from the plasticity which
it gives to infinitely rigid space. In experimenting upon a rec
tangular beam, cut from a round piece of timber, we can readily
determine its strength when set edgewise ; but cannot tell what
the strength would have been had the sides been in different pro
portions. The rectangular parallelopiped inscribed in a cylinder
is as absolutely fixed in its dimensions as the hewn timber, but by
expressing those dimensions in language borrowed from the science
of time, we can imagine them changing in their proportions, and
the strength changing with them. Thus we can determine the
precise proportion they must bear in order to give the strongest
possible rectangular beam that could be cut from a round log.
This illustrates, by a simple example, the power given to geome
try by Newton's conception of fluxions, his introduction of the
idea of velocity into the consideration of form.
The appearance of the same algebraic law in the creation, under
the two forms if time and space, has already been alluded to as
proof of unity of design ; the angles of leaves and the angular ve
locity of planets being expressed by the same series of fractions.
Other examples confirm the sublime induction. The elasticities of
gases, strings, and rods are so fundamentally different in kind that
we see no connection between them. The elastic force of the
stretched string we need not determine ; that of the rod, and that

THE CALCULUS. 23
of the gas, can be determined only by experiment,' and when deter
mined they have no very apparent connection or relation with each
other. Nevertheless, each of the three has a peculiar relation to
the force of gravity ; of which it is, nevertheless, entirely indepen
dent. The velocity of a sound traveling in the air, near the earth,
would be, were no heat developed in the action, equal to the velocity
acquired by a body falling from a height equal to that which the
atmosphere would have could it be all compressed to the density
of that near the earth's surface. The velocity of a wave traveling
on a string is equal to that which would be acquired by a body fall
ing from a height measured by the length of the same cord equal
in weight to the tension of the string. And if we take a very
fine glass thread by its two ends, the infinitely varied and beauti
ful forms which it can be made to assume, of waves and folds and
kinks and loops, the figure eight and the circle, are all expressed
in mathematical language by the same forms as those which ex
press the motions of an ordinary pendulum, under the forces of
gravity. The genetic connection, between these forms and these
motions, we do not see, any more than that between the times of
the planets and the angles of the leaves, but the intellectual con
nection we detect, and it leads us to recognize with reverential
awe the presence of Intellect in the disposition of the particles
of both gaseous and solid bodies.

V.

APPLIED MATHEMATICS.
Perfect symmetry belongs only to the ideal, not to the actual.
The algebraic conditions are exactly fulfilled by points of space,
in an invisible and eternal reality ; to this real form, conforming
to the algebraic ideal, the material embodiment makes at least a
rude approximation. The algebraist devises conditions of various
degrees of complexity, delighting chiefly in the simplest ; and
especially in those giving, with simplicity of conditions, the
greatest variety of resulting forms.
The symmetrical forms of nature suggest to man the invention
of laws of symmetry, at first simply to explain nature, then to an
ticipate her work ; leading to new examinations of that work.
Thus the great mathematical sciences have been alternately the
creation and the creators of physical science. The physicists
have been prone to deny that the mathematics constitute a sci
ence; they have inclined to pronounce them only a key to science,
a convenient language wherein to discuss the problems of matter
and motion. The mathematicians, on the other hand, whom we
should naturally consider the best judges of what their own work
is, have declared that geometry is the science of space, algebra
the science of time, and that these are simply the first subjects
handled by the human intellect with sufficient freedom, vigor, and
precision to enable us to draw necessary conclusions. As for
geometry and algebra being mere keys to physics, the mathema-
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APPLIED MATHEMATICS. 25
tician would sooner declare the whole visible creation a mere set
of models and diagrams wherewith to illustrate the laws of space
and time. Whichever of these conflicting views is right, it is un
questionable that the highway to the temple of truth leads alter
nately from mathematics to physics. Observation alone can lead
to nothing, without insight, — without that clearness of inward
vision which sees more than the outward fact, sees the divine ideal
which the fact partially embodies.
Now in this sublime ascent to knowledge the first steps are
easiest, and the way to them has been made exceedingly plain
and attractive. " In the beginning the Creating Spirit embodied,
in the material universe, those laws and forms of motion which
were best adapted to the instruction and development of the
created intellect." The circle and the ellipse are among the sim
plest of figures, defined by the simplest laws. Accordingly the
Creator has strewn examples of the circular form around us on
every side ; and, by the pictured alphabet of the heavens, called
our attention to the consideration of elliptical orbits. When, in the
course of ages, some of the comparatively easy problems of as
tronomy had been successfully solved, problems of more difficulty
were gradually brought into view ; and phenomena which were
not obvious, not pictured alphabet, but the fine print of creation,
led men into the hidden knowledge of optics, electricity, chem
istry, and other forms of molecular physics. The course of
history and of scientific progress has been precisely what it might
have been had God designed to educate men ; to reason with
them and teach them the sciences ; for there has been a con
stant presentation of simpler truths, whereby men have been
led to the acknowledgment of those less obvious ; and this is
essentially reasoning.

26 GEOMETRY AND FAITH.
Four centuries before the Christian era, the mathematicians of
Greece were lured into the study of the conic sections ; and this
prepared the way for the mathematicians of later ages to discuss
fully certain equations of the second degree. These were suf
ficient for all the more obvious phenomena of astronomy and me
chanics ; and as the demand for higher mathematics has been
made by physics, the supply has been granted. The faith, which
prompts the scientific investigator to his labor, he may never have
expressed in words, but his actions show us what it is, — an in
born, ineradicable conviction that the outward universe is intelligi
ble, and shall at some day be understood. But that day ever re
cedes, into the glorious future, as we approach it ; the rate of
scientific progress increases from decade to decade, and yet the
new problems, and the new instruments for their solution, in
crease more rapidly. The Divine Intellect can never be ex
hausted by the human.
A more detailed examination of the history of the separate
sciences would only confirm our conclusion, that, in the selection
of laws under which to subject the universe, God has chosen, for
those things which would first press themselves upon man's atten
tion, those which are most readily interpreted by man's intellect ;
and employed more intricate laws for things which would natur
ally escape man's notice until the state of mathematical science
enabled him to take higher problems ; in which we recognize evi
dence of that kindness and foresight, that care for our education
and our growth in knowledge and wisdom, which is an inspiring
pledge to us that we are indeed children of the Most High.

VI.

MOTION.
The universe about us is in motion. Nothing on which the eye
can fall, or the existence of which the hand of science can demon
strate, is at rest. The sun rises and sets, the moon waxes and
wanes, the very stars are in motion, to the telescopic eye. Clouds
drive over the heavens, and billows roll over the deep ; vapors rise
from the ocean, rivers run to the sea, and the free winds play
around the globe. Plants are ever growing or decaying ; and
animals maintain their waste, or their waste consumes them. Our
modern theories show that the sensible properties in inanimate and
apparently motionless matter, such as temperature, color, weight,
are really modes of motion in the particles of matter ; and this
re-echoes the sublime statement of the earliest seer, that the intro
duction of motion into the universe was the first act of creation.
For, upon a closer examination of motion, and more accurate
investigation of its laws, what do we find ? That the first law of
motion is this : A body, free from external influence, moves with
uniform velocity in a straight line forever. This is the first law of
motion, derived from the widest generalizations, by legitimate in
duction from observations, on an immense variety of motions, in
nature, and in the laboratory. But to what an astonishing result
does this law lead us when we apply it to the case of a body at
rest, the velocity of which is nothing. A body at rest, free from
external influence, would remain at rest forever. In other words,
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28 GEOMETRY AND FAITH.
the first result from the scientific observation of motion in matter
is, that matter cannot move. Hence follows the inevitable con
clusion, that the cause of all the motion in the universe, is some
thing else than matter. Higher than this the investigation of
motion itself cannot lead us ; but this is high enough for a most
valuable stepping-stone.
Why do we ask the cause of motion ? Whence do we derive
the idea that there is a cause for it ? It is not simply the impos
sibility of our imagining a beginning ; the beginning of motion we
often see. But the motions which we most narrowly examine are
those produced by our own will ; we are conscious that our own
volition is the cause of such motions ; and this consciousness is the
foundation of our faith that motion always has a cause. Is this
foundation trustworthy ? Beyond all question it is. Nay, it is
the foundation of all possible physical science ; no man can ex
tend a generalization beyond the particular instances for which he
drew it, unless he leans on this consciousness of causing. To
return to motion, — matter cannot move, our will can move it ;
there is nothing to suggest any other origin for motion than voli
tion ; hence we naturally, and legitimately, infer that the motion
which we see, everywhere in the universe, is produced by a will,
independent of matter, and superior to all the phenomena.
Thus the first law of motion, established in the earliest revival
of science, demonstrates not only the existence of God, but his
perpetual presence and action. Every moving thing in the
heavens, or on the earth, bears the same sort of testimony to his
being and presence, as that borne by the human voice and action
to the presence of a man. Whenever we see anything in motion,
God is the mover. In the ancient tongues this was one of his
names. The winds blow at his command, the sun rises because it

MOTION. 29
is his will, the falling rain and running stream are his gift ; and
each beating pulse, each breath that we unconsciously draw, is
a proof that this machine of the body is, each moment, depend
ent on the sustaining love and power of its Creator.
Since we thus refer all motion, even that in our own frames, to
the will of God, it may be thought that we are destroying man's
freedom, — making him a mere machine, kept in motion by the
Maker's supervision. But this objection to the doctrine of man's
present dependence, forgets that the consciousness of our freedom
is the very basis on which we have built our faith in the existence
of God. It is from our own consciousness of power, to cause mo
tion at our own will, that when the first law of motion has ex
cluded us from ascribing it to powers inherent in matter itself, we
ascribe all motion to his will, rather than to any unconscious
natures. This consciousness of our own power, our own will, may be de
nied in words ; but it will presently betray itself, lurking in the
mind ; it cannot be really denied ; it is the foundation of all phi
losophy and faith. The body, in all its molecular changes, by
which ultimately the free movement of the limbs is produced, is
moved by the will and power of God ; the first law of motion
proves that ; yet the direction of the movement in the limbs is
with man, consciousness testifies directly to this ; man is free, and
cannot heartily believe himself to be otherwise.
Our muscular power is not ours, but it is, to a certain extent,
under our control. We cannot lift a finger without the aid of him
who formed us ; and yet it is we who move our hands. So the
engineman, who has not, in his own muscles, strength to drive a
single loom, yet, by controlling the valves of his engine, keeps the
machinery of many spindles and looms in motion. Thus, with all

30 GEOMETRY AND FAITH.
man's frailty, and his absolute dependence upon other powers, he
yet remains a cause, — free and efficient to control and direct the
engine of his body, wonderfully framed and intrusted to his care.
Our argument has been, thus far, drawn only from the uniform
velocity of motion ; but the second clause in the first law would
lead to the same result. A moving body, free from external in
fluence, moves, not only with uniform velocity, but in a straight
line forever.
As we have no apparent examples in nature of a uniform veloc
ity, so we have none of uniform direction. External influences
perpetually accelerate or retain the velocity and change the direc
tion of moving bodies. But as the first part of the law is derived,
not from actual examples or instances, yet by irresistible induction
from observed facts, so the second part follows by like unavoida
ble inference from phenomena ; indeed, both parts are defended,
by some mechanicians, as axioms, needing no other proof than
Leibnitz's principle of the sufficient reason.
As the varied motions of the universe cannot have sprung from
the action of matter, that being inert, so the constant changes
of direction in the motions prove that the forces, independent of
matter, are still acting. The rebounding of a solid from a solid
shows that the particles of the solid adhere by some form of force
different from a cohesion of contact, — elasticity implies that the
particles are held together by some force which permits their dis
tances from each other to vary within certain limits. When the ball
leaves the muzzle of the gun its path instantly begins to be con
cave toward the earth ; and would be so at any conceivable degree
of velocity. The meteor passing the earth at eighty miles a
second bows to her as he passes. Thus the moon also is perpet
ually deflected from its path by the earth, and the earth by the

MOTION. 31
moon, and both are turned constantly aside from their straight
course by the sun ; and the whole host of heaven is constantly
moving in a rhythmic dance, wherein each star influences the
motions of the whole, and is influenced by the movements of
each of the others.
Our consciousness that we cause motion leads us to ascribe all
change of velocity to force, all force to will. The same conscious
ness bears witness also that all change of direction implies the
influence of will. The weight of bodies, the attraction of gravita
tion, the correlated forces of the universe, these are but reverent
forms of words in which we speak of that which can only be re
ferred to the Divine Will. The untaught man, the poets of the
earlier ages, were more true to reality when they used more re
ligious forms of speech. It is not so much figurative, as literally
true, to say that He who formed the Seven Stars and Orion
still guides them on their way. Their circling orbits by their
figure, and the golden orbs themselves by their motion, continu
ally manifest the presence of His guiding hand. The forces
of cohesion and repulsion, of electrical and chemical change, of
heat, of light, — all of the forces by which the existence of a
particle of matter can possibly make itself known to our human
senses, are but manifestations of the living action of the Most
High. Thus the first law of motion leads us to see God in all things,
and all things as the present creations of his hand. It might
lead us astray, it might lead us to Pantheism, were it not that it
first leads us to perceive that force is an attribute of will, and in
dependent of matter ; thus keeping us to the conclusion that the
Creator and Governor of all things is free, living, — and our
hearts add, good.

VII.

MUSCULAR ACTION.
We \ ave spoken of the human frame as an engine of wonder
ful construction, whase movements are made dependent on the
human will. Yet it is manifest that more of its motions are in
dependent of the will than are dependent upon it. The involun
tary muscles, and the involuntary movements not muscular, are
those which are essential to the very existence of the body. The
circulation of the blood, and its purification through the alternat
ing expansion and contraction of the chest, are obvious instances
of these vital, involuntary actions. Not less important is each one
of a thousand hidden operations, — capillary movements, glandular
secretions, the removal of the effete and the replacing of the liv
ing molecules; to say nothing of more muscular actions, — the
peristaltic motions, and the wonderful unconscious artifices of
swallowing, coughing, sneezing, and the like.
The voluntary muscles are also capable of involuntary action.
This is shown not only by occasional convulsive twitchings, or
more violent convulsions, but, in a still more instructive manner,
by inveterate habit, operating in sleep, or even when the will op
poses. But, although the action which has become habitual is not
done from distinct, conscious volition, the habit is originally formed
by acts in obedience to the will. The law by which the voluntary
action becomes involuntary habit, although it reduces, too fre
quently, man to slavery, is truly beneficent in its design, and in
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MUSCULAR ACTION. 33
its best effects. The simplest statement of the law would, perhaps,
be found in saying that actions which have been associated in vo
lition become associated in execution. In other words, when we
have done several things at the same time, or in quick succession,
the attempt to repeat one of these actions will tend to produce an
involuntary repetition of the others. For an illustration of the
beneficent action of this law, we may take the skillful player upon
a musical instrument, who is conscious of a volition only at the
commencement of each musical phrase : the fingering of the sep
arate notes comes from associated execution. In a familiar piece
his volitions would be even less frequent, being necessary only at
the commencement of a new strain.
This case of the skillful player, being less frequent, seems the
more striking ; but there is scarcely an action in life which is not
aided by the beneficent operation of the same law, just as there is
scarcely a mental action which does not illustrate the kindred law
of the association of ideas. The child, just learning to walk,
makes a painful effort at each successive movement of each mus
cle called into action. It requires all the concentrated energy of
his will to make the successive volitions necessary for simple step
ping from chair to chair. But in a few months he is able, by as
sociated execution, to set in action, by a single volition, a series of
alternate motions, that carry him forward, without his attention, in
a given course, at a uniform speed. No power of will is required
in walking, except when we wish to alter the velocity, or the direc
tion, of our movement.
When the successive movements, dependent on associated ex
ecution, are connected, as in walking, by a law of simple alterna
tion, the case is not difficult of explanation ; and the physiologists
show us how the will relieves itself from duty by a switch, turning

34 GEOMETRY AND FAITH.
off the currents of sensation and command from entering the main
offbe in the brain.
Other cases, in which the operation of the law is no less impor
tant to our comfort and convenience, require, however, much more
intricate combinations of movement. In many familiar occupa
tions we require our hands to guide an instrument rapidly and
freely through outlines of complex; but definite, forms ; as, for ex
ample, in writing, or in free-hand drawing. All men have more or
less of this power to execute ideal figures, or to imitate given
forms. This power has been gained, like that of walking, only
through repeated and laborious efforts. When moving the hand
in one direction, we need a new volition to change its direction,
or to alter its velocity ; hence our first attempts at curvilinear mo
tion produce polygonal lines. In order to produce a curve, as in
the ordinary forms of the capital letters, we must produce, by sev
eral muscles acting at once, motions in several directions at the
same time, each movement varying in velocity according to defi
nite laws. To draw, for example, a circle, by any conceivable set
of muscles operating on the arm, would require at least three sets
of muscles, each acting in a different direction, and no two ex
actly opposed ; and these would be obliged to accelerate and re
tard their action by peculiar laws. The circle, however, is the
simplest of all curves : in the ordinary operations of writing and
drawing, the rates of acceleration and retardation must follow
more complicated laws. Of these laws we think nothing, we know
nothing : we see the curve which we would form, and a single im
pulse of the will sends the pencil along the waving outline.
Charles Babbage, a successor of Sir Isaac Newton in the
Lucasian chair, has won an immortality of fame by inventing a
machiuj which will tabulate in numbers the results of any al^e-

MUSCULAR ACTION. 35
braic law which it may be set to obey. But how much mjre won
derful is this calculating engine of the human body, which is not
confined to arithmetical results, nor does it require that its director
should be learned in algebraical notation to set it at its appointed
task, but which is set by the artist, with his delicate perception of
the beauty of form, to embody his divine ideal ! and it obeys, and
places before us on the canvas, those figures, which, unconsciously
fulfilling algebraic or numerical law, reach far higher, and express
the spiritual thoughts and purposes of the Master. Is not the
Maker of this wondrous engine of the human body worthy of grat
itude and adoration V
Consider how wonderful is the phenomenon of a boy's throwing,
successfully, at a mark. The epicycloidal theories of Hipparchus,
the Newtonian theory of gravitation, the resolution of centripetal
and centrifugal forces, the conic sections of Apollonius, the modi
fications of those curves by the resistance of the air, — all these
are involved in the problem, and must be practically solved, with
considerable accuracy, before the school-boy can give his fellow a
good ball, or catch one on the fly.
It may be observed that the mechanical contrivance by which
the human hand is enabled to go through all imaginable motions,
and strike, at a free sweep, any curve, however complicated or
however beautiful, is an embodiment of one of the most celebrated
of mathematical conceptions, discussed in the writings of Plato
and Aristotle, constituting, in its development, one of the chief
triumphs of Hipparchus, and brought by modern mathematicians,
through the arithmetic of sines, and the canon mirificus of Napier,
into a form capable of reducing to a regular curve the most vari
able and irregular table of observations. In this method of epi
cycles, as used by the modern computer, a series of arms is sup-

36 GEOMETRY AND FAITH.
posed to be carried, each on the extremity of the preceding, and,
during the revolution of the first, each to revolve once oftener than
the preceding ; that is, while the first arm of the series revolves
once, the second revolves twice, the third three times, and so on.
It only remains for the computer to fix the length of these arms,
and determine their original position, in order to make the end of
the third or fourth, or, in cases of difficulty, of the fifth and sixth,
describe any path he wishes. In the human limb, the upper arm
is the first, the fore-arm the second, the hand the third, and the
fingers the fourth, fifth, and sixth of these rotating arms ; and
the fixedness in the ratio of their length is more than compensated
for, by our ability to graduate the ratio of their revolution at will.
Is there no meaning in the fact that the most cunning device of
human ingenuity for making a point travel, under simple laws,
through the greatest variety of paths, should thus prove to be sub
stantially the same with that adopted in the very creation of the
human frame, for enabling the hand to guide its tools with freedom
and accuracy ?

VIII
GEOMETRICAL INSTINCTS.
Since our fellow citizens of all the animal kingdom are, like
ourselves, dwellers in space and time, it is necessary for them,
also, to have ideas of distance and direction in space, duration
and lapse in time. Ideas gained by sense-perception seem to fur
nish them, as us, the data for reasoning ; but ideas of direct intui
tion do not appear to afford to them, as to us, objects whereon to
reason ; but merely serve, as certain of the kind do for us, as the
stimulus of desire, and the guide of unreflective action. These
intuitive ideas, perceived by inward sense, but not, perhaps, dis
tinctly eliminated in consciousness from co-existent ideas, are, in
the lower animals, called instincts ; and when used in like manner
by us, not as propositions for conscious reasoning, but as the
grounds of instantaneous judgment or action, they have the vari
ous names of instincts, feelings, promptings, conscience, or genius,
according to the nature of the objects to which they relate.
Geometrical instincts are common to us with all the animate
races. That instinctive trigonometry, for instance, by which a
child, of a few months old, learns to tell the position of any object,
to which his two eyes are directed, is probably exercised by all
animals with two eyes capable of being turned upon a single object.
The most striking instance is popularly believed to be the young
^uail, which is said to run, as soon as hatched, freely about, peck
ing at minute objects with as true an aim as its mother's. I have
(37)

38 GEOMETRY AND FAITH.
seen a Setter's pup, sired by a Pointer, when a few weeks old,
point at a piece of anthracite with all the accuracy of its father,
which it had never seen. In 1843, a toad, frequenting the gar
den at Burgoyne's Headquarters, in Cambridge, and losing by an
accident the sight of one eye, was for a long time unable to aim
his tongue, with certainty, at the overloaded bees, who, returning,
missed the threshold of the hive, under which the toad, expecting
such misfortunes of his insect neighbors, was accustomed to sit and
await their fall. In time he, like human beings who have had like
accidents befall them , learned to substitute optics for trigonometry,
and instead of solving triangles, with a base and adjacent angles
given, decided on the position of objects requiring a certain focal
adjustment and direction of his remaining telescope.
But how complicated the action by which he proved this, if I
may use the expression, unconscious knowledge of the position of
the bee ! A single conscious volition, and his tongue, which is
rooted in the front of his mouth, with the tip lying far down within
his throat, flies out and back like an electric spark, having taken
the bee up on its tip, and thrust her down the throat of the toad.
His calculating engine, set by the adjustment of his eyes, not only
computes the exact curve in which the tip of his tongue must
move, but the exact force and velocity with which it must be
sent, in order to accomplish its mechanical errand. The same
marvelous unconscious calculation is proved when a boy hits the
mark, at which he aims a stone, or when the expert player at bill
iards strikes his ball on exactly the right part of the ball, in
exactly the right direction, and with exactly the right force, in
order to make it pursue a long course, partly curved and partly
straight, with rebounds from the cushion, and rebounds from other
balls, and come to rest at a determined place. He does not know

GEOMETRICAL INSTINCTS. 39
the difficulties of the problem he has solved ; he does it with as
little of conscious calculation as that with which the toad snaps
up the bee ; but this only renders the more striking the wonderful
perfection of the muscular and nervous organization, as machinery
adapted to describe geometrical figures and solve mechanical prob
lems of great perplexity.
The architectural or nest-building instincts of animals show the
geometrical and mechanical knowledge of the Creator of animals
in a very conspicuous manner. Men invented and used the arch
long before human mathematicians solved its theory. Many other
of our mechanical inventions, and some of them, as the barrel and
the potter's wheel, for example, of a wonderful kind, have an
antiquity that long antedates abstract mathematical thought. We,
reasoning, discover the principles underlying our inventions, and
thus improve science, whieh again suggests new inventions ; so
that human art and human science stimulate and foster each other
to endless competition and endless progress. The lower races have,
apparently, no abstract thoughts, no intuitions, that are brought,
with consciousness, among the data of their reasoning ; in other
words, they appear to have no science, and hence their progress
in the arts is so slow as to appear stationary. But their instinc
tive judgments appear, frequently, more accurate and wonderful
than those of men. To see the republican swallow, coming through
the air, fold her wings at precisely the right moment, and when at
precisely the right speed, in order to enter softly and smoothly
her earthen bottle, makes the art of the most skillful coxswain
seem rude in comparison. The weaving of the bird's nest is in the
case of the African grosbeak carried to a degree of perfection that
vies with that of the nicest works of man, unaided by machinery.
But the architectural work of insects is most wonderful, and

40 GEOMETRY AND FAITH.
none more so than the familiar honeycomb. Always admired by
men, from the earliest ages, it was, at the beginning of the last
century, discovered by Maraldi, of Nice, to embody distinctly the
complicated geometrical conception, of forming cells to contain a
fluid mass, with the greatest strength, the greatest economy of
space, and the greatest economy of material. The paper-making
wasps make rude approximations toward the solution of the same
problem, but inasmuch as the larger part of their material is
cheaply gathered from the surface of wood, there is no call for so
strict an economy. The bee, needing a water-proof material, yet
finding the resin of trees too adhesive to be worked with facility,
confines her use of such resin to the places in which she needs
especial strength, or especial resistance to moisture ; and, for her
ordinary work of cell building, uses a material wholly secreted by
the glands of her own body. Her cells are approximately hexa
gons, which hold more, and have shorter, and therefore stronger,
sides than any other figures which could be packed without waste
of room. They are set, with economy of room, base to base ; and
still further strengthened on the bases, by being set one against a
part of three, so that the bottom of each cell is supported by three
partition walls on the other side. Finally, and most curious, the
bottom of each cell is depressed in the centre to about that degree
which will save most by diminishing the height of these supporting
partitions without increasing too much the area of the floor which
rests upon them. I say about that degree ; and the accordance
of the average cells, in a normal piece of comb, with theoretical
perfection as determined by the calculus of Newton, is very close.
AVe should not expect perfection, because the perfection of the
artisan is to be measured, not by the perfection of his results, but
by the perfection of their adaptation to his end. He were a poor

GEOMETRICAL INSTINCTS. 41
farrier who polished his shoes with the care that a dentist bestows
upon his gold filling. Nor would the bee be a wise economist if
she wasted time in bringing to theoretical perfection her saving of
wax. What the bee's conscious aim is, in the construction of the
cell, we may or may not at some time discover. That she has con
scious aims is evident, from her adaptation of the form of the
comb to circumstances, and from her ingenious contrivances, not
only to repair mischief, but to guard against threatened evil. But
it is equally evident that in the formation of the bee, and in the
inspiration of her instincts, a knowledge and wisdom presided, to
which the whole question of maximum and minimum lay open
countless ages before human thought grappled with its problems.
It only requires a more intimate acquaintance with the habits
of any animal to discover the adaptation of its instincts to its or
ganization. The apparent instances to the contrary arise from
want of patience and thoughtfulness in the observer. I stood, one
evening, at early dusk, watching the movements of a curious insect
on the inside wall of an open shed. Its body, a little over an inch
in length,, and very thin, seemed, nevertheless, too heavy for its
long and delicate legs, which swayed and trembled under the
weight, as it slowly stepped along, with long pauses between each
step. It walked on four legs, holding the other two, which were
shorter and stouter, extended in front. I presently perceived that
it was making toward a fly which had settled, apparently to sleep,
upon the board within three inches of my insect. I wished to see
what its designs were upon the fly, but so slow were its motions,
that I was obliged to wait fully twenty minutes before being grati
fied. As the insect approached the fly, he slowly extended a very
long and exceedingly slender antenna, and touched the fly gently,
in various parts, as if to ascertain more precisely its position. He

42 GEOMETRY AND FAITH.
then made a detour, and brought himself, at length, exactly in
front of the sleeping victim, with his own head nearly over the fly's
head, and began very slowly to raise his raptorious legs high above
the fly. I was growing tired of his slow and awkward motions,
when, in an instant, my feelings were changed to those of the
highest admiration for his great engineering skill ; the fly was
aloft in air, with the beak of the insect thrust into its back calmly
imbibing its juices ; while the fly's feet could touch nothing, its
wings were both dislocated, and firmly pinioned in the captor's
raptorious legs ; which, coming down suddenly between the wings,
had parted them, dislocated them, and pinned them between the
wrist-spurs of those legs and their sharp extremities ; then, with
out an instant's pause, lifted the fly from his feet, and impaled
him upon the ploiaria's beak.

IX.
MOTION ETERNAL IN DURATION.
The motion of bodies is not observed to be with uniform ve
locity. We see bodies at rest beginning to move, and bodies in
motion coming to rest. Let us consider these cases a little more
closely. We have said that motion implies force, and that force implies
will. Force is the energy of will acting upon matter. But how
does the will affect matter which is foreign to will, and over which
will, we might therefore suppose, would have no control ? To this
question we answer that the human will never affects the material
thing which it determines to move, except through the agency of
material agents of whose existence and motions it may be uncon
scious. Not to speak of the unconsciousness of all earnest labor,
the absorption of the mind in its object, take the case in which
the mind is seeking to analyze its volitions, and it will be found
that we cannot reach, in our analysis, the first effect of the volition
upon the physical frame. In the movements of my hand, although
I know it is effected by the movement of certain muscles, and
that this is effected through the nerves of volition, yet I cannot
trace my will behind the command issued to the hand itself.
Hence it may be said that the human will moves even the human
body through physical agencies, and the manner of its control over
these agencies is known only by Him who breathed into us the
breath of life. '43)

44 geometry and faith.
In like manner is it with the natural motions which we see on
every side. All bodies are moved through the agency of other
bodies, and we see nowhere a motion which is not dependent upon
physical causes, that is, which is not produced by physical agents.
Doubtless, He by whose will all things are moved is not restricted
from any mode of action, and He can move bodies independently
of all law, and without any intervention of means. Nevertheless,
such motion would be miraculous, and out of the course of nature.
Our will employs, unconsciously, the aid of nerve and muscle ; the
Supreme Will employs, with wise designs, the intervention of the
laws of impulse, attraction, and repulsion.
But when a body at rest receives motion through impulse, it
evidently continues the motion of the impelling body. So that, if
the impelling body is put to rest by its contact, the motion is not
lost, but only transferred. And this is true independently of the
size of the two bodies. The earth is not immovable, and the small
est grain of dust that falls upon it strikes with a certain amount of
force. Again : when bodies act upon each other by attraction or repul
sion, the force acts upon both bodies. As surely as two vessels,
floating on still water, would both move when one attempted to
draw or push the other, so surely must each moving thing move
those attracted by it, and all that attract it.
This motion, also, exists, whatever be the relative size of the
bodies. In the case of motion produced by direct attraction, —
for instance, in the fall of bodies toward the earth, — the motion
is in simple inverse proportion to the masses. It is, therefore,
capable of easy arithmetical calculation. The weight of the earth
in milligrams may be nearly represented by the figure 6 followed
by thirty cyphers, or six nonillions. Hence the falling of a little

motion eternal in duration. 45
insect, weighing six milligrams, would move the earth the non-
illionth part as much as the insect fell. That is, the thirtieth
place of decimals is capable of representing the motion given to
the earth by the fall of the smallest bodies.
There is, then, in nature, no provision for the destruction of
motion, but only for its transference. Motion in a body free from
external influences is uniform in velocity and direction ; it can be
retarded, and apparently destroyed, only by external influences,
by impulse, or by attraction or repulsion. But these, we have
shown, cannot really destroy ; they simply transfer the motion to
the interfering body. Hence all motion is eternal ; it is communi
cated to an ever-increasing amount of matter ; it is, in each suc
cessive particle affected, less in quantity, but never becomes noth
ing, since the sum of the motion in all the particles remains the
same. It is sometimes affirmed, since the demonstration of the theory
that heat and light are undulatory motions, that all mechanical
motions finally take the form of heat, or of light. The sound of a
tuning-fork, placed upon a mass of caoutchouc, is inaudible, and
a delicate thermo-galvanic test shows that the temperature of the
rubber has been raised. But a still more delicate test, if it were
possible to apply one, might show a quality in this heat that would
demonstrate it to have arisen from a fork ; just as Clairault's nicer
calculation showed the comet of 1770 to have been, before it had
its short period within the orbit of Jupiter, a wanderer outside the
Jovian sphere.

X.
MOTION OMNIPRESENT IN SPACE.
Charles Babbage, in the " Ninth Bridgewater Treatise," has
a chapter concerning the permanent impression of our words upon
the air, — a chapter which none have ever read without a thrill of
mingled admiration and fear ; and which closes with an eloquence
that were worthy the lips of an orator, though coming from a
mathematician's pen.
Would that Babbage had touched, in his fragmentary treatise,
upon some of the inferences which may be drawn from the New
tonian law of gravity, — inferences which would probably have
been as new to most of his readers as those which he, with so
much acuteness, draws from the law of the equality of action and
reaction. The motion of which Babbage speaks, in the chapter to which we
refer, is undulatory, communicated by impulse, and requiring time
for its transmission ; and the startling result of his reasoning comes
from the never-dying character of the motion, keeping forever a
record of our words in the atmosphere itself, always audible to a
finer sense than ours ; reserved against the day of account, when,
perchance, our own ears may be quickened to hear our own words
yet ringing in the air.
But motion is not only enduring through all time, it is simulta
neous throughout all space. The apple which falls from the tree is
met by the earth : not half way, but at a distance fitly proportioned (46)

motion omnipresent in space. 47
to their respective masses. The moon follows the movement of the
earth with instant obedience, and the sun with prompt humility
bends his course to theirs. The sister planets with their moons
are moved by sympathy with earth, and the stars and most distant
clusters of the universe obey the leading of the sun. Thus through
out all the fields of space, wherever stars or suns are scattered,
they move for the falling apple's sake. Nor is the motion slowly
taken up. The moon waits for no tardy moving impulse from the
earth, but instantly obeys. The speed of light which reaches the
sun in a few minutes would be too slow to compare with this.
Electricity itself, coursing round the earth a thousand times an
hour, can give us no conception of the perfectly simultaneous
motions of gravity. There are stars visible to the telescopic eye,
whose light has been ages on its swift-winged course before it
reached this distant part of space ; but they move in instant
accordance with the falling fruit.
True it is, that our senses refuse to bear witness to any motion
other than the apple's fall, and our fingers tire if we attempt to
write the long list of figures, which our Arabic notation requires
to express the movement thereby given to the sun. Yet that mo
tion can be proved to exist, and the algebraist's formula can rep
resent its quantity. The position of every particle of matter at
every instant of time, past, present, or to come, has been written
in one short sentence, which any man can read. And as each
man can understand more or less of this formula of motion, accord
ing to his ability and his acquaintance with mathematical learning,
so we may conceive of intelligent beings, whose faculties are very
far short of infinite perfection, who can read in that sentence the
motions not only of the sun, but of all bodies which our senses
reveal to us. Nay, if the mind of Newton has advanced in power

48 geometry and faith.
since he entered heaven with a speed at all proportioned to his in
tellectual growth on earth, perhaps even he could now with great
ease assign to every star in the wide universe of God the motion
which it received from the fall of that apple which led him to his
immortal discoveries.
Every moving thing on the earth, from the least unto the great
est, is accompanied in motion by all the heavenly spheres. The
rolling planets influence each other on their path, and each is influ
enced by the changes on its surface. The starry systems, wheel
ing round their unknown centre, move in harmony with each other,
and bend each other's courses, and each is moved by the planets
which accompany it in its mighty dance. Thus does this law of
gravitation bind all material bodies in one well-balanced system,
wherein not one particle can move but all the uncounted series of
worlds and suns must simultaneously move with it.
Thus may every deed on earth be instantly known in the farthest
star, whose light, traveling with almost unbounded speed since cre
ation's dawn, has not yet reached our eyes. It only needs in that
star a sense quick enough to perceive the motion, infinitely too
small for human sense, and an analysis far reaching enough to
trace that motion to its cause. The cloud of witnesses that ever
encompass this arena of our mortal life may need no near approach
to earthly scenes, that they may scan our conduct. As they jour
ney from star to star, and roam through the unlimited glories of
creation, they may read, in the motions of the heavens about them,
the ever-faithful report of the deeds of men.
This sympathetic movement of the planets, like the mechanical
impulse given by our words to the air, is everduring.
The astronomer, from the present motion of the comet, learns
all its former path, traces it back on its long round of many years,

motion omnipresent in space. 49
shows you when and where it was disturbed in its course by plan
ets, and points to you the altered movement which it assumed from
the interference of bodies unknown by any other means to human
science. He needs only a more subtle analysis, and a wider grasp
of mind, to do for the planets and the stars what he has done for
the comet. Nay, it were a task easily done by a spirit less than
infinite to read in the present motion of any one star the past mo
tions of every star in the universe, and thus of every planet that
wheels round those stars, and of every moving thing upon those
planets. Thus considered, how strange a record does the star-gemmed
vesture of the night present ! There, in the seemingly fixed order
of those blazing sapphires, is a living dance, in whose mazy track
is written the record of all the motions that ever men or nature
made. Had we the skill to read it, we should there find written
every deed of kindness, every deed of guilt, together with the fall
of the landslide, the play of the fountain, the sporting of the lamb,
and the waving of the grass. Nay, when we behold the super
human powers of calculation, exhibited sometimes by sickly chil
dren, long before they reach man's age, may we not believe that
men, when hereafter freed from the load of this mortal clay, may
be able in the movement of the planets or the sun to read the rec
ords of their own past life ?
Thou, who hast raised thy hand to do a deed of wickedness, stay
thine arm ! The universe will be witness of thine act, and bear
an everlasting testimony against thee ; for every star in the re
motest heavens will move when thy hand moves, and all the tear
ful prayers thy soul can utter will never restore those moving orbs
to the path from which thy deed has drawn them.

XI.
THE SPHERE OF HUMAN INFLUENCE.
The conclusions of the last chapter would need but slight modi
fication, should any future observations reveal the fact that the
motions of gravity are not absolutely simultaneous. If gravity
should prove to be a mere resultant of the undulations of light
and heat, we should gain, indeed, a magnificent illustration of the
inspired wisdom which begins its account of creation with recording
the fiat, " Let there be light; " but we should not lose the spir
itual lessons drawn from the fact that the material universe is
bound by gravitation into one sensitively-balanced whole, so that
each deed of man is felt in the farthest star, and a perpetual rec
ord thereof is kept in the movement of the heavenly orbs.
" Every natural fact is a symbol of some spiritual fact." As
motion is propagated throughout all space, and endures through all
time, so each change in the spirit of each man affects the state of
the spiritual universe, and its influence remains through all eter
nity. As matter by the law of gravity, so spirits by a law of sym
pathetic attraction are all bound in one harmonious whole, whereof
" if one member suffer, all the members suffer with it." Liebig's
law, that a moving particle communicates its motion to adjacent
particles, was announced in defence of a mistaken theory of trichi
nosis, but the law itself is true, and universal in physics, in physi
ology, and in psychology.
The law of attraction holds the same place of primary impor-
(So)

THE SPHERE OF HUMAN INFLUENCE. 51
tance in considering man, as in considering matter. Our great
economist makes the Unity of Law a fact of primal importance in
the development of his grand and cheering theses. The eagle
saint of the Christian Church declares that God is love ; and all
the highest religions teach that man is made in God's image. He
is the sun of infinite magnitude, the origin of all forces, but not
moved by any reaction ; we are the particles moved by Him both
immediately and mediately through each other. Love is the funda
mental law ; the sympathy between human souls is always greater
than the antipathy ; even when, through disturbing forces, the
sympathy is for a time neutralized, and the antipathy is developed
into hatred.
The influence which a man exerts does not cease with the effect
that he has upon his most intimate friends ; nor does it flow from
the power of his word alone, nor from the mere force of his
example. Whatever a man does, or thinks or feels, even in soli
tude, has an effect upon the world. For, in the first place, it
affects himself and his own character; and that character must
influence, in some manner, those with whom he comes in contact ;
influence them in proportion to the strength of his power to affect
them, and to the weakness of their power to resist him. A cheer
ful countenance carries a gleam of sunshine into the darkest alley ;
a sad face throws a shadow over the hearts of those who pass it,
even on a crowded thoroughfare ; thus every shade of thought and
feeling, expressed in the countenance, or in word, or gesture, or
action, produces some corresponding change, slight though it may
be, in all souls that recognize, however dimly, the expression.
And this change transfers itself, in varying proportions, to ever-
widening circles. Thus the spirit and tone of the age is the sum
of the individual thoughts, and thus also the individual character

52 GEOMETRY AND FAITH.
of each man is to some extent the product of all the preceding
ages of the race.
By the manners of a man, or by his speech, we know whether
his companions have been Galileans or Athenians. A close ob
server in the city can tell, of the majority of strangers he may
chance to notice, their age, their character, their calling, the
place of their residence, and of their nativity, or that of their
ancestors. It would only need a nicer observation, a closer in
sight, a more searching analysis, to detect in a stranger's heart
both the original traits of his character and the modifications due
to the influence of all with whom he has been associated. It might
be a task no more above a Shakespeare's grasp, than the creation
of a Hamlet is above the power of an ordinary man, to trace, in a
man's present character, the influence of every person and every
circumstance that have ever acted upon him to repress or to de
velop his powers. It would not require an absolutely infinite intel
lect to trace the effect of any humble act of an honest man, until
it had been seen to have blessed millions of his fellow-men ; nor to
show the loss or suffering that have flowed to thousands from an
evil deed. It may hereafter be possible for some higher intelli
gence than ours to read the record of my interior life, written in
positive or negative characters, upon the soul of some poor man
whom I have never seen, but whom I must, nevertheless, have
helped or hindered by my every act and word.
Thus the spiritual universe is bound, by the law of love, under
its wider enunciation of sympathetic attraction, into one finely-
constituted whole, so that not one heart can throb but all hearts
must throb with it. " There is joy in the presence of the angels
of God over one sinner that repenteth ; ' ' and to every man who
falls into sin, we may say, with deeper meaning than that of the

THE SPHERE OF HUMAN INFLUENCE. 53
prophet, " Hell from beneath is moved for thee, to meet thee at
thy coming."
As by the law of gravity the material universe, and by that
of love the spiritual world, so by the association of ideas the
world of thought is bound into one whole, whereof you cannot find
one thought that is not connected, more or less directly, with all.
Nothing known, nothing thought, nothing done, nothing felt, fails
to leave a clew by which it may be recalled to memory. Each
moment's state of consciousness is connected in a train which
reaches back to the earliest moments of life, and shall reach on
unbroken through eternity, so that it must ever be among the pos
sibilities of memory to recall the thoughts of any instant. And as
the rare occurrence of unusual power, developed by accidental
excitement, suggests hopes of an indefinite increase of power,
when we shall have laid aside this frame, subject to accidents, so
the preternatural manifestation of memory, in certain states of
health, warns us that this possibility of recalling all things may
become an actual reality in the future life.
Then, as the soul surveys the past, with memory presenting its
record of every thought and word and deed, and with an eye
quickened to see the influence which each has had, she may sit in
judgment on her own character ; and the word, which shall be
the final judge, may speak through the soul itself. Then, also, as
she enters the company of cherubim and seraphim, they will need
no record of her good or evil deeds other than that written upon
herself; from their eyes, as from her own, there shall, 'when she
is present, be no past sin hidden, and no good thought concealed.
In the anticipation of standing before such a tribunal, who can fail
to find strength to resist the tempter, and encouragement in striv
ing after good ?

XII.
MAGNITUDE.
In the minds of some who have read certain of the preceding
chapters there has, doubtless, arisen an objection to accepting the
results obtained, — the objection that the results are infinitesimal,
and therefore non-existent. Be minimis non curat lex ; and that,
it may be said, will be the opinion of the court of conscience con
cerning the " permanent impression of our words upon the air ; "
concerning the effect of our motions upon the distant stars ; con
cerning the influence of our character upon the tone of the spiritual
universe. These influences will be infinitesimal, and to a large
part will balance and destroy each other.
In reply to these objections, I will begin by observing that the
balancing of two forces is a real effect in nature, not to be for a
moment confounded with the non-existence or destruction of the
forces. If the earth do not rise to meet this falling rain-drop, it
demonstrates that another drop, or its equivalent, is falling on the
other side of the globe.
Let me also concede, at once, that, in the form in which the
arguments have been put, there is an assumption of certain laws
of physics, which, being laws deduced from observation, may be
subject to perturbations not yet discovered. Thus Babbage, in the
chapter alluded to, assumes that the wave of sound runs friction-
less through the air, the heat developed by the compression bein<»
absorbed in the expansion. But the experiments of Uriah Boyden
(54)

MAGNITUDE. 55
have since shown that there is a slight amount of heat developed
by the friction of the wave ; and this would slowly, but constantly,
diminish the force employed in propagating the sound as sound.
Again, I have assumed that gravity is a force acting at a distance,
and requiring no time whatever for its transmission ; but it may
possibly be hereafter shown that its speed is not thus absolutely
infinite. With regard, however, to the main objection, that the infinitesi
mal may be neglected, the objection appears to me not valid, and
to arise from the weakness of the human imagination. " Time and
space are great only with reference to the faculties of the beings
which note them." In space and time themselves there is no
natural unit, or scale of magnitude ; these are given only by
thought, manifesting itself through material phenomena. Hence
every scale of magnitude is relative to the mind employing it, and
it is only the Unlimited Mind that can be free from the fetters
imposed upon thought by the scale employed.
In some of the nebulas we have examples of the spira mirabilis
of Bernouilli, drawn on a gigantic scale, so that the part visible to
the telescope is many millions of leagues in extent. This spiral
may be drawn, on a smaller scale, by tracing upon a circumpolar
map the path of a ship keeping steadily upon any one course, not
to a cardinal point. Let us imagine this map extended until its
radiating meridians stretch out among the stars, and let us trace
upon it a spira mirabilis, which, beginning at a distance of one
billion kilometers from the pole, is running thirty degrees north of
east. The length of this spiral will be two billion kilometers, and
it will make an innumerable number of revolutions in reaching the
pole. Let us, next, imagine that we have come in upon the spiral,
until we are but one kilometer from the pole ; the length of spira]

56 GEOMETRY AND FAITH.
yet remaining will be two kilometers, and the number of revolu
tions still to be made around the pole will remain innumerable.
Let us again approach until we are within one millimeter from the
pole ; the remaining length of spiral will be two millimeters ; but
the number of revolutions yet to be made about the pole will still
be innumerable. This little central part of the spiral, all lying
within a circle two millimeters in diameter, will be precisely simi
lar in shape to the whole spiral, two billion kilometers in diameter.
If it were possible to look at this central part with a lens that
should magnify a quadrillion times, it would appear precisely of
the same size and shape as the whole spiral. By your approach to
the centre, the scale alone would be altered ; the central part of
the spiral would be a reduced picture of the whole ; its linear
dimensions would be the fifteenth place of decimals of the dimen
sions of the original spiral.
That is, it is the nature of the spira mirabilis, that whether you
move inward toward the pole, or outward away from it, the part
between your position and the centre remains always exactly of
the same shape, differing only in scale. Let us then approach to
within the hundredth of a millimeter from the pole ; the central
portion becomes now too small to be seen distinctly by the naked
eye ; but it is of same shape as the whole ; and although its total
length is only the fiftieth of a millimeter, it still makes an infinite
number of convolutions about the pole. On the .other hand, if we
should run out to the distance of a trillion kilometers, we should,
probably, reach the distance of the furthest star visible to the
naked eye. That is to say, the unaided eye enjoys a range of
vision through about twenty places of decimals in linear dimension.
And if we should revise Archimedes' calculation, on the number
of sand- grains requisite to bury the universe, we should, conse-

MAGNITUDE. 57
quently, see that sixty places of figures will express the number of
grains of the finest silt requisite to fill all space, out to the stars
of the sixth magnitude. The sixtieth place of decimals is, there
fore, not zero ; it is the ratio of the space occupied by a minute
grain of fine sand, to the space in which the fixed stars lie. In
that minutest space the same forms may lie concealed as those
which are illustrated in the whirlpool nebulae, and which might be
conceived as filling vaster spaces. Twenty places of figures, in
linear dimension, carry us beyond the limits of sight ; but space
does not end there ; and we might affix figures forever, without
arriving at a magnitude so great as to be impossible.
Turning our thoughts again to the minute central portion of the
spira mirabilis, we could, by the aid of the microscope, see an
object whose linear dimensions would stand in the seventh or
eighth place of decimals of a meter. In that central speck, visible
only to the best microscope, the wonderful spiral would still exist
in all its perfection ; still making its proper angle with a line to
the absolute centre ; still being in its fixed proportion of length
to the length of that line ; and still making an infinite number of
convolutions around the pole. And this would hold could we, by
the aid of more and more powerful lenses, continually approach
the centre until our distance from it stood in the ninth or ninetieth,
the nine hundreth or the nine millionth, place of decimals. We
may write cyphers, after the decimal point, at the rate of nine
million a second, for nine million centuries, but when we finally
write a significant figure, that figure is not a cypher ; it is signifi
cant ; and if it signifies the distance which yet remains between
our moving point and the pole which it is approaching, then, incon
ceivably small as the distance is, it has its relations : it is one-half
the length of the remaining portion of the spiral ; and that portion

58 GEOMETRY AND FAITH.
of the spiral, nearly as it may be without any length or size, still
makes its infinite number of convolutions about the pole.
It has been inferred that, because the scale is thus capable of
indefinite expansion and contraction, without any destruction of the
form, or law of the curve, the contraction might proceed to reduce
the spiral into the absolute point at the centre ; and that afterward
the point might be considered as an absolute nonentity, without
destroying the spiral. Thus space and time, it has been said, may
be shown to be purely subjective.
But the failure in this attempt to demonstrate the subjective
character of space is twofold. A point is not a nonentity ; it is a
zero of magnitude ; yet it has position, or is a position. It is not
in the mind, it is in space, fixed in its position, although without
magnitude. But although thus real, and indestructible in space,
yet being without dimensions or parts, it is incapable of containing
a curve except by a figure of speech ; by which we either attribute
to the point the potentiality of the subjective law ; or else speak
of a point when we simply mean an infinitesimal space. As we
diminish, for example, the scale of the spira mirabilis, — by run
ning in upon it towards the centre, and considering only the part
yet remaining, which is always similar to the original whole, —
it remains real, so long as we have not arrived at the actual
centre ; but, when that point is reached, the spiral has vanished ;
there is nothing remaining between us and the pole, for we are at
the pole. That pole does not then become subjective ; it is a real
point ; but it does not contain the spiral, except potentially.
In dealing with the infinitesimal and the infinite, the practice
of geometers varies ; some delight in stating truths in forms which
seem false and self-contradictory ; others carefully avoid such
forms. Both classes are liable to error ; the impossibility of

MAGNITUDE. 59
clearly imagining the infinite and the infinitesimal is not destroyed
by any forms of language ; and the difficulty of reaching true con
clusions is not necessarily increased by the use of forms of speech
concerning them, literally incorrect. The calculus of Leibnitz is,
in a majority of its propositions, literally false ; yet it always leads
to true conclusions, in the hands of one who can distinguish
between the letter and the spirit.
The best geometer can, however, err ; as, for example, some
have said that the point approaching the pole, in a spira mirabilis,
can never reach the pole, because it will always have an infinite
number of revolutions to make before reaching it. The premise
is true, but the conclusion false. The point will always, until it
reaches the pole, have an infinite number of revolutions to make
before reaching it. But if the motion in the spiral be with uniform
velocity, then the revolutions will be with increasing velocity ; and
finally with infinite velocity ; so that the infinite number of revolu
tions will be accomplished in a finite time. To take the particular
example we have been considering, — the point moving at a con
stant angle of sixty degrees with the line to the pole, — if the
point moves uniformly at the rate of a slow walk, say four kilo
meters an hour, then it will reach the pole in precisely one hour
from the time that its polar distance is two kilometers. But, while
the motion of the point is uniform, at four kilometers an hour, and
its approach to the pole uniform, at half that rate, the angular
velocity constantly increases, being always such that (if it could
remain uniform) it would carry the point around the pole in about
three and five-eighths the time that remains of the hour. Thus,
when at the distance of a meter from the pole, the revolution
would carry it around nine times a minute ; while at the distance
of a millimeter the revolution would be at the rate of one hundred

60 GEOMETRY AND FAITH.
and fifty circuits a second ; the rotation thus increasing in rapidity,
in direct proportion to the approach to the pole. But in less than
the one five-hundredth of a second the hour has expired ; the centre
is reached ; the whole spiral has been passed over, and the point,
having made innumerable revolutions, is at the pole. The pole
does not contain the spiral ; although it may be said to contain it,
in the sense that the spiral, with an infinite number of coils, is con
tained in any portion of space around the pole, however small, even
if smaller than any portion that can be measured, named, or im
agined. Thus the point may be said to contain the spiral poten
tially, so that the spiral would become actual, could the point be
magnified. This illustration of the logarithmic spiral has been chosen and
fully expanded, not only because of its peculiar adaptation to the"
illustration of similarity, and its curious property of always having,
even when reduced to an infinitesimal size, an infinite number of
coils, but for the historic associations with that curve which Ber
noulli would fain have had carved upon his tombstone. The same
conclusions would be reached did we consider the shrinking of any
other forms in just proportion. It is conceivable that the entire
universe might be altered in size, and if proportional changes went
on, in every part, in all the forces acting upon it, and in the facul
ties of all creatures, it would not be in the power of human beings
to discover that the change was made. Were the universe thus
reduced to the twentieth place of decimals in linear dimensions,
and the requisite changes made in the forces of nature and in the
faculties of man, the whole stellar system would be contained in
the space now occupied by a grain of sand. A second reduction
might take place, and a third, and so on forever ; and, unless the
rate at which the changes were made increased, the universe would

MAGNITUDE. 61
gtill remain to its inhabitants as large and grand as before. But
let the rate increase so as to bring the universe to a point, and its
actuality would be gone, and its potentiality alone .remain. Still
that potentiality would be objective to the Creative Mind ; objec
tive in the point. Sweep the point away, and the universe would
exist only, as in Erigena's " Second Division of Nature," subjec
tively in the Divine Thought.
We thus reach sublimer conceptions of the immeasurable gulf
between the human and the Divine Mind by holding to the
veracity of that intuition which pronounces space and time inde
structible objective entities ; and renouncing the philosophic wis
dom, made popular by Teufelsdreck, which accounts them mere
modes of human thought. To the human mind there is no unit
of space or of time, save those given in the creation ; we cannot
even imagine any unit not thus given. To the Divine Mind alone
belongs the possibility of deciding on the scale of creation, and
deciding what men shall consider large or small, brief or lasting.
As the human eye requires the aid of the telescope at one end of
its range of vision, and the microscope at the other, so all our fac
ulties are limited to that which is neither too great nor too small
for us. But, in regard to the works of nature, we neither discover
limits, nor are we compelled by any mental necessity to suppose
that there are limits. The bounds of the universe are independent
of the weakness and limitations of our powers of imagination.
There is, therefore, no impossibility in the speculations of Lov-
ering, published in the " Cambridge Miscellany ' ' in 1842 ; that
the atoms of our universe may be stars and suns of a smaller one,
composed in like manner of infinitely smaller stars and suns ; while
our constellations and solar systems may constitute only molecules
in a vaster world. The infinity of space would almost seem to

62 GEOMETRY AND FAITH.
demand such an arrangement to utilize its wastes. The human
mind, fettered by the body, seems in such speculations to show its
kindred to the Infinite Spirit, to whom
" There's nothing great appears,
. . . There's nothing small."
And these speculations are not confined to a few learned men,
whose studies lead them naturally to the theme. Fifteen years
before the publication of Lovering's paper, I myself, a child, heard
other children supposing that this universe might be the atoms in a
crumb let fall by a giant ; and that the slow precession of the
equinoxes might be the rotation of that crumb, in the air, as it fell
to the ground, in an infinitely larger universe.
The true greatness of a work is in the thought which it embodies,
not in the scale on which it is wrought. The idea or law of the
spira mirabilis has a fascinating beauty to a geometer ; but to such
a mind it is a matter of perfect indifference whether that spiral be
illustrated in the minutest shell or in the largest nebula. I have
been sometimes as much moved at a slight wood-cut outline of a
mountain range as at the sight of the vast masses themselves. The
Dead March from Saul will express grief on a grand scale, a sense
of human weakness resting in unshaken confidence on the Divine
strength, whether played on a single instrument or with a full
orchestra. Our speculations on the scale of magnitude become, therefore.
unimportant. Whether there are, or are not, infinitesimal world.
included in the atoms of our worlds, and infinite worlds in which
our systems are atoms, the grandest and most inspiring object for
our contemplation is the law, plan, or thought, on which the uni
verse, within reach of our faculties, is built. We need not reduce

MAGNITUDE. 63
it to a point, or dissolve it in ideal subjectivity, in order to show
its unity. The universality of gravity, the co-extensive universality
of light, which by the spectroscope has shown that some, at least?
of our chemical elements are universally diffused, the correlation
of forces, the adaptation of all parts of the universe to each other,
all tend to confirm the sublime conclusion that the universe is the
expression of one infinitely complex, yet infinitely simple, connected
thought, in which all was foreseen, and all comprehended at a sin
gle glance by the Intelligence which framed it. The aim of science
is to develop and trace the connection of the parts of this intel
lectual whole ; the end of religion is to interpret for the heart
and soul the lessons given through this intellectual form.

XIII.
CHANCE AND AVERAGE.
When two phenomena arise from entirely independent causes,
the relation of one to the other is said to result from chance. The
disposition to consider chance an actually existing cause is so
great, that men have, in all ages, personified, and in some nations
even deified it.
In the highest contemplation of the universe, as the realization
of one grand conception of the Divine Mind, it might be thought
that the idea of chance would be excluded, because all phenomena
would then be regarded as springing from a single cause ; all the
minutest events would be considered not only as foreseen, but as
intended ; as the necessary results of the original thought made
actual in the universe.
But the idea of chance, of relations in events springing from
independent causes, is so positive in its character, that we are
unwilling to concede it to be a mere result of the weakness of the
human mind, of our inability to rise to a habitual contemplation of
one First Cause. It seems more like a direct gift of power ; a
power to apprehend some really occurring phenomenon in nature.
As such, it forms the basis of a distinct and valuable calculus,
applicable to important economic questions of assurance and annui
ties, and to weighty scientific problems, as a test of hypotheses,
and a criterion for rejecting doubtful observations. The successful
application of this calculus of probabilities to so many actual prob-
(64)

chance and average. 65
lems in the universe is a demonstration that, however difficult it
may be to reconcile the conception with our ideas of ' ' foreknowl
edge absolute " in the single Creative Will, we must, nevertheless,
admit into our theory of the world the conception of independent
causes, leading to what may be justly called accidental results.
The reconciliation of this contradiction, so far as reconciliation is
possible in our finite minds, is probably to be found in the con
sideration of averages. In our human work we frequently act
upon a multitude of individual objects, without special designs in
regard to each, but with a general regard to the average action
and to the total result. The sower does not consciously choose
the position in which a single grain of his wheat shall fall, yet
designs and accomplishes an even cast of a given quantity of seed
to the acre. The causes which determine the position of each
grain are so numerous, and their connection so remote, that they
may be considered, for one grain, independent of those for another.
In throwing a die repeatedly, in like manner, the causes which
determine its position after one throw are so numerous, and so re
motely connected with those that determine its next position, that
they may be considered independent. Yet the throws are so
governed by our will that we may decide, beforehand, how many
to make in each minute, — and the positions are so determined by
the shape and material of the die, that if it be a homogeneous
cube the tendency will be, as the throws are multiplied, to have
each side uppermost one-sixth of the time.
This is the law of chance, as applied to averages. And as
chance has been personified, and even deified, so average has, by
some writers, had divine powers ascribed to it. It has been gravely
asserted that the saving of a man from criminal courses only drives
another man into crime to keep up the average ; as though the

00 GEOMETRY AND FAITH.
present average had an inherent power to perpetuate itself; as
though dice could not be loaded without producing a counter-
loading in the other player's dice ; as though the sower could not
vary the size or cast of one handful without immediately varying
another handful to keep his field from having more or less seed
upon its surface.
The average is a result, not a cause. It is the result of rela
tions that exist between the various causes producing the effects,
and may be changed at any time by interfering between those
relations ; by the dice being loaded, or the sower walking at a dif
ferent pace ; by Jenner's introduction of vaccination, or the dis
covery of America putting quinine into the physician's hand ; or
John Howard visiting the prisons, or the Apostle Paul receiving
his commission. Great changes thus take place in natural aver
ages ; and small changes may be made at any moment by the
action of the human will.
It seems not unworthy our highest conceptions of the divine plan
to suppose that certain groups of phenomena in nature may, like
the sowing of seed by man, be directed and intended for average
results, without special design for each individual case. The im
passable gulf between the finite and the Infinite Mind would still
remain, in the ability of the Divine Providence to select at will
any one of the innumerable cases, and employ it as a means to
higher and further ends. The winds from the Mediterranean, for
example, bring a fixed average of vapor to the summits of the
Alps, where it is showered down in countless crystals of snow.
These, under slight changes of temperature, contract into minute
globes ; and these particles of ice, piled up in the mountain basins,
press themselves into an almost solid mass, and push themselves,
or their earlier companions, down the valleys, grinding off the

CHANCE AND AVERAGE. 67
rocks into powder, which is washed by the melting ice into the
rivers and into the sea. The magnitude of these glaciers is limited
by the quantity of snow ; and, in its turn, limits the quantity of
gravel and sand, and the size of the boulders formed by them.
To these results, a thousand causes which I have not mentioned
conspire. A theist, believing that the glacial system of the mod
ern Alps is part of a divine plan, is not obliged to suppose that
this plan includes the position of every individual grain of silt, in
the ocean bed, brought from the Alps ; is not obliged to make this
supposition, even if to his theism he adds faith in a special Provi
dence, and thinks that a grain of sand may be a providential
instrument in effecting some great result.
But whatever our explanation of the occurrence of so-called
chance among the averages of nature, these chances and averages
are frequently adapted to each other with a harmony that seems
to admit of no other solution than a reference to the Divine plan
which fits each to all and all to each. Enthusiastic students of the
calculus of probabilities sometimes represent all human judgments
as the result of a calculation of chances ; and our certainties are
said by them to be merely propositions, the truth of which is in
finitely probable. Many of the arguments of natural theology, so
called, can be very conveniently put into this form. In the forma
tion of planets around the sun, according to the nebular hypothesis, '
the chances were small against an order which should fail to pre
serve the stability of the system ; and the present harmony of dis
tances must be referred, directly or indirectly, to presiding thought.
In the formation of the solar system, the chances were small that
this particular planet should have its elements mingled in precisely
that proportion which has resulted in so full a development of life
and of human activity; and the arguments of Prof. Cooke's " Re-

68 GEOMETRY AND FAITH.
ligion and Chemistry" derive from this consideration a demon
strative force.
In the course of this successive development of vegetable and
animal life upon the earth, there has been, with frequent muta
tion, a general permanence. Scientific speculation at the present
time busies itself with the question whether the permanence has
been real, and the changes sudden ; or whether the stability is
seeming, and the mutations have always been going on with
stealthy step. Whichever of these theories proves to be true,
that of Plato, or that of Democritus, there is a seeming stability
in the present species, which have lasted without sensible change,
except the extinction of some kinds, for thousands of years. Ac
cording to the theory of Democritus, as revived in our days, this
arises from the fact that the species at present existing are the
fittest for the existing epoch, and thus survive. According to the
rival theory this fitness arose from no blind struggle for life, but in
accordance with a Divine plan, fulfilled by divine power. In ordi
nary cases the judgment may possibly remain suspended, whether
to suppose the Divine Will acted in reference to the perpetuation
of a species by some general law, covering many species, or by
special adaptations to one. These cases may therefore be dis
missed from the argument. If we grant that a blind evolution by
natural variation and survival of the fittest will explain them, it
must also be conceded that an intelligent adaptation of the organ
ism to its medium will also explain them. But there are other
cases, in which the imagination runs riot in vain for any " suffi
cient reason ' ' outside of the will and purpose of the Creator, ful
filling an original plan. These cases are like " experimenta cru-
cis," — the theory that fails to hint at a possible explanation fails
to explain the universe.

CHANCE AND AVERAGE. 69
Such, it appears to me, are the cases in which the fecundity of
a creature is in inverse proportion to its chances of life. I would
by no means say that these are the only points in the animal and
vegetable economy which the evolution theories appear to me to be
utterly incapable of explaining ; but they are the cases which fall
under the head of average and chances, and demonstrate that the
Eternal Thought which planned this present world comprehended
all and more than all which is included in our modern calculus of
probabilities. If the ovaries of the Dodo contained one thousand ova, and if,
on an average, less than one of these grew into an adult female
Dodo, with equal chances of propagating . her kind, it is evident
that the Dodo must become extinct. If, on the other hand, two
of these ova were impregnated, and came to maturity, it would
take but a few generations of the bird to cover the earth, and
exclude all other beings. This is prevented, it may be said, by
the struggle for life. But the fecundity of each species must be
exactly proportioned to the chances of failure in that struggle.
The horse-hair eel is said to lay several millions of e^gs ; let us
say five million. Why this enormous fecundity ? Because the
chances of the eggs coming to maturity, as eels, is so small. In
order to keep the species in existence, two in five millions (if the
sexes are of equal numbers) must succeed in escaping all the
dangers which beset the eggs and the young in the brook, and
then succeed in finding, near the brook, crickets or grasshoppers
into which they may penetrate. These grasshoppers must escape
their enemies, and survive the depredations of the hair eels, until
the latter reach maturity, when they must escape near enough to
a brook to find their way there, and meet hair eels of the opposite
sex. The chances are two in five millions, let us say, and the

70 GEOMETRY AND FAITH.
creature lays five millions of eggs. Did she average but four
millions, the race would in a few years become extinct ; did she
average six, the creatures would multiply in a few years beyond
all bounds. The permanence of the species for so many years
demonstrates the accuracy with which its fecundity is proportioned
to the slimness of its chance in the falsely-called struggle for life.

XIV.
PHYLLOTAXIS.
The reader may be interested in a more detailed development
of the arguments briefly alluded to upon pages 5 and 6, 35 and 36.
There is a rule in arithmetic called the rule of False, or the
rule of Position. It is the most general, and the most useful of
all rules in the art of computation. Its method renders it appli
cable to every problem in which the accuracy of an answer can
be tested. The rule may be briefly stated as follows : Guess at an
answer, and test by numerical computation the accuracy of its
results. If the results are accurate, the answer stands the test
and was correct. And, if the results are not accurate, the error
affords data for estimating the error of the assumed answer ; and
making a second better guess, to be tested as the first was ; and
thus to afford data for a third guess, still nearer the truth.
This method of hypothesis and verification is applicable not
only to arithmetical problems but to all questions of practical
science. It may even be said to be the general rule by which the
finite mind approaches every truth which it can approach ; and
thus reaches every truth which it can reach. In some minds the
process is rapid and with a very obscure and evanescent con
sciousness of its operation ; in others slower and with distinct
knowledge of the steps ; and in all minds the rapidity and ease
of the process vary with the nature of the problem. But in
regard to ail phenomena, which require a cause or theory to
account for them, we are not able to draw the theory out of the
facts ; the theory comes from our own minds ; we place it among
71

72 GEOMETRY AND FAITH.
the facts ; and then adopt, modify or reject it, according to its
agreement with the facts and ability to explain them. Some
times the theory explains the facts perfectly, but goes no further;
as, for example, when we assume a centre for a circumference
passing through three points ; the final verification of the centre,
by proving it equidistant from the points, gives us nothing fur
ther to do ; we have found the centre, that is all. But, in more
complex cases, it frequently happens that the theory proposed
and tested, for the explanation of one set of facts, proves to be
also a satisfactory explanation of other facts ; its value and
authority as a theory are thereby immediately greatly enhanced.
Nor is it in the physical sciences alone that this increase of cer
tainty and value may be found ; the operations of the intellect are
similar, on whatever class of objects it is operating; and theology
itself may be lawfully treated, to a certain extent, as Spinoza
treated it, by mathematical methods.
Thus Maupertuis assumed it as a fundamental principle that
the Divine Being, being unerringly wise, would waste no energy ;
that everything in nature must therefore be done with perfect
economy of force. This theological dogma is called, in mechanics,
the principle of the least action ; and is, in that science, an invalu
able and fruitful principle ; as may be readily shown even to
those not skilled in mathematical analysis. Take for example the
mechanical theory of light. How shall light move in a uniform
medium, according to this principle of the least action? Evi
dently in a straight line, since that is the shortest distance between
two points. How shall light be reflected from a polished surface?
Evidently so as to make the sum of the incident and reflected
rays the shortest possible in going from the point giving light to
the point on which the reflected ray shines ; and a simple calcu
lation shows that this requires the incident and reflected ray to
make the same angle with the reflecting surface. How shall light

PHYLLOTAXIS. 73
be refracted in passing from one medium to another? Here,
again, we must have the sum of the incident and refracted ray
such as to make the whole power expended as small as possible ;
and a simple calculation proves that this requires the sines of the
angles of incidence and reflection to be in proportion of the dif
ficulty of moving in the two media. These instances show the
usefulness of the principle of least action in physical inquiry. These
results concerning light are amply confirmed by experiment ; and
shown to give the manner in which a ray of light actually behaves.
But when we turn the argument about, and from this agree
ment of the experimental results of observations on light, with
the principle of the least action, would argue that the creator of
light uses a perfect economy of force, and is therefore unerring
in wisdom ; we feel at once that the argument is not absolutely
conclusive. It creates a presumption, but does not force a con
viction. We see that the motion of light in a straight line, reflec
tion at an equal angle, and refraction by Snell's law of sines,
involve perfect economy of force ; and yet these may be neces
sary results of the constitution of the luminiferous ether, not
foreseen when the ether was constituted. In order to make the
argument from morphology or from teleology conclusive, the
instances of the divine thought and forethought must be such as
cannot be explained from mechanical or mathematical necessities.
The fact that Maupertuis's theological doctrine of the divine
economy of force has led, or could have led, to such numerous
discoveries of the actual laws of physics, certainly creates a
strong presumption in favor of its theological truth. It shows
that Bacon's sneer at barren virgins consecrated to God is wholly
uncalled for ; here is a religious tenet, a purely theological doc
trine actually giving to mathematicians and physicists more
knowledge of the universe than they could get from observation
of nature, without its aid.

74 GEOMETRY AND FAITH.
There are various other points in which this theological hypothe
sis of an infinite wisdom directing the world can be subjected to
tests ; and in some of them it stands the test, so perfectly as to
rise rapidly toward a settled and demonstrated theory; even
upon these lower grounds of the understanding, and independent
of the intuitions of the higher reason from which Maupertuis and
the theologians announce it. Let me briefly touch upon two of
them. In the early history of astronomy it was assumed, that the law
of the movement of the heavenly bodies must be a perfect law.
The planet, swinging free in space, is subject to no interruption
from finite hindrances, but moves under the influence of universal
law alone ; therefore its motion is perfect. This grand theologi
cal conception is worthy a place beside Maupertuis's principle
of the least action. But, in attempting to verify this hypothesis,
the ancient astronomers made another assumption not so fortu
nate and trustworthy ; the assumption that circular motion is the
only perfect motion. They assumed that the orbit of a planet is
circular. But on putting this to the test it failed to account for
the appearances in the heavens. Still this did not shake their
faith in the perfection of circular motion. They only thought
that there must be a combination of circles. Instead of a single
arm carrying the planet, they put an arm upon the end of an
arm ; and conceived it as rotating twice, while the first arm was
rotating once. If the second arm be very short, in comparison
with the first, the path of the planet carried by it would differ
but slightly from a circle ; while if the arms are nearly equal in
length the path would be very different. But they soon found
that no imaginable proportion between the lengths of the two
arms would enable them to represent well the planetary motions.
Still adhering to the circular movement, they place a third arm
at the end of the second. This third arm was to revolve three

PHYLLOTAXIS. 75
times, while the second revolved twice, and the first arm once.
By giving proper proportions to the lengths of these three arms
the orbit of any planet could be described with tolerable accu
racy. By the addition of fourth and fifth arms, to revolve re
spectively four and five times, while the first revolved once, the
positions of the planets for a single revolution in their orbits
could be given with the utmost nicety.
These epicycles of Hipparchus are no longer used in astron
omy ; but they have been used in modern days, in other depart
ments ; in what might be called the statistics of physics, —
tabulated observations. It has been shown that with from three
to six arms, rotating each with a rapidity proportioned to the
number of the arm, beginning at the centre ; the end of the outer
arm may be made to move in any path required, provided we
may fix the length of the arms as we choose, and put them in
what position we choose, at the beginning of the motion. And
it is justly esteemed a grand work of Hipparchus and of modern
analysts (including our own Peirce) that they should have invented
and perfected by two thousand years of study so remarkable a
result, as that of describing any outline whatever, by the simple
device of rotating, simultaneously, three or more radii linked
together by their ends. Yet in nature a similar device had been
used in a rude form at least from the eozoic ages ; and in a per
fected form, even better than that of Hipparchus and incompara
bly more practical and rich in results, from the very advent of
man upon the planet. The difference between the epicycles of
Hipparchus and those of nature, is that in the former, the ratio
of the rapidities of rotation is fixed, and the ratio of the lengths
of the rotating arms is varied at pleasure ; while in nature it is
the proportion of their lengths that is fixed, and the proportion
ate velocity of rotation is varied at pleasure. For what is this
right arm and hand of man but a linked series of radii in which,

76 GEOMETRY AND FAITH.
from the mechanical necessities of the skeleton, the length of the
spokes is fixed, but the resulting stiffness is much more than
compensated by the variable proportions allowed to the veloci
ties of rotation. The end, free motion in any direction, could of
course be attained without a skeleton ; as for example the tip of
the human tongue can readily be trained to move in any conceiv
able path within the buccal chamber. But when strength and
dignity have been given to the frame by a skeleton, and mobility
by articulations, the articulations are exquisitely adapted to the
psychic needs of the species. No mechanical necessity can be
conceived as producing this adaptation ; nor can I see the proba
bility, or even possibility, of this adaptation having been pro
duced by the mutual reaction of the psyche of the animal and its
environment. In the case of man, his immensely varied mental and spiritual
powders and capacities ; his need of actions incalculably more
varied than those needed by the brute; whether we regard his
movements as aimed to produce physical or mental effects ; require
that he should have much greater freedom of movement under
definite control. And this is accomplished for him, partly through
the perfection of his general form, so " express and admirable ; "
but more particularly by the epicycloidal movements of the hand.
The shoulder is the centre of these movements, but it is not
rigidly fixed, it has a proper motion of its own by the sway and
torsion of the trunk, by the movement also of the lower limbs,
by which the whole body has a capability of motion. But assum
ing, for the moment, this moveable shoulder-blade to contain in
its socket the centre of the epicycloidal movement of the hands;
we have first the humerus, swinging freely in all directions,
within a cone of about 120° ; which is accomplished by what is
called a ball and socket joint. The second link consists of two
bones in the fore arm, hinged to the first link by a hinge joint;

PHYLLOTAXIS. 77
but this joint receives an equivalent for ball and socket freedom,
by the humerus rotating about 90° on its own axis. The third
link is the metacarpus, jointed to the arm by a compound hinge,
which, however, has an equivalent for ball and socket freedom by
the peculiar semi-revolution of the two bones of the forearm
about each other. For ordinary movements these three links
suffice ; the phalanges being kept in a fixed position with reference
to the metacarpus ; just as three arms suffice for the easier curves
in the Hipparchian epicycles. But for the nicest delicacy there
are the three joints of the phalanges, perfecting the instrument,
just as additional short radii are sometimes needed to perfect the
epicycles. Every one is familiar with the fact that a trained hand can
sweep, with a crayon, any outline or figure whatever upon the
blackboard. What I am endeavoring to show is, that this ability
is furnished by an adaptation of the skeleton and muscular system
to freedom of motion, and to freedom under control ; by a system
similar to the Hipparchian epicycles, famous in astronomy and
physics. In fact, every motion of the finger ends, so long as the
feet remain in one spot, is an epicycloidal motion ; the multitude
of links giving it a practical infinity of possible forms. Even the
six links from the scapula to the finger-ends give an incalculable
variety of possible forms, which our imagination cannot distin
guish from an infinite variety. Here then is a wisdom and skill in
the adaptation of the body to the mind of man, which seems to
me an irrefragable proof of the existence of a wise designer of the
animal kingdom.
Let us turn now to the vegetable kingdom for the second in
stance by which I would test the hypothesis of the world being the
work of an infinitely wise Builder. Suppose such a Builder to
design a part of the vegetable kingdom to grow by leaves and
buds on an ascending axis ; the leaves to be the lungs and stomach

78 GEOMETRY AND FAITH.
of the plants, and to require light for the fulfilment of their
functions. In the usual growth of such plants it will be manife.st
that light from the zenith will be most valuable. Side lights will
be more apt to be cut off by neighboring plants or other obstacles ;
and, even if not cut off, may have and will be likely to have the
highest pitched and most valuable vibrations absorbed and
destroyed or diverted by passing horizontally through so much
more of the lower atmosphere. The first necessity of a leaf,
therefore, will be zenith light. And it will be expected of uner
ring wisdom, we might almost say it would be expected of the
divine justice, that each leaf should have the fairest possible
chance to have zenith light. In other words, it will be expected
that the leaves of plants should not be placed one over another ;
but they should be scattered around the stem in such way as to
give each the least shade, and the most light possible. Of course
it will not be essential that this should be done with perfect ac
curacy ; practical ends do not need theoretical exactness ; on the
contrary, it is a mark of a poor workman to have him give a
degree of finish and exactness incommensurate with the nature of
the work ; finishing, for example a kitchen clock as one would
finish a clock for an astronomical observatory ; that is waste, not
economy, of power. But we should expect in plants built by an
infinitely wise creator to find distinct evidence that a general
plan, for the accurate distribution of the leaves around a vertical
stem, was in operation ; not distributing them in exact conformity
with the plan, but near enough for practical purposes ; and evi
dently showing a perfect knowledge of the perfect plan.
But what would be a perfect plan ? Like the old astronomers,
we are assuming that the plan of creation is perfect; let us not too
hastily assume, as they did, that we know what a perfect plan
would be. But, as near as we can see it, a theoretically perfect
plan ought to be a symmetrical plan ; to go on by a general law ;

PHYLLOTAXIS. 79
and to place the leaves always as far apart laterally as possible, as
we go up the stem ; so that they shall shade each other as little as
possible. The first leaf being at the bottom, the second one going
up must be nearly opposite, and thus divide the circumference of
the stem nearly in halves. Yet when the third leaf comes out still
higher, the circumference, as you look down upon the plant, must
be nearly divided into thirds. When the fourth leaf is added, the
circumference must be nearly divided into quarters; when the
fifth leaf comes out, nearly into fifths ; and so on. The distance
round the stem, the angular distance, as you look straight down
on the end of the stem, between any leaf and the one next above,
or next below it, must evidently be between y$ and % a circum
ference ; or else between y2 and ^ ; were it not so, the first three
leaves could not fulfil the requirements of the problem.
These requirements with reference to the first three leaves can
easily be put in a simple algebraical equation. The second leaf
must divide the circumference in halves as nearly as the third leaf
makes thirds of it. That is to say : The remainder of the cir
cumference, after two angles have been taken out, must be to the
remainder after only one was taken, as that one was to the
whole. But by a well-known proposition in proportion this yields
at once the proportion that the angle is to the remainder of the
circumference, as that remainder is to the whole circumference.
In other words the circumference should be divided in extreme
and mean ratio ; a curious expression familiar in geometry, signi
fying the division of a thing into two parts, such that the smaller
part is to the larger as the larger is to the whole.* Division in
extreme and mean ratio was invented by the early geometers,
before the Christian era, as a means of inscribing a five-sided
figure in a circle ; but it was never suspected that it occurred in
* Let the arc AB be x ; the circumference, 1. Let BC and CD each = AB = x. That
the halves AB, and BCA, may be in the same proportion as the thirds AB, BC,

80 GEOMETRY AND FAITH.
nature until 1849 yearf? after that era. The smaller part is nearly
382 thousandths of the whole ; and, if any one wishes to calculate
it more nicely, he may extract the square root of 5 as far as he
pleases, subtract it from 3, and divide the remainder by 2. It is
a peculiar fraction ; added to its own square root it produces
unity ; it is also one-third of the sum of 1 added to its square.
But, as the square root of 5 is inexpressible in numbers, this pecu
liar fraction is also inexpressible. Its presence in nature, were it
there, could not be detected by any microscope. All that we
could hope to find would be approximations to it. If a plant had
such a law of growth that the successive leaves had a tendency to
follow each other at equal intervals, each .381966 of a circumfer
ence horizontally from its neighbor, we could not measure the
angle from centre to centre of the leaves, accurately to any thing
like the one millionth ; we should be content with finding it about
382 thousandths.
If the perpendicular distance between two successive leaves
(called in botany an internode), is large, we can readily trace a
helical line around the stem, passing in succession through the
foot of every leaf. This maybe called the main or principal helix.
and CA, we will put AC : CB = AB : BCA ; or ^T— = — — .
x 1— x
By proportion, we can add each denominator to its own
1  x i
numerator giving  = — — ; and this, by transposing
U the members, is extreme and mean ratio, 1 : 1 — x = l — x : x.
Again, that the quarters AD and AC should be as near
. equality as the thirds, we put AD : AC = AC : CB. or — — i =
i» 1 — 2x
— - — ; and again adding the denominators r^v,— = ~* ; then adding the numera-
x 1 — x
tors to the denominators,  = — — which is extreme and mean ratio as before.
The same result would follow at every step ; showing that this length of step would
always make the nearest approach possible to equal division into as many parts as there
had been points placed, whatever the number of steps.

PHYLLOTAXIS. 81
And it is evident that we may take the very same set of leaves
and pass a helix in an opposite direction through every leaf ; only
in that case, the angular distance between two leaves will be 618
instead of 382 thousandths of the way round.
Let us now inquire, what further visible and easily observable
evidence we should have, if a plant was actually constituted with
this law of extreme and mean ratio, as the ideal plan of its distri
bution of leaves ; as seems to be demanded by perfect wisdom and
justice, and by symmetry. In case of the internodes being com
paratively short, this principal helix will wind round the stem
with its threads so close, and the leaves so crowded that it will be
difficult for the eye to follow the helix, or discover order in the
arrangement of the leaves. Let us imagine the leaves on a piece
of stem arranged in this ideal order, and numbered from zero,
upward in the order of the stem. Split down the stem on the
opposite side to the zero leaf, take off the bark, and spread it flat.
If the internodes are short, the numbers over the zero will be
arranged as those in figure A are arranged, when you turn the fig
ure cornerwise, so as to have 0 at the bottom, and 55 almost over
it, a shade to the right of the vertical. If the internodes are
longer, the numbers are better represented by B.
FIGURE A.
15 23 31 39 47 55
10 18 26 34 42 50
5 13 21 29 37 45
0 8 16 24 32
. 3 11 19
. . 6 14

82 GEOMETRY AND FAITH.
FIGURE B.
15 23 31 39 47 55
10 18 26 34 42 50
5 13 21 29 37
0 8 16 24 32
3 11 19 27
6 14 22
In either figure the principal helix, joining consecutive num
bers, is transformed into a series of parallel straight lines. But
we find also that straight lines from any selected leaf in any direc
tion, join equidistant numbers ; and every straight line, when the
bark is rolled back into its cylindrical form, will become a helix.
Thus the lines in figure A, joining 3 to 5, 6 to 8, and 10, 14 to 16
and 18, etc., would become two secondary helices, steeper than
the principal one.
Again, it will be observed that the numbers 0, 3, 6, etc., 5, 8, 11,
14, etc., are in parallel straight lines ; and would make still steeper
tertiary helices on the stem.
Turning once more in the opposite direction, we find yet steeper
parallel lines joining every fifth number; such as 0, 5, 10, etc.; 3,
8, 13, etc.
Still closer to the perpendicular, and in the direction of the
original helix, we trace the lines 0, 8, 16 and 5, 13, 21, joining
every eighth leaf.
With a still greater compression of the length of the stem, and
a diminution of the size of the numbers, lines would become con
spicuous, joining the leaves 0, 13, 26, etc.; or the leaves 0, 21, 42;
with very great compression and very small numbers, even the

PHYLLOTAXIS. 83
lines 0, 34, 68, and 0, 55, 110, almost perpendicular, would become
visible and conspicuous.
Observe how curiously these numbers are generated from each
other. Starting from zero on B, to go straight to 3 you pass be
tween 1 and 2 ; but 1 + 2 = 3. Passing from 0 to 5 we go close
in between 2 and 3, and 2 + 3=5. Again the passage from 0 to
8 is between 5 and 3, and 5 + 3 = 8. The gateway from 0 to 13
is between 8 and 5 ; but 8 + 5 = 13.
Observe, also, how readily these auxiliary helices will enable us
to number the leaves. Suppose this piece of bark, with its foot
prints of the leaves on it, is before us, but the leaves not numbered.
Assume any one as zero. Now here are three parallel helices
taking in every leaf. Then upon the one passing through 0 you
can certainly write the numbers 0, 3, 6, etc. Here are five other
helices crossing the first and including every leaf. Then upon the
one passing' through 0 you can write 0, 5, 10, 15, etc. You can
now pass to any leaf whatever by adding or subtracting 5 or 3,
as you move up or down on either one of either set of helices.
Were the stem crowded with a greater multitude of leaves, so
that the more perpendicular and more numerous helices were most
prominent, the same thing could be done. For example, were
there 21 helices in one direction and 34 in the other, and you
wished to find the 8th leaf from one which you had chosen as
zero, you have simply to ascend two steps, on one of 21 leaf lines,
and descend one step on the 34 leaf line ; since 42 — 34 = 8.
It will at once be seen that if the stem of a plant be tough, and
of even texture, so that it will bear twisting evenly ; then, if the
leaves were arranged in this fashion, a twist in the stem would
straighten up every alternate set of helices, and flatten down the
others. A very gentle twist might, for example, bring the 34th
leaf over the zero. Every leaf, the twist being uniform, would
then be exactly over the 34th below it. The 34 leaf helices would

84 GEOMETRY AND FAITH.
be vertical lines, and the leaves would be in 34 perpendicular
rows. A little harder twist would bring the 13th leaf over the
zero ; and the leaves would then stand in 13 vertical rows. Twist
still harder ; the 5th leaf comes over the zero ; the five helices be
come five vertical rows. If your stem is tough enough, twist vio
lently enough to bring 2 over the 0, and your leaves are in two
rows, alternately opposite.
Let us now imagine our gentle twist at the beginning to be in
the opposite direction, and bring the 21st leaf over the zero ; this
wrould give us 21 perpendicular rows ; a little more twist in that
direction would develop 8 rows ; a hard twist reduce them to
three. This supposed twisting of the stem would increase or diminish
the theoretically perfect angle, .381966 of the circumference,
until, in the case of the two rows, the angle was .5 ; or, in the
case of the three rows, it was .333333. Between these ex
tremes, y^ and Yi of the circumference, lie all those arithmetical
approximations to the perfect angle, which we have developed by
twisting ; 1 : 2, 1 : 3, 2 : 5, 3 : 8, 5 : 13, 8 : 21, 13 : 34, 21 : 55, 34 : 89,
55 : 144, 89 : 233, 144 : 377. These approximate angles are formed
each from the two preceding, by simply adding the two numera
tors, and adding the two denominators.
One half is larger than .381966, and Ys is smaller; and through
out the series they are alternately larger and smaller than the
perfect angle. No other vulgar fraction stands in the line of
direct advance. Take for example 7 : 19, it is not as near as 3 : 8,
nor is 7 : 20, nor 5 : 14, nor any fraction with a denominator under
21, except 5 : 13. This series settles toward the extreme and
mean ratio like a vibrating needle settling to the magnetic merid
ian ; the last one given above differs from the true angle by only
.001 of 1°.
Suppose we should now take the case of an actual stem crowded

PHYLLOTAXIS. 85
with leaves, and should find that an actual twist to the left or to
the right would bring the leaves into 3, 5, or 8 rows, what would
it prove ? From the diagrams A and B it would prove that the
leaves were actually arranged at equal angles around the stem, and
that the angle was between 1 : 3 and 2 : 5 of a circumference. Sup
pose, however, that a slighter twist would bring the leaves into 13
or 21 rows ; or suppose that the 34 and 55 leaf helices were already
conspicuous, without any twist ; and suppose that by counting up
by 34s and 55s we should find the 377th leaf was, more exactly
than any other, vertical over the zero leaf. This would prove the
leaves to be actually arranged not only evenly around the stem,
but at an angle almost precisely that of the theory.
These suppositions are actually verified. Taking up, for ex
ample, a stalk of broad-leaved plantain, crowded with flower buds,
or with seed pods ; you can, by twisting in one direction, bring
out 8 or 3 rows ; by twisting in the other, 5 rows, and with a hard
twist, 2. Taking up any pine or spruce cone, and numbering a
few scales by first counting the conspicuous parallel helices, in
each of the two conspicuous intersecting sets, you will find the
13th, 21st, 34th, or 55th leaf come most directly over the leaf
taken as your zero.
When the cylindrical stem is contracted into a strobile, cyme or
rosette, or into the nearly flat head of a composite flower, the
helices are transformed into other curves. Were the stem trans
parent, and the helices drawn in perspective upon a plane at right
angles to it, the eye being the axis of the stem, the helices would
become hyperbolic spirals. But the spirals of sunflower seeds
make no approach to that form. Were a short portion of the
stem, the height being equal to the radius, folded in toward the
centre, the distance of the helix from the circumference being un
changed, the helix would become a spiral of Conon, or Archimedes.
But the sunflower does not conform to that law.

86 GEOMETRY AND FAITH.
The helices each make a constant angle with the meridian of
the cylindrical stem ; and as far as I have yet observed, this is
true of the transformed curves ; so that on a globular strobile the
helix becomes a rhumb line ; and on the sunflower head, a spira
mirabilis of Bernouilli. In the rosettes of young iEnotheras, Cap-
sellas, etc., the spiral is developed by a different process ; and may
possibly be a different spiral, but the sunflower certainly approxi
mates (according to my measurement) very closely to a logarith
mic spiral ; the spira mirabilis.
In the heads of composite flowers these spirals are beautifully
conspicuous, and afford an easy method of determining the de
gree of approximation. The vertical lines have here become radii.
There will always be two spirals (one running in each direction),
plainly to be seen, crossing at each seed. The value of each of
these can be determined by counting round the stem and seeing
how many similar spirals there are in each direction. Thus, if
there are 8 spirals, as it were, parallel to each other, running round
to the right, there they must be the 8 leaf helices ; and the 5
leaf helices will be found running to the left, 5 in number. As
suming now any seed as a zero leaf, the adjacent seeds running
up the spiral to the right will be the 8th, 16th, 24th, etc. ; and
running to the left the 5th, 10th, 15th, etc. By running to the
left and right alternately you can thus determine the number of
any seed as you please. By determining as nearly as possible the
centre of the head, you can draw a radius, from your assumed
zero and inward. If that radius, after leaving the zero, strikes
first on the centre of number 34, the angle is 13 : 34 ; but if it
steers between 34 and 55, grazes 89, and strikes a seed centrally
first at 144, then the angle must be 55 : 144.
As I was writing this, I picked up three heads of dandelion in
seed, blew off the seeds, and counted the pits in the receptacle.
One of them was on the 34 : 89 arrangement ; the other two each

PHYLLOTAXIS. . 87
on the 55 : 144 arrangement. The largest of .these was half an inch
across ; and the error in the position of any two consecutive seeds
in the outer row was, therefore, less than one 30,000th of an inch.
One of the heads gave an angle a trifle too large ; the other two,
an angle a little too small ; the average was almost exact.
Subsequently, I counted the first ox-eye daisy, and the first sun
flower which I saw. They were equally exact as the dandelion,
— the sunflower even nearer. It was upon a 144:377 approxi
mation ; that is, the angle, instead of being 137°.508, was 137°.507,
only about a thousandth of a degree too small. The head was
about 20 centimetres or 8 inches in diameter ; so that two con
secutive seeds near the circumference would have an- arc of 9j4
inches or 24 centimetres between ; and this arc was actually about
one 500th of a millimetre, less than the ten thousandth of an inch,
too small. It would surely be unreasonable to ask for any closer
conformity of observation to theory.
Nor is it easy to imagine any cause which necessitates the ar
rangement. It has been shown that if a cell generate cells on a
horizontal plane at equal intervals of time ; and each cell begin to
generate in the same manner, as soon as it is two intervals old :
and the generation be always on a plane, and at the right hand
side ; then the cells will be arranged like the seed pits in the dan
delion receptacle. But this goes very little way, — nay, I do not
see that it even starts on the road, — toward showing us the gen
esis of the ascending helix.
Again it has been suggested that as leaves grow by light and
air, they will naturally grow where they have the best chance at
getting them ; and this in the course of generations would lead
them to come out exactly at the right spot. Unfortunately for
that explanation, the leaves of plants which need the light and
air, and for whose benefit we have invented the law, do not con
form to it in such wise as to make it a physical benefit. They

88 GEOMETRY AND FAITH.
grow almost universally on the lowest approximations, 2, 3, and 5
ranked. Practically, the physical benefit is not felt. And in the part
where the highest approximations are reached, in the heads, cones,
cymes, etc., the physical benefit is lost through the crowding. We
have evidently over-estimated, at the beginning of our speculation,
the merely physical necessity of the arrangement.
Of course we can set no bounds to the discovery of physical
causes for physical effects ; and it is therefore possible that the
botanist may, at some day, discover the physical agencies by
which this physical arrangement of leaves is effected. But when
he has done so, he will not have in the least shaken the theological
inferences. The preponderance of the ruder approximations,
1 : 2, 1 : 3, 2 : 5, and 3:8, (1:3 and 3 : 8 giving too small, and 1 : 2
and 2 : 5 too large, an angle) shows that the perfect phyllotactic
law is not of practical importance in the growth of plants ; they
live and flourish on the rudest approaches to it. But the tracing
of these approximations up, in such very numerous instances, to
the highest degree of accuracy, such as 55 : 44 and 34 : 89, one
above, the other below the perfect, shows that the law of extreme
and mean ratio is actually incorporated into the vegetable king
dom. The builder of the plant knew that law untold ages before
the geometer invented it, to inscribe a pentagon. These succes
sive approximations point out more clearly and strikingly than
absolute conformity to it could have done.
And as its efficient cause thus lay in the divine wisdom and
divine power, so its final cause lies also in the spiritual realm.
We have come upon it by an assumption that the leaves are
treated justly ; that each is given the best possible chance at light
and air. But while we have learned from our examination of it
that the divine Architect knew this need of the leaf, and in pro
viding for it took this absolutely perfect law ; we learn also that

PHYLLOTAXIS. 89
he knew that perfect conformity was not physically needed ; and
he therefore allowed these continued and great variations by
which the law is suggested rather than thrust upon us ; he made
the symmetry of the plant potential, rather than actual, and this
suggestion, rather than actualization, of the perfect, makes the
plant a more valuable teacher and companion of man. The sug
gestion of infinite perfection, that is beauty.
The Lethe of nature
Cannot trance him asjain,
Whose soul sees the perfect,
"Which his eyes seek in vain.
The outward eye cannot directly see the division of the circum
ference in extreme and mean ratio ; half the leaves are hidden,
and even if we see two consecutive leaves, we cannot tell the
precise angle. But the secondary helices of approximation are
constantly visible, and give a great geometric fascination to the
fructification, sometimes even to the foliage ; while in the larger
growth of the plant the law secures general symmetry ; the varia
tion and concealment through various causes, prevent monotony
and give an endless charm of variety.

XV.
NUMBER AND PROPORTION.
It is only at a comparatively late period, in the development
of the human mind, that number comes into view, as a distinct
object of thought. The idea of number is evolved from things
of imperfect unity ; as an abstraction from things concrete, tan
gible and audible. The two hands, the ten fingers, the mother,
the nurse, the window-panes, suggest the idea ; and it is slowly
brought into the field of distinct intellection, by more or less
laborious effort. A child usually attains the age of three or four
years, before it gives evidence of attaching clear ideas to the
names of numbers. Not until adult life do men usually perceive
that persons are examples of the most complete and absolute
unity. But this slowness with which the idea of number rises to the
surface of consciousness, only shows how very deeply it is im
bedded in the soul. At the beginning of conscious life our atten
tion is fixed upon the individual objects presented in sensation.
The child at that period,
Nescio quid meditans nugarum : totus in illis :
thinks only of the direct lessons of the outward world. The
abstraction of number, and the invisible realities of space, of
time, and of the spiritual world escape his attention, until he
arrives at a mature condition. But the mature mind perceives
that this historical order of the development of ideas is almost
invariably precisely the reverse of the logical order of their de-
90

NUMBER AND PROPORTION. 91'
pendence. For example, space and time seem at first to be
abstractions from the observed facts of matter and motion ; and
it is hastily assumed that the experience of the outward world
comes, at first, independently of any perception of space and
time; and that these ideas are derived from that experience.
' This may be true in the chronological, or historical, succession of
our distinct analytical attention to the ideas ; the actual sequence
of distinct conscious attention is, that we first perceive motion, or
rather matter in motion ; and this leads us to the consideration of
space and time.
Subsequent thought will, however, always show that, in the
very first perception of matter in motion, we quietly take for
granted the existence of space, occupied by matter ; and of time
taken up in the motion. The ideas of matter, motion, space and
time, actually enter the field of consciousness at the same instant ;
that is, the historical order only relates to the sequence of the acts
of attention, by which we separate the ideas from each other.
In strict logical connection, the conception of space and time must
precede the conceptions of matter and motion ; since space and
time are the conditions on which motion is alone possible.
In like manner it will be found that although number is a late
object of conscious attention ; and can be developed as an object of
distinct consecutive thought only in a mind of some maturity ; it
nevertheless stands logically antecedent to every act of intellec
tion. The conscious subject is conscious of an object ; and these
with the act of consciousness form a tri-unity at the very begin
ning of any conscious life. If we dare venture so high a flight of
thought, we may even say that as the creative Mind must be pos
ited as a logical antecedent to creation ; so even in that infinite
Mind, considered even as its own object, that same tri-unity ex
isted antecedent to any creation. Of course we speak of logical,
and not of chronological antecedence. Whether the latter ever

92 GEOMETRY AND FAITH.
really existed ; whether there was creation, in the widest sense of the
word, is a matter beyond our most daring flight. But in the first
act of intellection, the conception of subject and object implies
the conception of number. As with the idea of self so with the
idea of space and time ; they also necessarily imply number the
moment that they are made objects of thought. Even infinite
space, although absolutely homogeneous and without distinction
of parts, except as such parts are created by thought, has its two
elements of distance and direction ; and the element of direction
has a manifoldness, which by no artifice of ingenuity can be re
duced to less than three dimensions. Some modern geometers,
assuming that the conception of space is derived from, as well as
suggested by experience, have S]3eculated upon the possibility
that, in those parts of the universe which are beyond the range
of our experience, space may have other properties ; more dimen
sions than three, or a curvature by which two straight Hues might
include a surface ; and so on. Pursuing this speculation they
have investigated certain algebraical sentences, expressing these
impossible conceptions ; and have found the language capable of
self-consistent interpretation. They have urged this possibility of
self-consistent interpretation as an evidence that the conceptions
themselves may be realized in the infinite distance. In spite of
this ingenuity Reason sturdily maintains that the properties of
space, as we see it here, are invariable throughout the whole ex
tent of absolutely boundless distances; and although we may
technically express, in algebraical language, the conception of a
circle having unequal diameters, and deduce logically self-consist
ent results from the conception, yet we can neither make any
picture of such a circle in our imagination, nor believe that such
a circle can exist in any regions beyond the telescope.
Number, inhering in the primal act of consciousness, follows
every step of thought. All intellection, all thinking is the per-

NUMBER AND PROPORTION. 93
ception or creation of differences and distinctions, unities and
resemblances. The definition of chemistry, that it is the identifi
cation of the one in the many, and the detection of the many in
the one, may be considered also as a definition of all science and
of all thought. Number is more prominent in Chemistry, just as
Space is more prominent in Mechanics, and Time in Biology ; but
Number, Space and Time are all three involved in every finite act
of intellection. All language bears witness to this presence of the
three ideas in every thought ; take any word of any language
and analyze it carefully, trace back its history and you find in it
some more or less apparent reference to number and motion, taken
perhaps as typical of spiritual things.
But number, although thus involved in every act of conscious
ness, even the primal, is not the highest genus ; it is a species of
relation. The highest unity is the person ; and the highest Per
son, although we speak of Him as Absolute and Unconditioned,
stands logically related to His own attributes and to His own cre
ation. Every act of finite intellection involves not only the per
ception of number, but of other relations also. There is more in
the consciousness of subject, object, and relation between them,
than the mere perception of tri-unity in the act. There is the
perception of more than the numbers three, two, and one. Of
course in the more complex acts of intellect, it is still more em
phatically true, that the perception of numerical relation does not
constitute the whole contents of consciousness. Number is, bow-
ever, the relation by which the relation of quantity becomes
amenable to thought and calculation.
At first, number is generated by the distinction of things dis
crete, and perhaps different in kind. It immediately becomes in
itself an object of thought ; and its distinction of many or few
becomes the most firmly grasped and clearly comprehended of all
relations of quantity ; which is at once applied to the measure-

94 GEOMETRY AND FAITH.
ment of all things that may naturally be counted. But there are
many kinds of quantity which are, absolutely or relatively, contin
uous ; and the measurement of the greater or less in these kinds
is accomplished by the analogy of the greater or less quantity to
large or small numbers. So indispensable to all clear intellection
is this relation of numbers to each other, that the Greeks called it
loyog, that is word or wisdom ; and the Latins called it ratio, or
reason ; and this is its technical name among mathematicians to
the present hour. The ratio of two numbers is their relation of
magnitude, not as estimated by the excess of one over the other ;
but estimated by how many times one is larger than the other.
Thus the excess of 6 over 2 is the same as that of 8 over 4 ; but
the ratio of 6 to 2, is 3 to 1 ; and that of 8 to 4 is but 2 to 1.
When we seek to find the ratio of one continuous quantity to
another of the same kind, we simply seek to find two numbers
having the same ratio as the two quantities. One quantity is con
sidered as the unit, to which to refer the other. Usually as a pre
liminary step, both are first referred to some artificial unit ; such
as an inch, a meter, a quart, a liter, a degree of the thermometer,
a degree of angle, an hour, a dollar, etc., etc.
Yet many of the most interesting ratios are found not to be
equal to the ratio of any two numbers whatever. For example,
the diagonal and side of a square, although nearly in the propor^
tion of 17 to 12 are not exactly in the ratio of any two numbers
whatever. And in general wre may say that the mean proportional
between 1 and another number, will very seldom be in a ratio ex
pressible in numbers. Two is a mean proportional between 1 and 4 ;
because 1 is to 2, as 2 is to 4 ; 3 is a mean proportional between 1
and 9 ; and so on. This mean proportional, or geometric mean,
may be illustrated by letting fall a perpendicular from any point
in a semi-circumference upon the diameter ; the length of the
perpendicular is then a mean proportional between the two parts
into which it divides the diameter.

NUMBER AND PROPORTION. 95
In the attempt to measure quantities, it is frequently necessary
to assume a starting point, and measure in both directions ; —
one is then able in subtracting to obtain quantities less than noth
ing ; like south latitude, east longitude, temperature below zero,
deficit in a treasury ; such quantities are called negative quanti
ties ; they lead us at once to the useful fiction of negative num
bers. But this fiction requires the additional fiction of negative
ratios. For example, the ratio of 20 above zero to 10 below zero,
is not simply that of 2 to 1, which would only give 10 above. It
must be expressed by saying it is negative 2 ; — meaning that it
is twice as far from the zero mark, but in the other direction.
But out of this conies a very remarkable case of impossibility ;
namely the impossibility of finding the geometric mean between
a positive and a negative number. The geometric mean between
1 and 2 cannot be expressed in numbers ; but the fraction |J is
nearly it ; and no error can be named so small that a fraction can
not be named differing from the geometric mean less than that
error. But the geometric mean between 1 above and 1 below
zero cannot be expressed by any degree of the thermometer or by
any conceivable numbers. It must be 1, but it can be neither pos
itive 1 nor negative 1 ; since 1 is not to positive 1, as positive 1 is
to negative 1 ; neither is 1 to negative 1, as negative 1 is to nega
tive 1. Certainly zero is not the geometric mean ; for zero is not
as many times negative 1, as 1 is times zero. What then is this
temperature neither above, below, nor at zero? this latitude
neither north nor south of the equator ? It is called in mathemat
ics (lucus a non lucendo) the imaginary ; because it is unimagina-
ble. The mathematicians have various symbols and various
names for it, and use it freely in their calculations. They have
endeavored, with great success, to reduce all quantitative and
geometric impossibilities to this one ; that is to say, they will give
you a correct answer to any absurd question you may devise,

96 GEOMETRY AND FAITH.
provided you allow them to introduce into their answers a sym
bol standing for the unimaginable mean proportional between pos
itive and negative unity. In geometrical questions this symbol
may often be interpreted as signifying the rotation of a line about
some point, but no general interpretation has been discovered. It
is, however, a singular tribute to the ingenuity of the mathema
tician, that he has reduced all absurdities, all impossibilities, to
this one, — the finding of an operation upon a quantity which
shall neither increase it, diminish it, nor leave it unchanged.
Give him a symbol, say i, for that operation, and he is as compe
tent to deal with the impossible, as with the possible. But the
advantage and power, gained by the use of i, extends also into
the realm of the possible and of the actual; and enables the math
ematician to assist in delicate and complicated researches of mod
ern physical science, otherwise beyond the reach of man.
If we multiply 1 by 1.00000001, one hundred million times, the
product is 2.7182818. It is the second of three famous quantities,
pertaining to number alone, yet incapable of exact expression in
number ; since exact expression would require the decimal part
of the constant multiplier to be infinitely small, and the number
of multiplications to be infinitely great.
This peculiar number is known in mathematics by the name of the
base ; (the base) ; and may be symbolized by b. It may be defined
as the product of 100,000,000 factors, each equal to 1.00,000,001;
but is susceptible of various other definitions. Like i, it furnishes
the key to many physical problems, otherwise insoluble ; and it is
very difficult to express, in terms not technical, any reason why it
should exert this beneficent power.
The third of these famous quantities is the ratio of the circum
ference to its diameter ; which is nearly 22 to 7, still more nearly
355 to 113 ; but which cannot be expressed exactly either in num
bers, or in mean proportionals between numbers ; it is a peculiar,

NUMBER AND PROPORTION. 97
unique ratio ; it may be symbolized, if you please, by c ; and its
immense value in calculation arises partly from the fact that the
difficulty of measuring not only circles, but every conceivable
kind of curved lines, surfaces and solids, is generally reducible to
the calculations of b- and c. Add i and you have the means of
calculating the inconceivable also.
These three famous ratios, so entirely different in their origin,
and so utterly incapable of exact expression in number, may be
connected by a very simple bond. If we multiply 1 by 2 ten
times we obtain 1024 ; and if we multiply 1 by 1024 we obtain
the same. Since ten equal multiplications by 2 are equal to one
multiplication by 1024 ; we may say that multiplying by 2 is mul
tiplying by 1024 one-tenth of a time ; multiplying by 8 is multi
plying by 1024 three-tenths of a time, and so on. Using the like ex
pression we may say that b multiplied by itself half c times gives
4.8148. Divide 1 by this number and we get .20788. Next we
take the impossible, inconceivable, inexpressible i, and multiply it
by itself i times ; and can readily demonstrate that it produces
identically that same .20788. Here then are the three famous and
inexpressible ratios of numbers, — base, circumference, and the
imaginary, — bound together in this simple law ; to wit : Multiply
base by itself, half circumference times ; then multiply the pro
duct by imaginary, imaginary times; and the final product is
unity. The object of this long discussion of such abstruse numerical
relations is to set in a more striking light the marvellous power of
the human body as an unconscious calculating engine, alluded to
in more than one of the preceding chapters. Number, the crea
tion of conscious spirit, created in the act of consciousness, cannot,
in its utmost reaches of abstruseness, go beyond the power of the
Infinite Mind. But in our finite thoughts we reach conclusions,
like the above, binding together what is conceivable, but inexr

98 GEOMETRY AND FAITH.
plicable, with what is neither conceivable nor expressible, in one
inexplicable bond.
Yet feeling, or emotion, is a higher, deeper state of conscious
ness than thought; and often carries us into regions of the
Divine thonght where finite thought is unable to follow. In some
instances the reality of this flight into higher regions has afterward
been verified, by the slower laborious ascent of the finite thought,
in paths over which feeling had flown. Thus early thinkers de-
clar_d that in the perception of musical harmony, the soul un
known to herself calculated secretly the numerical ratios of the
undulations of the air. Modern musical statics" has proved that
these secret, unconscious numerical calculations by the musical
ear, by the mere aesthetic feelings, have been far more numerous,
complicated and precise than the older thinkers had supposed ;
and that elaborate comparison, calculation and experiment demon
strate the course and progress of music, from barbaric times to
the present, to have been in perfect agreement in all respects
with laws of number, not revealed until very recently. It is
certainly difficult to imagine modes by which experience and
habit could have led to this progress, had not the human body
been in its very creation formed in exquisite adaptation to those
laws. I have, in another chapter, alluded to Hay's law that beauty of
geometric form depends upon the division of the right angle in
harmonic ratio. Burke argues with a great deal of misplaced in
genuity to prove that proportion has nothing to do with the cause
of beauty ; but Hay simply takes the Parthenon, the Temple of
Theseus, the Lincoln Cathedral, and other universally recognized
types of beauty, and shows that their actual and potential angles do
always stand in simple harmonic ratio to the right angle. Here,
therefore, is a second instance in which the secret unconscious
calculations of the beholders have always recognized by feeling

NUMBER AND PROPORTION. 99
what the labors of Hay have shown to be numerical proportions
in architecture.
But I think I have proved that Hay's law is incomplete ; and
that the eye for beauty recognizes sometimes a far more delicate
and involved proportion than that which he assumes. He illus
trates proportion of geometric stationary angle by the proportion
of vibrations in time, producing a musical concord ; and draws his
law from an examination of architectural and plastic art. But the
beauty of the vegetable world led me to consider that the phyllo
tactic ratio fully developed in the preceding chapter might also
give beauty in artificial forms. The approximations J, •£, f , and f ,
would be also included in Hay's series. To test this question I
have tried various experiments with satisfactory results* A single
one will show their nature. I drew two semi-ellipses ; in one, the
angle (from the end to the middle, and to the mid side) was f ; so
that either by phyllotaxy or by harmony it should be beautiful.
The other was drawn exactly on the perfect phyllotactic angle ;
which, as harmony, would be rudely discordant. Showing these
two ellipses, without explanation, to many persons, in private and
in schools, and taking a vote on the merits of their shape, the vote
was unanimous on their both being good, very many preferred
the phyllotactic angle. They were so nearly alike in their propor
tions that some observers mistook, in attempted conscious calcula
tion, and thought the higher one flatter, even although joining in
the judgment that it was more beautiful.
How much more wonderful, if possible, is that unconscious cal
culation of numerical relations not yet confirmed by figures, but
proved by experiment to be correctly performed in higher depart
ments of art. The harmony of colors unquestionably depends on
numerical relations of wave lengths. The workman at a Bessemer
steel factory knows when to draw the charge by a secret uncon
scious calculation of the rapidity of vibrations, millions in a

100 GEOMETRY AND FAITH.
second, taking place in the flame. All the world have by similar,
but more delicate computations, confirmed and endorsed Titian's
judgment in the harmony of colors; What is a great painting
but the conveyance of great thoughts and feelings from the painter
to the beholder through medium of numerical ratios of angles in
the drawing and of vibrations in the coloring. It is ratio ; it is
the logos, the incarnate word of the painter, imitating the Al
mighty Logos by which the heavens and the earth were made,
and by which the painter's hand had received the skill to make,
the beholder's eye the skill to interpret, the imitation.
The same questions may be asked, the same answer must be
given, concerning the higher ends of music. I have demonstrated
by hundreds of carefully conducted experiments upon hundreds
of persons, sometimes upon large classes in schools, that fully
three-fourths of an audience receive from a musical composition
the very same moral mood or tone of feeling which filled the com
poser at the time of composing it. The closing chorus in Beetho
ven's " Mount of Olives," played upon an organ, without words,
gives to a hearer who has not heard it before, and who is not in
formed of what it is, immediately and irresistibly an impression
of solemn awe, of inexpressible majesty, of penitence, yet of the
peace of forgiveness ; of gratitude for forgiveness, and a sense of
reconciliation through mediation. All this is accomplished sim
ply through rhythmic modulations ; but not, as in spoken words,
through any conventional or artificial associations with them. It
is the direct natural language in which Beethoven utters his pro
foundest, deepest faith. If the hearer have knowrn anything con
cerning the tenets of the Christian faith, he will be certain that
the composer was a Christian believer, uttering through those
chords a thanksgiving, in the name of all worlds, for the recon
ciliation of the world through the cross. What explanation can
be given of this high power of the artist, in whatever department,

NUMBER AND PROPORTION. 101
to pierce direct to the heart without the aid of conscious intellec
tion in the head ? I see none, except to admit that given in
Emerson's " Problem," namely, to admit that the forces of nature,
partially under our conscious guidance, are wholly under the con
trol of that same infinite wisdom and love which inspires the
artist, and interprets his work to his audience.

XVI.
THE DEVELOPMENT OF FORMS.
It is sometimes said that classification always proceeds by a
process of dismissing from attention that which is peculiar to the
individuals ; and fastening the mind upon some arbitrarily chosen
points of resemblance. Examples of this method are found in
Linne's artificial system of botany, and in Agassiz's proposed
classification of fish by their scales. All classification does not,
however, proceed upon this method. Again, it is said that the
classification of the organic kingdoms is an arrangement accord
ing to the degree of development ; that development has been a
continuous process ; and the degrees have been accidentally, or
else arbitrarily marked out. A little examination will show that
this view, also, is erroneous.
The classification of the organic kingdoms is unquestionably
based upon form, in the geometric sense of the word. An accurate
drawing, an accurate plaster-cast, the impression left in a rock, —
these are sufficient data for a naturalist to decide the identity of a
species. If, therefore, classification is built upon development, it
is upon the development of forms in space. All these forms have
a certain amount of symmetry ; their outlines and surfaces corre
spond, more or less perfectly, to the geometric law. We have
already shown that this conformity to law is obedience to thought.
But not only does the form of each individual conform to law ;
the series of forms, also, as Agassiz has shown, indicates a law
pervading the series ; and this fact is inconsistent with the opin
ion that evolution, if there has been evolution, has proceeded con
tinuously, by insensible gradations.
102

THE DEVELOPMENT OF FORMS. 103
All scientific men, at the present day, admit the reign of iaw in
organic nature ; even those who believe in continuous develop
ment. The origin of the universal, invariable law of inorganic
nature cannot, however, be discovered by an investigation con
fined to nature herself. That question lies outside the realm of
science. The scientific man may consider it, and may answer it,
wisely or unwisely ; but in thus doing he has left the domain of
science, and entered that of philosophy. Science infers the uni
versality and invariability of natural law, by a " simple enumer
ation" of the increasing number of observed phenomena con
forming to law; philosophy takes a firmer ground. She con
ceives the universality of law to flow as a necessary consequence
from the grandest and most certain conclusion of human thought ;
the being of an infinitely wise Creator. The infinite Wisdom
foresaw from eternity the best possible modes of action, and
adopted them. No occasion can arise to make Infinite Wisdom
change its plan. " By the determinate counsel of the Lord," says
the wise Hebrew, " were his works from the beginning ; " " He
gave eternal order to his works, and to the atoms thereof, for all
generations ; " " nor to eternity shall they disobey His word." No
scientific writer of the nineteenth century can more distinctly
affirm the universality and invariability of law than this religious
teacher more than twenty centuries ago. Historically, as well as
philosophically, the doctrine is the direct outcome of theistic faith.
To one whose mind has grasped the conception of the existence
of an infinite Deity, of infinite wisdom and power, the whole
universe becomes the expression of a single Divine Thought,
almost infinitely full of detail, and thus giving endless occupation
to our finite intellect ; but also possessing perfect unity, and thus
flooding us with the fulness of its beauty. In this theological
form the doctrine of the correlation of forces, and interdependence
of all sciences, was familiar to metaphysical and theological writ-

104 GEOMETRY AND FAITH.
ers, long before its tardy confirmation by the induction of
physicists. The natural sciences, like the mathematics, have their
postulates ; among them is that of the invariability of law ; they
need also the postulate of the universality of law. The doctrine
of a continuous development will be found difficult to reconcile
with either.
This doctrine is a virtual denial of the existence of law in a
department in which the whole history of science would lead us
most surely to expect the discovery of law. Evolution itself has
strong antecedent probabilities in its favor ; it is only against the
mode of evolution by continuous change and accidental arrest, at
certain accidentally determined stages, that the present objections
lie. The whole course of scientific study up to the middle of this
century had revealed more and more clearly the presence of law,
governing the atoms, and molecules, and masses of the inorganic
world. In order to do this, it had called in the aid of the mathe
matician. By his technical language, alone, can any problem of
time and space, matter and motion be exactly and definitely
solved. Even in the organic world, he had begun to render
the botanist and zoologist important services. He had shown
that, in the plant, there is a law of extreme and mean ratio ; that
this is the nearest approach to a uniform distribution of the leaves;
he had shown that, in the zoologist's classification of animals, the
first great division rests upon laws of mechanical equilibrium in
the embryo, as imperious as the law of an arch. The natural ex
pectation was that mathematical bases would in like manner be
found for the division of classes, orders, families and genera, if not
the species.
But under the theory of continuous gradual development this
course of scientific progress seems arrested. From the fact that
between any two forms of nature we can always, if we have
numerous specimens, find intervening links, it is assumed that

THE DEVELOPMENT OF FORMS. 105
there is no real line of demarcation between the forms. There is
a double fallacy in this assumption. Between any chestnut and
any oak, for example, it may be possible to find an intermediate
form. But it does not follow that it would be possible to find a
form of which a thoroughly trained botanist would say, " This is
either an oak or a chestnut, but I cannot tell which." It may
easily be true that no definition, in words, of an oak would ex
clude every chestnut ; nor any definition of a chestnut exclude
every oak. It does not follow that the senses would not distin
guish; a sharp observer sees many differences not easily to be
described in words ; as those between an oak and a chestnut, an
apple and a pear, a plum and a cherry. The history of science is
full of instances in which a more acute observer has learned to see
and point out distinctions between things which had been con
founded. And, secondly, if it were possible to show that between
two groups the transition is so gradual that no eye can detect the
dividing line, it would not follow that no dividing line existed,
nor that the groups had a common origin. The intellect can some
times clearly distinguish things, indistinguishable by sense. The
elastic curve has, at the extremities of its series of variations, a
straight line and a circle ; indistinguishable by the eye, or by the
imagination, from those produced by varying the eccentricity of
the ellipse. Yet reason, transcending imagination, shows them to
be entirely different ; they cannot be made alike, except by a
process which would confound all intellectual distinctions, and de
stroy the possibility of science.
The ellipse and the elastic curve belong to genera further re
moved from each other than the oak and the chestnut ; yet each
may pass by variation into the form of a circle. Similar instances
abound in the curves investigated by geometry. An observer
unacquainted with mathematics might think it easy to pass the
elastic curve into the form of a circle, and then elongate it to an

106 GEOMETRY AND FAITH.
ellipse ; or easy to frame a definition of the one curve, which should
also include the other ; the geometer knows that neither is possi
ble. When the forms of the chestnut and the oak are as thor
oughly understood by the botanist, as those of the ellipse and the
elastic curve are by the geometer, he will probably wonder that
the two genera were ever considered difficult to define and separate.
It may even be that the mathematician will demonstrate the dif
ference of the forms. It is the mathematician alone who has in
troduced precision and certainty into the other physical sciences ;
and he will probably, at some day, introduce them into biology.
The botanist and the zoologist may rebel, but they will rebel in
vain. The numbers of Pythagoras j,nd the axioms of Euclid are
inexorable. The fates themselves cannot violate the laws of geom
etry, arithmetic, and algebra ; much less can fluttering theorists
break through those adamantine bars.
The erratic genius of De Maillet, and of Erasmus Darwin, has
built an ingenious theory, perfected under the magic influence of
Charles Darwin, by the aid of a host of writers, which flatters
itself that it explains all organic forms in complete independence
of geometrical and arithmetical law. It makes the labors of
classification as empty of real meaning as though they had been
expended upon the forms of clouds or upon the disposition of the
settlings of cups of tea. This theory of insensible, accidental
variation, modified and arrested by the surroundings, is a virtual
assertion that the whole problem of classification is a delusion.
Expand the theory to its most complete form, and it becomes self-
contradictory and self-destructive. It would declare all personifi
cation of objects, and all entification of attributes, mythical and
illusory, and make monotheism the last step of the illusion pre
vious to awaking and discovering that it is all a dream. Thus its
evolution contains in itself an illogical breach of continuity.
When a mathematical series tends through an indefinite series of

THE DEVELOPMENT OF FORMS. 107
terms toward unity, it is illogical to assume either that it ends in
zero or that it becomes indefinite. As well might the modern
correlation of forces, tending constantly to show that all phenom
ena are modes of motion, be held to show that there is no motion,
or at least that we can not know that there is motion, as the
universal tendency of cultivated thought to reduce all super
natural forces to the will of one God be held to lead properly to
atheism or to agnosticism. If the reduction of personification in
things and ideas to smaller and smaller numbers logically leads
to the denial of real being in any entitled idea, it should lead
also to the denial of personality in any thing; that is, to the
denial of personality in our fellow-animals, and even in our fel
low-men. In fact, this theory goes further; it ignores the ex-
stence of personality in one's self ; it makes the theorist himself
a non-existent being, knowing only that one thing, that he does
not exist.
The students of botany and zoology have been laboring for
nearly twenty-five centuries in one direction, with steady pro
gress ; there has been a substantial agreement among them as to
the proper divisions of the animal and vegetable kingdoms ; their
divisions have been concerning the grounds of the division. For
the final settlement of the questions of classification the aid of
the mathematics is necessary. This has been the destiny of
the other physical sciences ; it is that of biology also. There is no
breach of continuity in nature ; geometric and algebraic law rules
absolute over all that can be bounded in space and time. The
vagueness of arbitrary variation and survival of the fittest is a
poetical dream ; it must give way to the intellectual, scientific
sternness of invariable law, bounded by invariable limits. As the
four primitive forms of the embryo flow from necessary mechani
cal conditions, inflexible as the law of equilibrium in arches ; so
the classes, orders, families, genera, will be found to be formed

10§ GEOMETRY AND FAITH.
by conditions sharply defined in nature ; and hereafter to be,
through the aid of mathematics, sharply defined in human
thought. No geometer will willingly relinquish his hope of great triumphs
in the future ; when the new mathematical methods of the nine
teenth century shall have been as faithfully applied to the prob
lems of organic form, as the methods of the seventeenth century
have been to those of inorganic matter. If the melodious wail of
Darwin's bugle leads the naturalists to retreat from the grand
problem of classification ; and if the trumpet of the elder Agassiz
fails to rally them ; the mathematicians will press forward and
gain the high honor of victory, on the noblest field of natural sci
ence. Mathematical science cannot admit the possibility that the
rhythm and symmetry of the organic kingdoms is an accidental
result of accidental variations ; there must be algebraic and geo
metric law at the basis, not only of each organic form, but of the
seriefe of forms. The series has a unity ; capable, when men have
attained a fuller comprehension of it, of expression in terms of
thought. The rhythm arid harmony of a symphony reveal not only the
skill of the orchestra and its conductor ; but the great mind and
noble heart of the composer. The rhythm and harmony of the
organic world reveal the power, the wisdom, and the love of God.
So long as man is less than the universe, his wisest and best course
is to seek everywhere, not for discords and maladaptations, but for
harmonies, correlations, adaptations. The universe is the sum of
all symmetries ; and contains all geometries, architectures, sculpt
ures, and pictorial arts. It is the sum of all rhythms, melodic or
harmonic ; and contains all algebra, poetry, music and dance.
The Divine Word, which created it, is wisdom and love; and
manifests wisdom and love in every syllable and tone in which
it utters itself; not least in the wondrous series of the forms

THE DEVELOPMENT OF FORMS. 109
of plants and animals; swaying, in the responsive rhythm of
growth and decay, sleep and activity, generation and succession,
to the periodic march of the earth, the moon, the planets and
the sun.

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