Atlantic Monthly, Volume 3, No. 20, June, 1859 by Various
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Various >> Atlantic Monthly, Volume 3, No. 20, June, 1859
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But let us not forget, meanwhile, that within its own sphere this same
Human Reason is an apt conjuror, marshalling and deftly controlling the
powers of the earth and air to a degree wonderful and full of interest.
And nowhere have all its possibilities so fully found expression in vast
attainment as in those studies preeminently called the mathematics, as
embracing all [Greek: mathaesis], all sound learning. Casting about for
some sure anchorage, drifting hither and thither over changeful seas
of phenomena, a large body of men, deep, clear thinkers withal, some
twenty-four centuries since, fancied that they had found _all_ truth
in the fixed, eternal relations of number and quantity. Hence that
wide-spread Pythagorean philosophy, with its spheral harmonics and
esoteric mysteries, uniting in one brotherhood for many years men of
thought and action,--dare we say, our inferiors? Why allude to the old
fable of the dwarf upon the giant's shoulders? Let us have a tender
care for the sensitive nature of this ultimate Nineteenth Century, and
refrain. They were not so far wrong either, those old philosophers; they
saw clearly a part of the boundless expanse of Truth,--and somewhat
prematurely, as we believe, pronounced it the true Land's End, stoutly
asserting that beyond lay only barren seas of uncertain conjecture.
But mark what followed! Presently, under their hands, fair and clear of
outline as a Grecian temple, grew up the science of Geometry. Perfect
for all time, and as incapable of change or improvement as the
Parthenon, appear the Elements of Euclid, whose voice comes floating
down through the ages, in that one significant rejoinder,--"_Non est
regia ad mathematicam via_." It is the reply of the mathematician,
quiet-eyed and thoughtful, to the first Ptolemy, inquiring if there were
not some less difficult path to the mysteries. But the Greek Geometry
was in no wise confined to the elements. Before Euclid, Plato is said to
have written over the entrance to his garden,--"Let no one enter, who is
unacquainted with geometry,"--and had himself unveiled the geometrical
analysis, exhibiting the whole strength and weakness of the instrument,
and applying it successfully in the discussion of the properties of
the Conic Sections. Various were the discoveries, and various the
discoverers also, all now at rest, like Archimedes, the greatest of them
all, in his Sicilian tomb, overgrown with brambles and forgotten, found
only by careful research of that liberal-minded Cicero, and recognized
only by the sphere and circumscribed cylinder thereon engraved by the
dead mathematician's direction.
Meanwhile, let us turn elsewhere, to that singular people whose name
alone is suggestive of all the passion, all the deep repose of the
East. Very unlike the Greeks we shall find these Arabs, a nation
intellectually, as physically, characterized by adroitness rather than
endurance, by free, careless grace rather than perfect, well-ordered
symmetry. Called forth from centuries of proud repose, not unadorned by
noble studies and by poesy, they swept like wildfire, under Mohammed and
his successors, over Palestine, Syria, Persia, Egypt, and before the
expiration of the Seventh Century occupied Sicily and the North of
Africa. Spain soon fell into their hands;--only that seven-days' battle
of Tours, resplendent with many brilliant feats of arms, resonant with
shoutings, and weightier with fate than those dusty combatants knew,
saved France. Then until the last year of the Eleventh Century,
almost four hundred years, the Caliphs ruled the Spanish Peninsula.
Architecture, music, astrology, chemistry, medicine,--all these arts,
were theirs; the grace of the Alhambra endures; deep and permanent are
the traces left by these Saracens upon European civilization. During
all this time they were never idle. Continually they seized upon the
thoughts of others, gathering them in from every quarter, translating
the Greek mathematical works, borrowing the Indian arithmetic and system
of notation, which we in turn call Arabic, filling the world with wild
astrological fantasies. Nay, the "good Haroun Al Raschid," familiar to
us all as the genial-hearted sovereign of the World of Faery, is said to
have sent from Bagdad, in the year 807 or thereabout, a royal present
to Charlemagne, a very singular clock, which marked the hours by the
sonorous fall of heavy balls into an iron vase. At noon, appeared
simultaneously, at twelve open doors, twelve knights in armor, retiring
one after another, as the hour struck. The time-piece then had
superseded the sun-dial and hour-glass: the mechanical arts had
attained no slight degree of perfection. But passing over all ingenious
mechanism, making no mention here of astronomical discoveries, some of
them surprising enough, it is especially for the Algebraic analysis that
we must thank the Moors. A strange fascination, doubtless, these crafty
men found in the cabalistic characters and hidden processes of reasoning
peculiar to this science. So they established it on a firm basis,
solving equations of no inconsiderable difficulty, (of the fourth
degree, it is said,) and enriched our arithmetic with various rules
derived from this source, Single and Double Position among others.
Trigonometry became a distinct branch of study with them; and then, as
suddenly as they had appeared, they passed away. The Moorish cavalier
had no longer a place in the history of the coming days; the sage had
done his duty and departed, leaving among his mysterious manuscripts,
bristling with uncouth and, as the many believed, unholy signs, the
elements of truth mingled with much error,--error which in the advancing
centuries fell off as easily as the husk from ripe corn. Whether the
present civilization of Spain is an advance upon that of the Moors might
in many respects become a matter of much doubt.
Long lethargy and intellectual inanition brooded over Christian Europe.
The darkness of the Middle Ages reached its midnight, and slowly the
dawn arose,--musical with the chirping of innumerable trouveres and
minnesingers. As early as the Tenth Century, Gerbert, afterwards Pope
Sylvester II., had passed into Spain and brought thence arithmetic,
astronomy, and geometry; and five hundred years after, led by the old
tradition of Moorish skill, Camille Leonard of Pisa sailed away over the
sea into the distant East, and brought back the forgotten algebra and
trigonometry,--a rich lading, better than gold-dust or many negroes.
Then, in that Fifteenth Century, and in the Sixteenth, followed much
that is of interest, not to be mentioned here. Copernicus, Galileo,
Kepler,--we must pass on, only indicating these names of men whose lives
have something of romance in them, so much are they tinged with the
characteristics of an age just passing away forever, played out and
ended. The invention of printing, the restoration of classical learning,
the discovery of America, the Reformation, followed each other in
splendid succession, and the Seventeenth Century dawned upon the world.
The Seventeenth Century!--forever remarkable alike for intellectual and
physical activity, the age of Louis XIV. in France, the revolutionary
period of English history, say, rather, the Cromwellian period,
indelibly written down in German remembrance by that Thirty-Years'
War,--these are only the external manifestations of that prodigious
activity which prevailed in every direction. Meanwhile the two sciences
of algebra and geometry, thus far single, each depending on its own
resources, neither in consequence fully developed, as nothing of human
or divine origin can be alone, were united, in the very beginning of
this epoch, by Descartes. This philosopher first applied the algebraic
analysis to the solution of geometrical problems; and in this brilliant
discovery lay the germ of a sudden growth of interest in the pure
mathematics. The breadth and facility of these solutions added a new
charm to the investigation of curves; and passing lightly by the Conic
Sections, the mathematicians of that day busied themselves in finding
the areas, solids of revolution, tangents, etc., of all imaginable
curves,--some of them remarkable enough. Such is the cycloid, first
conceived by Galileo, and a stumbling-block and cause of contention
among geometers long after he had left it, together with his system
of the universe, undetermined. Descartes, Roberval, Pascal, became
successively challengers or challenged respecting some new property of
this curve. Thereupon followed the epicycloids, curves which--as the
cycloid is generated by a point upon the circumference of a circle
rolled along a straight line--are generated by a similar point when the
path of the circle becomes any curve whatever. Caustic curves, spirals
without number, succeeded, of which but one shall claim our notice,--the
logarithmic spiral, first fully discussed by James Bernouilli. This
curve possesses the property of reproducing itself in a variety of
curious and interesting ways; for which reason Bernouilli wished it
inscribed upon his tomb, with the motto,--_Eadem mutata resurgo_. Shall
we wisely shake our heads at all this, as unavailing? Can we not see the
hand of Providence, all through history, leading men wiselier than
they knew? If not, may it not be possible that we have read the wrong
book,--the Universal Gazetteer, perhaps, instead of the true History?
When Plato and Plato's followers wrought out the theory of those Conic
Sections, do we imagine that they saw the great truth, now evident, that
every whirling planet in the silent spaces, yes, and every falling body
on this earth, describes one of these same curves which furnished to
those Athenian philosophers what you, my practical friend, stigmatize as
idle amusement? Comfort yourself, my friend: there was many a Callicles
then who believed that he could better bestow his time upon the politics
of the state, neglecting these vain speculations, which to-day are found
to be not quite unprofitable, after all, you perceive.
And so in the instance which suggested these reflections, all this eager
study of unmeaning curves (if there be anything in the starry universe
quite unmeaning) was leading gradually, but directly, to the discovery
of the most wonderful of all mathematical instruments, the Calculus
preeminently. In the quadrature of curves, the method of exhaustions was
most ancient,--whereby similar circumscribed and inscribed polygons, by
continually increasing the number of their sides, were made to approach
the curve until the space contained between them was _exhausted_, or
reduced to an inappreciable quantity. The sides of the polygons, it was
evident, must then be infinitely small. Yet the polygons and curves
were always regarded as distinct lines, differing inappreciably, but
different. The careful study of the period to which we refer led to
a new discovery, that every curve may be considered as composed of
infinitely small straight lines. For, by the definition which assigns to
a point position _without_ extension, there can be no tangency of points
without coincidence. In the circumference of the circle, then, no two
of the points equidistant from the centre can touch each other; and the
circumference must be made up of infinite all rectilineal sides joining
these points.
A clear conception of this fact led almost immediately to the Method of
Tangents of Fermat and Barrow; and this again is the stepping-stone to
the Differential Calculus,--itself a particular application of that
instrument. Dr. Barrow regarded the tangent as merely the prolongation
of any one of these infinitely small sides, and demonstrated the
relations of these sides to the curve and its ordinates. His work,
entitled "Lectiones Geometricae," appeared in 1669. To his high
abilities was united a simplicity of character almost sublime. "_Tu,
autem, Domine, quantus es geometra_!" was written on the title-page of
his Apollonius; and in the last hour he expressed his joy, that now, in
the bosom of God, he should arrive at the solution of many problems of
the highest interest, without pain or weariness. The comment of the
French historian conveys a sly sarcasm on the Encyclopedists:--"_On voit
au reste, par-la, que Barrow etoit un pauvre philosophe; car il croiroit
en l'immortalite de l'ame, et une Divinite, autre que la nature
universelle_."[A]
[Footnote A: MONTUCLA. _Hist. des Math_. Part iv. liv. 1.]
The Italian Cavalleri had, before this, published his "Geometry of
Indivisibles," and fully established his theory in the "Exercitationes
Mathematicae," which appeared in 1647. Led to these considerations by
various problems of unusual difficulty proposed by the great Kepler,
who appears to have introduced infinitely great and infinitely small
quantities into mathematical calculations for the first time, in a tract
on the measure of solids, Cavalleri enounced the principle, that all
lines are composed of an infinite number of points, all surfaces of
an infinite number of lines, and all solids of an infinite number of
surfaces. What this statement lacks in strict accuracy is abundantly
made up in its conciseness; and when some discussion arose thereupon,
it appeared that the absurdity was only seeming, and that the author
himself clearly enough understood by these apparently harsh terms,
infinitely small sides, areas, and sections. Establishing the relation
between these elements and their primitives, the way lay open to the
Integral Calculus. The greatest geometers of the day, Pascal, Roberval,
and others, unhesitatingly adopted this method, and employed it in the
abstruse researches which engaged their attention.
And now, when but the magic touch of genius was wanting to unite and
harmonize these scattered elements, came Newton. Early recognized by Dr.
Barrow, that truly great and good man resigned the Mathematical Chair at
Cambridge in his favor. Twenty-seven years of age, he entered upon his
duties, having been in possession of the Calculus of Fluxions since
1666, three years previously. Why speak of all his other discoveries,
known to the whole world? _Animi vi prope divina, planetarum motus,
figuras, cometarum semitas, Oceanique aestus, sua Mathesi lucem
praeferente, primus demonstravit. Radiorum lucis dissimilitudines,
colorumque inde nascentium proprietates, quas nemo suspicatus est,
pervestigavit_. So stands the record in Westminster Abbey; and in many
a dusty alcove stands the "Principia," a prouder monument perhaps, more
enduring than brass or crumbling stone. And yet, with rare modesty, such
as might be considered again and again with singular advantage by many
another, this great man hesitated to publish to the world his rich
discoveries, wishing rather to wait for maturity and perfection. The
solicitation of Dr. Barrow, however, prevailed upon him to send forth,
about this time, the "Analysis of Equations containing an Infinite
Number of Terms,"--a work which proves, incontestably, that he was in
possession of the Calculus, though nowhere explaining its principles.
This delay occasioned the bitter quarrel between Newton and Leibnitz,--a
quarrel exaggerated by narrow-minded partisans, and in truth not very
creditable, in all its ramifications, to either party. Newton, in the
course of a scientific correspondence with Leibnitz, published in 1712,
by the Royal Society, under the title, "Commercium Epistolicum
de Analysi promota," not only communicated very many remarkable
discoveries, but added, that he was in possession of the inverse problem
of the tangents, and that he employed two methods which he did
not choose to make public, for which reason he concealed them by
anagrammatical transposition, so effectual as completely to
extinguish the faint glimmer of light which shone through his scanty
explanation.[B] The reference is obviously to what was afterwards known
as the Method of Fluxions and Fluents. This method he derived from the
consideration of the laws of motion uniformly varied, like the motion of
the extreme point of the ordinate of any curve whatever. The name which
he gave to his method is derived from the idea of motion connected with
its origin.
[Footnote B: This logograph Newton afterwards rendered as follows: "Una
methodus consistit in extractione fluentis quantitatis ex aequatione
simul involvente; altera tantum in assumptione seriei pro quantitate
incognita ex qua ceterae commode derivari possunt, et in collatione
terminonim homologorum aequationis resultantis ad eruendos terminos
seriei assumptae."]
Leibnitz, reflecting upon these statements on the part of Newton,
arrived by a somewhat different path at the Differential and Integral
Calculus, reasoning, however, concerning infinitely great and infinitely
small quantities in general, viewing the problem algebraically instead
of geometrically,--and immediately imparted the result of his studies to
the English mathematician. In the Preface to the _first_ edition of
the "Principia," Newton says, "It is ten years since, being in
correspondence with M. Leibnitz, and having instructed him that I was
in possession of a method of determining tangents and solving questions
involving _maxima_ and _minima_, a method which included irrational
expressions, and having concealed it by transposing the letters,
he replied to me that he had discovered a similar method, which he
communicated, differing from mine only in the terms and signs, as
well as in the generation of the quantities." This would seem to be
sufficient to set at rest any conceivable controversy, establishing an
equal claim to originality, conceding priority of discovery to Newton.
Thus far all had been open and honorable. The petty complaint, that,
while Leibnitz freely imparted his discoveries to Newton, the latter
churlishly concealed his own, would deserve to be considered, if it were
obligatory upon every man of genius to unfold immediately to the world
the results of his labor. As there may be many reasons for a different
course, which we can never know, perhaps could never hope to appreciate,
if we did know them, let us pass on, merely recalling the example of
Galileo. When the first faint glimpses of the rings of Saturn floated
hazily in the field of his imperfect telescope, he was misled into the
belief that three large bodies composed the then most distant light of
the system,--a conclusion which, in 1610, he communicated to Kepler in
the following logograph:--
SMAISMRMILMEPOETALEVMIBVNENGTTAVIRAVS.
It is not strange that the riddle was unread. The old problem, Given the
Greek alphabet, to find an Iliad, differs from this rather in degree
than in kind. The sentence disentangled runs thus:--
ALTISSIMVM PLANETAM TERGEMINVM OBSERVAVI.
And yet we have never heard that Kepler, or, in fact, Leibnitz himself,
felt aggrieved by such a course.
But Leibnitz made his discovery public, neglecting to give Newton _any_
credit whatever; and so it happened that various patriotic Englishmen
raised the cry of plagiarism. Keil, in the "Philosophical Transactions"
for 1708, declared that he had published the Method of Fluxions, only
changing the name and notation. Much debate and angry discussion
followed; and, alas for human weakness! Newton himself, in a later
edition of the "Principia," struck out the generous recognition of
genius recorded above, and joined in terming Leibnitz an impostor,
--while the latter maintained that Newton had not fathomed the more
abstruse depths of the new Calculus. The "Commercium Epistolicum" was
published, giving rise to new contentions; and only death, which ends
all things, ended the dispute. Leibnitz died in 1716.
The Calculus at first found its chief supporters on the Continent. James
and John Bernouilli, Varignon, author of the "Theory of Variations," and
the Marquis de l'Hopital, were the first to appreciate it; but soon it
attracted the attention of the scientific world to such a degree that
the frivolous populace of Paris had even a well-known song with the
burden, "_Des infiniment petits_." Neither were opponents wanting.
Wrong-headed men and thick-headed men are unfortunately too numerous
in all times and places. One Nieuwentiit, a dweller in intellectual
fogbanks, who had distinguished himself by proving the existence of
the Deity in one of his works, made about this time what he doubtless
considered a second discovery. He found a flaw in the reasoning of
Leibnitz, namely, that _he_ (Nieuwentiit) could not conceive of
quantities infinitely small! A certain Chever also performed sundry
singular mathematical feats, such as squaring the circle, a problem
which he reduced to the single question, _Construere mundum divinae
menti analogum_, and showing that the parabola, the only conic section
squared by ancient or modern geometers, could never be quadrated, to the
eternal discomfiture and discredit of the shade of Archimedes. Leibnitz
used every means in his power to engage these worthy adversaries in
a contest concerning his Calculus, but unfortunately failed. Bishop
Berkeley, too, author of the "Essay on Tar-Water," devout disbeliever in
the material universe, could not resist the Quixotic inclination to run
a tilt against a science which promised so much aid in unveiling those
starry splendors which he with strenuous asseveration denied. He
published, in 1754, "The Minute Philosopher," and soon after, "The
Analyst, or the Discourse of a Mathematician," showing that the
Mathematics are opposed to religion, and cultivate an incredulous
spirit,--such as would never for a moment listen, let us hope, to any
theory which proclaims this green earth and all the universe "such stuff
as dreams are made of," even though the doctrine be ecclesiastically
sustained and backed with abundant wealth of learning. Numerous were the
defenders, called out rather by the acknowledged metaphysical ability of
Bishop Berkeley than by any transcendent merit in these two tracts; and
among others came Maclaurin.
Taylor's Theorem, based upon that first published by Maclaurin, is the
foundation of the Calculus by La Grange, differing from the methods of
Leibnitz and Newton in the manner of deriving the auxiliaries employed,
proceeding upon analytical considerations throughout. Of his "Theorie
des Fonctions," and that noblest achievement of the pure reason, the
"Mecanique Analytique," we do not propose to speak, nor of the later
developments of the Calculus, so largely due to his genius and labors.
These are mysteries, known only to the initiated, yet capable of raising
their thoughts in as sublime emotion as arose from the view of the
elder, forgotten mysteries, which Cicero deemed the very source and
beginning of true life.
We have seen how, and through whose toil, this mightiest instrument of
human thought has reached its present perfection. Now, its vast powers
fully recognized, it has become interwoven with all Natural Philosophy.
On its sure basis rests that majestic structure, the "Mecanique Celeste"
of La Place. Its demonstration supports with undoubted proof many
doctrines of the great Newton. Discovery has succeeded discovery; but
its powers have never yet been fully tested. "It is that field of
mathematical investigation," says Davies, "where genius may exert its
highest powers and find its surest rewards." Looking back through the
long course of events leading to such a magnificent result, looking up
to that choral dance of wandering planets, all whose courses and seasons
are marked down for us in the yearly almanac, can we not find in these
manifestations something on the whole quite wonderful, worthy of very
deep thankfulness, heartfelt humility withal, and far-reaching hope?
In an age of many-colored absurdity, when extremes meet and
contradictions harmonize,--when men of gross, material aims give
implicit confidence to the wildest ravings of the supernatural, and
pure-minded men embrace French theories of social organization,--when
crowds of dullards all aflame with unexpected imagination assemble in
ascension-robes to await the apocalyptic trump, and Asiatic polygamy
spreads unmolested along our Western rivers,--when the prediction is
accomplished, "Old men dream dreams and young men see visions," and the
most practical of the ages bids fair to glide ghostly into history as
the most superstitious,--it is well, it can but be well, to contemplate
reverently that Reason, which Coleridge, after Leighton, calls "an
influence from the Glory of the Almighty." In the contemplation of the
spirit of man (not your _animula_, by any means!) there is earnest of
immortality which needs not that one rise from the dead to confirm it.
In view of the Foresight which guides men, we may trust that all this
tumultuous sense of inadequacy in present institutions, this blind
notion of wrong, far enough from intelligent correction, is, after all,
better than sluggish inaction.
BULLS AND BEARS.
[Concluded.]
CHAPTER XXX.
The suspension of specie payments brought instant relief to all really
solvent mercantile houses; since those who had valuable assets of any
kind could now obtain discounts sufficient to enable them to meet their
liabilities. Among those who were at once relieved was the house of
Lindsay and Company; they resumed payment and recommenced business.
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