its correct value.
In 1682 was founded in Berlin the Adta Eruditorum, a
journal usually known by the name of Leipzig Acts. It
was a partialimitation of the Prench Journal des Savans
(founded in 1665), and the literaryand scientific review
publishedin Germ
kind can be compared together,and also their velocities of
increase and decrease ; therefore,in what follows I shall have
no regard to time formally considered, but I shall suppose
some one of the quantitiesproposed,being of the same kind,
to be increased
Newton's study of quadraturessoon led him to another
and most profound invention. He himself says that in 1665
and 1666 he conceived the method of fluxions and applied
them to the quadrature of curves. Newton did not cmoumn-icate
the invention to any of h
to Paris. Simply because they performed scientific work in
Paris, that work belongs no more to France than the cdiosv-eries
of Descartes belong to Holland, or those of Lagrange
to Germany, or those of Euler and Poncelet to Eussia. We
200 A HISTORY OF MATH
and renounced the study of mathematics for that of divinity.
As a mathematician, he is most celebrated for his method of
tangents. He simplifiedthe method of Fermat by ingtroductwo
infinitesimals instead of one, and approximated to the
course of reasoning
were the results of simplemultiplication without the discovery
of any law. The binomial coefficients for positivewhole enexnpot-s
were known to some Arabic and European mtaicthiaemnas-.
Pascal derived the coefficients from the method of
what is called the
eases such us y = ax" + hxi by performing the quadrature
for each term separately,and then addingthe results.
The manner in which Wallis studied the quadrature of the
circle and arrived at his expressionfor the value of rr is
extraordinary.He found that t
work. By the applicationof analysisto the Method of Ivindsi-bles,
he greatlyincreased the power of this instrument for
effectingquadratures. He advanced beyond Kepler by imangkmore
extended use of the " law of continuity" and placing
full reliance in it.
prejudicesof men.
It is most remarkable that the mathematics and philosophy
of Descartes should at first have been appreciatedless by his
countrymen than by foreigners.The indiscreet temper of
Descartes alienated the great contemporary French mathemar
tic
is the inventor. Then follows a treatment of accelerated
motion of bodies fallingfree,or slidingon inclined planes, or
on given curves, " culminating in the brilliant discoverythat
the cycloid is the tautochronous curve. To the theory of
curves he added t
equal roots. Had there been no common divisor,then the
originalequationwould not have possessedequalroots. Hudde
gave a demonstration for this rule.^
Heinrich van Heuraet must be mentioned as one of the easrtligeometers
who occupied themselves with succes
first he seems to have been very inattentive to his studies
and very low in the school; but when, one day, the little
Isaac received a severe kick upon his stomach from a boy
who was above him, he laboured hard till he ranked higher
in school than his ant
and y, after any indefinitely small interval of time, become
x + xO and y + yO, and therefore the equation,which at all
times indifferentleyxpresses the relation of the flowingqtiuatin-es,
will as well express the relation between a; -1-a;0 and
y + yO, as
" November 21, 1675, he found the equation ydx = dxy " xdy,
giving an expression for dcfw_xy), which he observed to be true
for all curves. He succeeded also in eliminating dx from
a differential equation, so that it contained only dy, and
thereby led to
Leibniz first introduces a new notation. He says :
" It will
be useful to write j for omn., as ( I for omn. I,that is, the
sum of the I's "
; he then writes the equation thus : "
2a-Jja
Erom this he deduced the simplest integrals,such as
222 A HISTOEY OF
Huygens was his principalmaster. He studied the geometric
works of Descartes,Honorarius Pabri, Gregory St. Vincent,
and Pascal. A careful study of infinite series led him to the
discoveryof the following expressionfor the ratio of the
circumference to the
gotten at the facts,since much of the analysisused by Newton
and a few additional theorems have been discovered among the
Portsmouth papers. An account of the four holograph muascn-ripts
on this subject has been published by W. W. Rouse
Ball,in the Transa
Newton's Arithmetica Universalis,consistingof algebraical
lectures delivered by him during the first nine years he was
professorat Cambridge, were publishedin 1707, or more than
thirtyyears after they were written. This work was lpiubs-hed
by Mr. Whiston.
have claimed that it yielded close approximations. When
Halley visited Newton in 1684, he requestedNewton to dmeinterwhat
the orbit of a planetwould be if the law of atitotnracwere
that of inverse squares. Newton had solved a
similar problem for Hooke in
labours. It is only a sketch of a much more extended etilaobnoraof
the subjectwhich he had planned,but which was never
brought to completion.
The law of gravitationis enunciated in the first book. Its
discoveryenvelops the name of Newton in a halo of perp
are not to be neglected." This is plainlya rejectionof the
postulatesof Leibniz. The doctrine of infinitelsymall quan212
A HISTOKY OF MATHEMATICS.
necessary to introduce into geometry infinitelsymall qtiuaens.t"iThis mode of differentiatingdoes not remove
in the mode of generatingquantities.*
We give Newton's statement of the method of fluxions or
rates,as given in the introduction to his Quadrature of Curves.
"1 consider mathematical quantitiesin this place not as
consistingof very small parts,but as desc
Newton's third case comes now under the solution of partial
differential equations. He took the equation2x " z + xy = 0
and succeeded in findinga particularintegralof it.
The rest of the treatise is devoted to the determination of
maxima and minima, the r
the perpendiculars, or more generally,straightlines at given
angles,drawn from the point to the given lines,shall satisfy
the condition that the product of certain of them shall be in
a givenratio to the product of the rest." Of this celebrated
problem, t
shows that he is sorry that I do not wish to studymore in egetormy,but I have resolved to quitonly abstract geometry, that
is to say, the consideration of questionswhich serve only to
exercise the mind, and this,in order to study another kind of
geometry,
Though he formulated the fundamental principleof statics,
known as the parallelogram of forces, yet he did not fully
recognise its scope. The principleof virtual velocities was
partlyconceived by Guido Ubaldo (died1607), and afterwards
more fullyby Galile
a pupil of Stevin, in 1629 uses the point on one occasion.
John Wallis in 1657 writes 12 1345, but afterwards in his
algebra adopts the usual point. De Morgan says that " to the
first quarter of the eighteenthcentury we must refer not only
the complete an
Cardan's. Orontius Finaeus (died 1555) in France, and Wlial-m
Buckley (died about 1550) in England extracted the
square root in the same way as Cardan and John of Seville.
160 A HISTORY OF MATHEMATICS.
The invention of decimals is frequentlyattributed to
arithmetical work of English authorship was published in
Latin in 1522 by Cuthbert Tonstall (1474-1569). He had
studied at Oxford, Cambridge, and Padua, and drew freely
from the works of Pacioli and Eegiomontanus. Eeprints of
his arithmetic appeared in En
VIETA TO DESCARTES. 157
religiousstrifes ; ttey concentrated their abilityupon secular
matters, and acquired, in the sixteenth century, a literature
which is immortalised by the genius of Shakespeare and
Spenser. This great literaryage in England was foll