An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 22-24.

Geometrical and historical notes.

43

nec fractas nee irrationales quantitates moratur, et singulare pro illis
calculi genus", which appeared in the periodical: Acta Eruditorum, at
Leipzig. Independently of him, Newton*) had already for years in
working mechanical problems been developing the same method of
calculation, as he has repeatedly mentioned and intimated in his letters
from about 1670 until he ultimately published it in 1687 in his great
work "Philosophiae naturalis principia mathematical as quite an indispensable resource for investigating continuously measurable phenomena**) Here Newton introduced the conception of a variable,
considering the independent variable as measure of the time. At the
very outset he etablishes the theorem for calculation with continuous
variables: "Quantities which in a given time continually approach
to equality and which before the end of that time can come nearer
to each other than any assigned quantity, are finally equal to each
another", which is only a different statement of our fundamental proposition of ~ 5. Taking the distance described by a movable point as
the dependent variable, the quotient of differences assigns the mean
velocity with which a finite length is described, while the differential
quotient measures the actual velocity at each point.
24. Geometrical corollaries and illustrations.
a) If the differential quotient for a finite value of x and of? is
determinately infinite, then the tangent of the curve at this point is
parallel to the axis of ordinates.
b) At points at which the progressive and regressive differential
quotients differ, the direction of the tangent changes discontinuously;
the curve forms an angle.
c) At points at which the function undergoes a break, provided
it is continuous towards one side, it can also possess a differential
quotient towards this side.
d) If the function f(x) be determinately infinite for a finite value
*) The inscription on Newton's monument in Westminster Abbey runs:
H. S. E. ISAACUS NEWTONUS, EQUES AURATUS, QUI ANIMI VI PROPE
DIVINA PLANETARUM MOTUS FIGURAS, COMETARUM SEMITAS, OCEANIQUE AESTUS, SUA MATHESI FACEM PRAEFERENTE, PRIMUS DEMONSTRAVIT. RADIORUM LUCIS DISSIMILITUDINES, COLORUMQUE INDE
NASCENTIUM PROPRIETATES, QUAS NEMO ANTE VEL SUSPICATUS
ERAT, PERVESTIGAVIT. NATURAE, ANTIQUITATIS, S. SCRIPTURAE SEDULUS, SAGAX, FIDUS INTERPRES DEI O. M. MAIESTATEM PHILOSOPHIA
APERUIT, EVANGELII SIMPLICITATEM MORIBUS EXPRESSIT. SIBI GRATULENTUR MORTALES TALE TANTUMQUE EXTITISSE HUMANI GENERIS
DECUS.
Natus XXV. Decemb. A. D. MDCXLII; obiit Martii XX MDCCXXVI. (N. S. 1727.)
**) The treatise: Methodus fluxionum et serierum infinitarum, cum ejusdem
applicatione ad curvarum geometriam, first appeared in 1736 after his death.