Elementary arithmetic, with brief notices of its history... by Robert Potts.

1'2

LOGARITHMS.

self:-" I consider mathematical quantities in this place not as consisting of very small parts, but as described by a continued motion.
Lines are described, and therefore generated not by the apposition of
parts, but by the continued motion of points; superficies by the
motion of lines; solids by the motion of superficies; angles by the
rotation of the sides; portions of time by a continual flux; and so in
other quantities. These geneses really take place in the nature of
things, and are daily seen in the motion of bodies. And after this
manner the ancients, by drawing moveable right lines along immoveable right lines, taught the genesis of rectangles. Therefore,
considering that quantities, which increase in equal times, and by
increasing are generated, become greater or less according to the
greater or less velocity with which they increase and are generated,
I sought a method of determining quantities from the velocities of the
motions or increments with which they are generated; and calling
-these velocities of the motions or increments Fluxions, and the
generated quantities Fluents, I fell by degrees, in the years 1655 and
1656, upon the method of fluxions, which I have made use of here in
the quadrature of curves." 1
Newton wrote his paper on Fluxions in 1665, the year before he
took his B.A. degree. It appears that the minds both of Napier and
Newton were led independently by the same ideas to effect the objects
they proposed; one to construct a table of logarithms, the other to
devise a calculus for dealing with variable magnitudes.
Edmund Gunter was appointed Professor of Astronomy at
Gresham College in 1619, while Henry Briggs was there the Professor of Geometry, and he held the office till his death, which
happened in 1626. In 1620 he published his I" Canon of Triangles,"
which contains the logarithmic sines and tangents for every minute of
the quadrant, calculated to seven places of figures. In the year 1623
he reprinted these tables in his work "De Sectore et radio," and
added the "Chilias Prima" of his colleague Briggs. He introduced the
use of the arithmetical complement into the arithmetic of logarithms,
and was the first who employed the word cosine for the sine of the
complement of an arc. In 1623 he also applied the logarithms of
numbers, of sines and of tangents, to straight lines divided on a scale
by which proportions in numbers and in trigonometry could be
resolved by means of a pair of compasses. His method of division
was founded on the property, that the logarithms of the terms of equal
ratios are equidifferent. This instrument, in the form of a two-foot
scale, has long been in use for navigation and other purposes, and is
1 Quantitates Mathematicas non ut ex partibus quain minimis constantes, sed ut
motu continue descriptas hic considero. Linese describuntur ac describendo generantru non per appositionem partium sed per motum continuum punctorum, superficies
per motum linearum, solida per motum superficierum, anguli per rotationem laterum,
tempora per fluxumn continuum, et sic in ceteris. Hee geneses in rerum natura
locumn vere habeat et in motu corporum quotidie cernuntur. Et ad hunc mocdun
veteres ducendo rectas mobiles in longitudinem rectarum immobilium genesin docucrunt rectangulorum.
Considerando igitur quod quantitates sequalibus temporibus crescentes et crescendo
genitse, pro velocitate majori vel minori qua crescunt ac generantur, evadunt majores
vel minores; methodum quaerebam determinandi quantitates ex velocitatibus motuuni
vel incrementorum quibus generantur; et has motuum vel incrementorum velocitates
nominando Fluxiones, et quantitates genitas nominando Fluentes, incidi paulatim
Annis 1665 et 1666 in Methodum Fluxionum qua hic usus sum in quadratura
curvarum. -Tractatus de Quadratura Curvarum.