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

LOGARITHMS.

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kcnown by the name of " Gunter's Scale." The logarithmic lines on
Gunter's scale were afterwards drawn in different ways. In 1627
they were drawn by Edmund Wingate on two separate scales sliding
against one another, to save the trouble of using the compasses.
And they were also in 1627 applied by Oughtred to concentric circles.
In 1627 and 1628 Adrian Ylacq reprinted at Gouda, in Holland, the
"Arithmetica Logarithmica" of Briggs, and added the logarithms
which Briggs had omitted, from 20,000 to 90,000. But these he had
computed only to ten places of figures. He added also a table of
natural sines, tangents, and secants to every minute of the quadrant.
Gregory St. Vincent, in his "Opus Geometricum Quadraturse
circuli et Sectionumn Coni," published in 1647, shows that if one
asymptote of a hyperbola be divided into parts in geometrical progression, and from the points of division ordinates be drawn parallel to the
other asymptote, they will divide into equal portions the spaces contained between the asymptote and the curve. It was afterwards
observed that by taking the continual sums of these parts, there would
be obtained areas in arithmetical progression corresponding to the
abscissae in geometrical progression, and that these areas and abscissae
would be analogous to a system of logarithms and their corresponding
numbers. On account of this analogy Napier's logarithms have been
named hyperbolic logarithms, but this analogy is not applicable to
Napier's logarithms only, but to all other possible systems of
logarithms. Nor does it illustrate his idea of the generation of
logarithms; Napier's exact idea, however, is completely illustrated
by another curve. If the lines which represent Napier's logarithms
be taken on a line as abscissae, and lines which denote the corresponding natural numbers be drawn at right angles as ordinates to
these abscissae, the curve passing through the extremities of the
ordinates will be the logarithmic curve. Its properties were first
described by Huygens in his "IDissertatio de Causa Gravitatis,"
where it is shown that its subtangent is constant. This curve was also
considered by Roger Cotes, the Plumian Professor at Cambridge, in
his "Hlarmonia Mensurarum," which was published in 1722. He
named the constant subtangent of the curve the modulus of the system
of logarithms, being a fourth proportional to the increment of the
ordinate, the increment of the abscissa, and the ordinate itself. But
under Briggs's idea of logarithms, being rather numeral than geometrical, the modulus may be considered as the natural number at that
point of the system where the increment of the number is equal to the
increment of the logarithm, or when the consecutive numbers and their
logarithms have equal differences. A-d it will appear that the
logarithms of equal numbers in any two systems are proportional to
their modulus. The ratio which connects the systems of Napier and
Briggs is 2-3025850 to 1.
Napier's theory, however ingenious in itself, was felt by mathematicians to labour under the objection of treating geometrically a
subject which was in its nature arithmetical, as Briggs had exhibited
in the tables he had calculated and published. Their objection was
founded on the definition of logarithms, "numeri rationem exponentes,"
and in course of time various improvements were made in the methods
of constructing them-more simple, and involving less labour than the
methods of Napier and Briggs. It was about fifty years after Napier's