Pre-Calculus and early calculus Views

Stevin: 1548-1620
Decimal Fractions

Although he did not invent decimals (they had been used by the Arabs and the Chinese long before Stevin's time) he did introduce their use in European mathematics. His notation was to be taken up by Clavius and Napier. Stevin states that the universal introduction of decimal coinage, measures and weights would only be a matter of time.

Stevin's notion of a real number was accepted by essentially all later scientists. Particularly important was Stevin's acceptance of negative numbers but he did not accept the 'new' imaginary numbers and this was to hold back their acceptance.

Napier: 1550-1617
Uses decimals to compute first tables of logarithms, based on ratios, not exponents.


At their meeting Napier suggested to Briggs the new tables should be constructed with base 10 and with log 1 = 0. Briggs wrote that Napier proposed (see for example [1]):-

... that 0 should be the logarithm of unity and 10,000,000,000 that of the whole sine, which I could not but admit was by far the most convenient. Indeed Briggs did construct such tables. He spent a month with Napier on his first visit of 1615, made a second journey from London to Edinburgh to visit Napier again in 1616, and would have made yet a third visit the following year but Napier died in the spring before the planned summer visit.

Briggs's first work on logarithms Logarithmorum Chilias Prima was published in London in 1617. The recent death of Napier is referred to in the preface as is Briggs reason to publish that work, namely:-

... for the sake of his friends and hearers at Gresham College. Briggs published an article A Description of an Instrument Table to find the Part Proportional, devised by Mr Edward Wright in Wright's English translation of Napier's Canon. After Wright's death, Briggs published two further editions of this work (1616 and 1618) with a preface he wrote himself. Briggs's mathematical treatise Arithmetica Logarithmica was published in 1624. This gave the logarithms of the natural numbers from 1 to 20,000 and 90,000 to 100,000 computed to 14 decimal places. It also gave tables of natural sin functions to 15 decimal places, and the tan and sec functions to 10 decimal places. In this book Briggs suggested that the logs of the missing numbers might be computed by a team of people and he even offered to supply specially designed paper for the purpose.

The completed tables were printed at Gouda, in the Netherlands, in 1628 in an edition by Vlacq in which Vlacq had added the logarithms of the natural numbers from 20,000 to 90,000. The tables were also published in London in 1633 under the title of Trigonometria Britannica. The printing of the London edition took place after Briggs had died but he had asked his friend Henry Gellibrand to look after the project on his behalf. Gellibrand was professor of astronomy at Gresham College and was particularly interested in applications of logarithms to trigonometry. He therefore added a preface of his own on applications of logarithms to both plane trigonometry and to spherical trigonometry.

Descartes: 1596-1650
Creates an algebra for measuring and analyzing lengths... analytic geometry. 
x, x2 , and x3  and higher powers of x can all represent (measure) lengths [as well as areas and volumes].

Newton1643-1727/ Mercator 1620-1687: 
Computation of Hyperbolic logarithms using decimals.

Newton  and Leibniz 1646-1716: 
use "limits and functions and numbers" with level of rigor typical of that period.