Vieta to Descartes

A B decreasing in such a way that when it arrives at any point C , its velocity is proportional to the remaining distance B C . While the first point moves over a distance A C , the second one moves over a distance D F . Napier calls D F the logarithm of B C .

Napier's process is so unique and so different from all other modes of presenting the subject that there cannot be the shadow of a doubt that this invention is entirely his own; it is the result of unaided, isolated speculation. He first sought the logarithms only of sines; the line $AB$ was the sine of $90°$ and was taken $={10}^7$; $BC$ was the sine of the arc, and $DF$ its logarithm. We notice that as the motion proceeds, $BC$ decreases in geometrical progression, while $DF$ increases in arithmetical progression. Let $AB=a={10}^7$, let $x=DF$,

$y=BC$, then $AC=a-y$. The velocity of the point $C$ is ${\displaystyle \frac{d(a-y)}{dt}}=y$; this gives $-\mathrm{\backslash natlog}y=t+c$. When $t=0$, then $y=a$ and $c=-\mathrm{\backslash natlog}a$. Again, let ${\displaystyle \frac{dx}{dt}}=a$ be the velocity of the point $F$, then $x=at$. Substituting for $t$ and $c$ their values and remembering that $a={10}^7$ and that by definition $x=\mathrm{\backslash Naplog}y$, we get $\mathrm{\backslash Naplog}y={10}^7\mathrm{\backslash natlog}\frac{{10}^7}{y}.$

It is evident from this formula that Napier's logarithms are

not the same as the natural logarithms. Napier's logarithms increase as the number itself decreases. He took the logarithm of $\sin 90=0$; i.e. the logarithm of ${10}^7=0$. The logarithm of $\sin \alpha$ increased from zero as $\alpha$ decreased from $90°$. Napier's genesis of logarithms from the conception of two flowing points reminds us of Newton's doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by Archimedes, Stifel, and others. Napier did not determine the base to his system of logarithms. The notion of a "base" in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napier's figures, owing to slight inaccuracies in the calculation of the tables. Napier's great invention was given to the world in 1614 in a work entitled Mirifici logarithmorum canonis descriptio. In it he explained the nature of his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute.

Henry Briggs (1556–1631), in Napier's time professor of

geometry at Gresham College, London, and afterwards professor at Oxford, was so struck with admiration of Napier's book, that he left his studies in London to do

homage to the Scottish philosopher. Briggs was delayed in his journey, and Napier complained to a common friend, "Ah,

John, Mr. Briggs will not come." At that very moment knocks were heard at the gate, and Briggs was brought into the lord's chamber. Almost one-quarter of an hour was spent, each beholding the other without speaking a word. At last Briggs began: "My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy, viz. the logarithms; but, my lord, being by you found out, I wonder nobody found it out before, when now known it is so easy."28 Briggs suggested to Napier the advantage that would result from retaining zero for the logarithm of the whole sine, but choosing 10 , 000 , 000 , 000 for the logarithm of the 10 th