ADDRESS. 23 
fundamental theorem on the subject. We have in the fifteenth and 
sixteenth centuries a series of mathematicians who with wonderful en- 
thusiasm and perseverance calculated tables of the trigonometrical or 
circular functions, Purbach, Miller or Regiomontanus, Copernicus, 
Reinhold, Maurolycus, Vieta, and many others; the tabulations of the 
functions tangent and secant are due to Reinhold and Maurolycus re- 
spectively. 
Logarithms were invented, not exclusively with reference to the calcu- 
lation of trigonometrical tables, but in order to facilitate numerical calcu- 
lations generally ; the invention is due to John Napier of Merchiston, 
who died in 1618 at 67 years of age; the notion was based upon refined 
mathematical reasoning on the comparison of the spaces described by 
two points, the one moving with a uniform velocity, the other with a 
velocity varying according to a given law. It is to be observed that 
Napier’s logarithms were nearly but not exactly those which are now 
called (sometimes Napierian, but more usually) hyperbolic logarithms— 
those to the base e; and that the change to the base 10 (the great step 
by which the invention was perfected for the object in view) was indicated 
by Napier but actually made by Henry Briggs, afterwards Savilian Pro- 
fessor at Oxford (d. 1630). But it is the hyperbolic logarithm which is 
mathematically important. The direct function e* or exp. «, which has 
for its inverse the hyperbolic logarithm, presented itself, but not in a 
prominent way. Tables were calculated of the logarithms of numbers, 
and of those of the trigonometrical functions. 
The circular functions and the logarithm were thus invented each for 
a practical purpose, separately and without any proper connection with 
each other. The functions are connected through the theory of imaginaries 
and form together a group of the utmost importance throughout mathe- 
matics: but this is mathematical theory ; the obligation of mathematics 
is for the discovery of the functions. 
Forms of spirals presented themselves in Greek architecture, and 
the curves were considered mathematically by Archimedes; the Greek 
geometers invented some other curves, more or less interesting, but re- 
condite enough in their origin. A curve which might have presented itself 
to anybody, that described by a point in the circumference of a rolling 
-carriage-wheel, was first noticed by Mersenne in 1615, and is the curve 
afterwards considered by Roberval, Pascal, and others under the name of 
the Roulette, otherwise the Cycloid. Pascal (1623-1662) wrote at the age 
of seventeen his ‘ Essais pour les Coniques’ in seven short pages, full of new 
views on these curves, and in which he gives, in a paragraph of eight 
lines, his theorem of the inscribed hexagon. 
Kepler (1571-1630) by his empirical determination of the laws of 
planetary motion, brought into connection with astronomy one of the 
forms of conic, the ellipse, and established a foundation for the theory of 
gravitation. Contemporary with him for most of his life, we have Galileo 
(1564-1642), the founder of the science of dynamics; and closely follow- 
