e. Save for
one discovery of the first magnitude, and two or three others of some
little importance, the work of the sixteenth century was that of
observing, describing and classifying facts. This was no small service
in itself, though it does not strike the imagination as do the great
new theories.

[Sidenote: Mathematics]

In mathematics the preparatory work for the statement and solution of
new problems consisted in the perfection of symbolism. As reasoning in
general is dependent on words, as music is dependent on the mechanical
invention of instruments, so mathematics cannot progress far save with
a simple and adequate symbolism. The introduction of the Arabic as
against the Roman numerals, and particularly the introduction of the
zero in reckoning, for the first time, in the later Middle Ages,
allowed men to perform conveniently the four fundamental processes.
The use of the signs + {610} and - for plus and minus (formerly written
p. and m.), and of the sign = for equality and of V [square root
symbol] for root, were additional conveniences. To this might be added
the popularization of decimals by Simon Stevin in 1586, which he called
"the art of calculating by whole numbers without fractions." How
clumsy are all things at their birth is illustrated by his method of
writing decimals by putting them as powers of one-tenth, with circles
around the exponents; _e.g._, the number that we should write 237.578,
he wrote 237(to the power 0) 5(to the power 1) 7(to the power 2) 8 (to
the power 3). He first declared for decimal systems of coinage,
weights and measures.

[Sidenote: Algebra 1494]

Algebraic notation also improved vastly in the period. In a treatise
of Lucas Paciolus we find cumbrous signs instead of letters, thus no.
(numero) for the known quantity, co. (cosa) for the unknown quantity,
ce. (censo) for the square, and cu. (cubo) for the cube of the unknown
quantity. As he still used p. and m. for plus and minus, he wrote
3co.p.4ce.m.5cu.p.2ce.ce.m.6no. for the number we should write 3x +
4x(power 2) - 5x(power 3) + 2x(power 4) - 6a. The use of letters in
the modern style is due to the mathematicians of the sixteenth century.
The solution of cubic and of biquadratic equations, at first only in
certain particular forms, but later in all forms, was mastered by
Tartaglia and Cardan. The latter even discussed negative roots,
whether rational or irrational.

[Sidenote: Geometry]

Geometry at that