[2] Lord Morley.
SECTION 4. SCIENCE
[Sidenote: Inductive method]
The glory of sixteenth-century science is that for the first time, on a large scale, since the ancient Greeks, did men try to look at nature through their own eyes instead of through those of Aristotle and the Physiologus. Bacon and Vives have each been credited with the discovery of the inductive method, but, like so many philosophers, they merely generalized a practice already common at their time. 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 time, as for long afterwards, was dependent wholly on Euclid, of whose work a Latin translation was first published at Venice. [Sidenote: 1505] Copernicus with his pupil George Joachim, called Rheticus, and Francis Vieta, made some progress in trigonometry. Copernicus gave the first simple demonstration of the fundamental formula of spherical trigonometry; Rheticus made tables of sines, tangents and secants {611} of arcs. Vieta discovered the formula for deriving the sine of a multiple angle.