In the sixteenth century several mathematician contributed to the advancement of the science, par ticularly by improvements in the form of tables. By far the most conspicuous among these was the celebrated Vieta. This mathematician, in a work entitled " Canon Mathematicus seu ad triangula cum appendicibus," gave a table of sines, tangents, and secants for every minute of the quadrant to the radius of 100,000, with their differences. At the end of the quadrant the tangents and secants are extended to eight or nine places of figures. The second part of this work contains an account of the construction of tables, plane and spherical trigo nometry, miscellaneous problems, Etc.
The necessity of having accurate tables for all computations in physical and mathematical sci ence, directed the attention of mathematicians at this period to this department of trigonometry in particular. Accordingly we find many elaborate computations, among which may be mentioned the work of Rheticus, published subsequently by Otho his pupil, and afterwards corrected by Pitiscus.
This work contains a trigonometrical canon for every ten seconds of the quadrant to fifteen places of figures. The sines and cosines were computed for every second in the first and last degrees of the quadrant.
With all the assistance derivable from tables, the labour of trigonometrical computation was still excessive, and consequently the chances of error proportionally great. Stimulated by the desire to remove or overcome this difficulty, the celebrated Napier applied the energies of a powerful mind to the subject, and the result was the invention of log arithms. For the particulars of this admirable invention and its history, the reader is referred to our article LOGARITHMS.
To Napier we are also indebted for those techni cal methods of retaining a large class of formula, which are necessary for the solution of spherical triangles, and which are known by the of Napier's rules and Napier's analogies. In the course of the following treatise, we shall have occasion to explain these.
Towards the end of the last century, trigonometry underwent a complete revolution,by being altogether transformed from a geometrical science, as it had heretofore been uniformly treated, into an analytical one. This change took its rise in that department of trigonometry called the theory of angular sections, and which was first successfully cultivated by Vieta.
The object of this part of the science is to express the sines, cosines, tangents, &c. of any proposed mul tiples or submultiples of an arc, as functions of the arc itself, or of its sine, cosine, tangent, &c. This class of problems from its nature not admit ting of geometrical investigation, the powers of analysis were necessarily resorted to, and its lan guage and principles once admitted within the pale of trigonometry, spread their influence through the whole science, so that at length they reached its most elementary parts, and have now left nothing geometrical except the definitions, if indeed we can admit the necessity of giving even these a geo metrical form.
To this happy subversion of the geometrical me thod and the substitution of the analytical, is due all that power and facility of investigation which the analyst receives from the formulae of this sci ence. To this is due the great generality of its theorems, the beautiful symmetry which reigns among the groups of results, the order with which they are developed one from another, offering themselves as unavoidable consequences of the me thod, and almost independent of the will or the skill of the author. The singular fitness with which the language of analysis adapts itself so as to represent, even to the eye, all this order and harmony, are effects too conspicuous not to be im mediately perceived. Nor is the elegant form which the science has thus received from the hand of analysis, a mere object pleasurable to contem plate but barren of utility. All this order and symmetry, which is given as well to the matter as to the form, as well to the things expressed as to the characters which express them, not only serves to impress the knowledge indelibly on the memory, but is the fruitful source of further improvement and discovery.
The advantages which have resulted from the conversion of trigonometry into an analytical sci ence have not been, however, confined to trigonom etry itself. Almost every part of physical science has felt the benefit of this change, and none in a greater degree than astronomy. Many of the most brilliant discoveries, with which modern times have enriched this science, could scarcely be ex pressed, much less discovered, without the aid of the language of analytical trigonometry.