The philosophy of mathematics · Auguste Comte
TRIGONOMETRY.
The difficulty in resolving algebraically such a question as the above, consists essentially in forming, between the angles and the sides of a triangle, three distinct equations; which, when once obtained, will evidently reduce all trigonometrical problems to mere questions of analysis. _How to introduce Angles._ In considering the establishment of these equations in the most general manner, we immediately meet with a fundamental distinction with respect to the manner of introducing the angles into the calculation, according as they are made to enter _directly_, by themselves or by the circular arcs which are proportional to them; or _indirectly_, by the chords of these arcs, which are hence called their _trigonometrical lines_. Of these two systems of trigonometry the second was of necessity the only one originally adopted, as being the only practicable one, since the condition of geometry made it easy enough to find exact relations between the sides of the triangles and the trigonometrical lines which represent the angles, while it would have been absolutely impossible at that epoch to establish equations between the sides and the angles themselves. _Advantages of introducing Trigonometrical Lines._ At the present day, since the solution can be obtained by either system indifferently, that motive for preference no longer exists; but geometers have none the less persisted in following from choice the system primitively admitted from necessity; for, the same reason which enabled these trigonometrical equations to be obtained with much more facility, must, in like manner, as it is still more easy to conceive _à priori_, render these equations much more simple, since they then exist only between right lines, instead of being established between right lines and arcs of circles. Such a consideration has so much the more importance, as the question relates to formulas which are eminently elementary, and destined to be continually employed in all parts of mathematical science, as well as in all its various applications.
It may be objected, however, that when an angle is given, it is, in reality, always given by itself, and not by its trigonometrical lines; and that when it is unknown, it is its angular value which is properly to be determined, and not that of any of its trigonometrical lines. It seems, according to this, that such lines are only useless intermediaries between the sides and the angles, which have to be finally eliminated, and the introduction of which does not appear capable of simplifying the proposed research. It is indeed important to explain, with more generality and precision than is customary, the great real utility of this manner of proceeding. _Division of Trigonometry into two Parts._ It consists in the fact that the introduction of these auxiliary magnitudes divides the entire question of trigonometry into two others essentially distinct, one of which has for its object to pass from the angles to their trigonometrical lines, or the converse, and the other of which proposes to determine the sides of the triangles by the trigonometrical lines of their angles, or the converse. Now the first of these two fundamental questions is evidently susceptible, by its nature, of being entirely treated and reduced to numerical tables once for all, in considering all possible angles, since it depends only upon those angles, and not at all upon the particular triangles in which they may enter in each case; while the solution of the second question must necessarily be renewed, at least in its arithmetical relations, for each new triangle which it is necessary to resolve. This is the reason why the first portion of the complete work, which would be precisely the most laborious, is no longer taken into the account, being always done in advance; while, if such a decomposition had not been performed, we would evidently have found ourselves under the obligation of recommencing the entire calculation in each particular case. Such is the essential property of the present trigonometrical system, which in fact would really present no actual advantage, if it was necessary to calculate continually the trigonometrical line of each angle to be considered, or the converse; the intermediate agency introduced would then be more troublesome than convenient.
In order to clearly comprehend the true nature of this conception, it will be useful to compare it with a still more important one, designed to produce an analogous effect either in its algebraic, or, still more, in its arithmetical relations--the admirable theory of _logarithms_. In examining in a philosophical manner the influence of this theory, we see in fact that its general result is to decompose all imaginable arithmetical operations into two distinct parts. The first and most complicated of these is capable of being executed in advance once for all (since it depends only upon the numbers to be considered, and not at all upon the infinitely different combinations into which they can enter), and consists in considering all numbers as assignable powers of a constant number. The second part of the calculation, which must of necessity be recommenced for each new formula which is to have its value determined, is thenceforth reduced to executing upon these exponents correlative operations which are infinitely more simple. I confine myself here to merely indicating this resemblance, which any one can carry out for himself.
We must besides observe, as a property (secondary at the present day, but all-important at its origin) of the trigonometrical system adopted, the very remarkable circumstance that the determination of angles by their trigonometrical lines, or the converse, admits of an arithmetical solution (the only one which is directly indispensable for the special destination of trigonometry) without the previous resolution of the corresponding algebraic question. It is doubtless to such a peculiarity that the ancients owed the possibility of knowing trigonometry. The investigation conceived in this way was so much the more easy, inasmuch as tables of chords (which the ancients naturally took as the trigonometrical lines) had been previously constructed for quite a different object, in the course of the labours of Archimedes on the rectification of the circle, from which resulted the actual determination of a certain series of chords; so that when Hipparchus subsequently invented trigonometry, he could confine himself to completing that operation by suitable intercalations; which shows clearly the connexion of ideas in that matter.
_The Increase of such Trigonometrical Lines._ To complete this philosophical sketch of trigonometry, it is proper now to observe that the extension of the same considerations which lead us to replace angles or arcs of circles by straight lines, with the view of simplifying our equations, must also lead us to employ concurrently several trigonometrical lines, instead of confining ourselves to one only (as did the ancients), so as to perfect this system by choosing that one which will be algebraically the most convenient on each occasion. In this point of view, it is clear that the number of these lines is in itself no ways limited; provided that they are determined by the arc, and that they determine it, whatever may be the law according to which they are derived from it, they are suitable to be substituted for it in the equations. The Arabians, and subsequently the moderns, in confining themselves to the most simple constructions, have carried to four or five the number of _direct_ trigonometrical lines, which might be extended much farther.
But instead of recurring to geometrical formations, which would finally become very complicated, we conceive with the utmost facility as many new trigonometrical lines as the analytical transformations may require, by means of a remarkable artifice, which is not usually apprehended in a sufficiently general manner. It consists in not directly multiplying the trigonometrical lines appropriate to each arc considered, but in introducing new ones, by considering this arc as indirectly determined by all lines relating to an arc which is a very simple function of the first. It is thus, for example, that, in order to calculate an angle with more facility, we will determine, instead of its sine, the sine of its half, or of its double, &c. Such a creation of _indirect_ trigonometrical lines is evidently much more fruitful than all the direct geometrical methods for obtaining new ones. We may accordingly say that the number of trigonometrical lines actually employed at the present day by geometers is in reality unlimited, since at every instant, so to say, the transformations of analysis may lead us to augment it by the method which I have just indicated. Special names, however, have been given to those only of these _indirect_ lines which refer to the complement of the primitive arc, the others not occurring sufficiently often to render such denominations necessary; a circumstance which has caused a common misconception of the true extent of the system of trigonometry.
_Study of their Mutual Relations._ This multiplicity of trigonometrical lines evidently gives rise to a third fundamental question in trigonometry, the study of the relations which exist between these different lines; since, without such a knowledge, we could not make use, for our analytical necessities, of this variety of auxiliary magnitudes, which, however, have no other destination. It is clear, besides, from the consideration just indicated, that this essential part of trigonometry, although simply preparatory, is, by its nature, susceptible of an indefinite extension when we view it in its entire generality, while the two others are circumscribed within rigorously defined limits. It is needless to add that these three principal parts of trigonometry have to be studied in precisely the inverse order from that in which we have seen them necessarily derived from the general nature of the subject; for the third is evidently independent of the two others, and the second, of that which was first presented--the resolution of triangles, properly so called--which must for that reason be treated in the last place; which rendered so much the more important the consideration of their natural succession and logical relations to one another. It is useless to consider here separately _spherical trigonometry_, which cannot give rise to any special philosophical consideration; since, essential as it is by the importance and the multiplicity of its uses, it can be treated at the present day only as a simple application of rectilinear trigonometry, which furnishes directly its fundamental equations, by substituting for the spherical triangle the corresponding trihedral angle.
This summary exposition of the philosophy of trigonometry has been here given in order to render apparent, by an important example, that rigorous dependence and those successive ramifications which are presented by what are apparently the most simple questions of elementary geometry. * * * * * Having thus examined the peculiar character of _special_ geometry reduced to its only dogmatic destination, that of furnishing to general geometry an indispensable preliminary basis, we have now to give all our attention to the true science of geometry, considered as a whole, in the most rational manner. For that purpose, it is necessary to carefully examine the great original idea of Descartes, upon which it is entirely founded. This will be the object of the following chapter.
Text from Project Gutenberg, public domain in the United States.