had been proved. The definitions themselves have been very differently enunciated by different authors. Here, therefore, it became necessary to make a choice. A sphere, for example, has been defined, in the manner of Theodosius, rather than in that of Euclid; and a spherical angle, instead of being called the Inclination of two Planes, has been described, in language accommodated to the idea which we form of it, from sight; it being, in reality, the mutual inclination, not of two planes, but of two curve lines, on a convex surface. But, the object chiefly aimed at, in the former part of the Treatise, was first to collect, and to confine within a very small number, such propositions as require any dissection of the sphere; and then to reduce all the rest to a Geometry, really practicable, on any spherical surface, and depending solely on the use of the rule and the compass. This, however, will hardly be understood, without some further explanation. All the ancient writers, then, it is well known, and perhaps all the modern writers, on Spherics, in the construction of their problems, as well as in the proofs of their theorems, have recourse to certain graphical operations, which, are to be performed in the interior of the solid sphere. Now, although in establishing the truth of a theorem, there seems to be no impropriety, in supposing such operations to have been executed, whenever they are evidently or demonstrably possible, and although, in demonstrating spherical theorems, there is often a kind of necessity for making this supposition, yet in the management of problems, the case is far from being the same. In this latter instance, a construction, which is really impracticable, can serve only, at most, to shew, what is seldom doubtful, that the problem, which it pretends to solve, involves no impossibility. In reality, the problems, which are thus treated, either admit of a different and a practicable solution, or else they are totally useless. To find the center of a given sphere, for example, is the first problem in Theodosius, and it has been copied by many succeeding writers. To effect what is thus proposed, the sphere is first to be cut by a plane; a perpendicular is next to be drawn, within the sphere, from the center of the section, and to be produced, both ways, until it meet the sphere's surface in two points: lastly, that straight line, so terminated, is to be bisected. Now this process is plainly impracticable, without wholly destroying the form of the given solid. But neither is it necessary, for the solution of any important spherical problem, to have actually found the sphere's center: the existence of such a point is established by a definition: and in all theorems, relating to a given sphere, it may fairly be considered as given. If, however, a sphere be put into the hands of a geometrician, he ought, without mutilating its figure, to be able to find the exact length of its diameter, and the exact places of the poles of any circles, the circumferences of which are in its surface: and all this he can do, by a series of common and easy operations, performed by a rule and compass; the whole solution of the problem, mainly depending upon this plain theorem, that if circles be described about triangles, which have the sides of the one severally equal to the sides of the other, the diameters of the circles are equal to one another. In the following pages, therefore, regard has been had to practicability, in the solution of this, and of all other spherical problems. In the treatment of them, nothing has been directed, but what can easily be performed, either on the sphere's surface, or else on a given plane. In the former part of the Treatise, not only the plan, but a considerable portion of the matter also, is original, at least, if it be not new. Here, therefore, it would be great presumption, the nature of the subject being considered, to suppose that no mistakes have been committed. If the genius of Copernicus failed to guard him from errors, in this particular walk of science, men of inferior powers can hardly expect to have been more fortunate; although they may, perhaps, be allowed to hope for indulgence and excuse. In the latter part of the work, a more beaten track has been pursued. Care, however, has been taken, to present to the reader the subject of Trigonometry, in the two different points of view, from which it ought always to be considered. The learner ought always to be apprized, that the great problem, in which Trigonometry, properly so called, is comprised, may be solved in two very different ways: it is important, that he compare the two methods with one another, in order fully to apprehend the nature of the advantage, which the Algebraical method has, over the Geometrical: and, whilst he is taught Trigonometry, according to the former method, his attention should be constantly directed to the several bearings of the principal problem, in all its cases, and to the suppositions, which are often tacitly made, in its solution: in a word, he should be instructed to keep in view, the absolute dependence, which that solution has, upon the nu merical values of sines, and tangents, having been previously calculated; and he should be shewn the mode, or at least the possibility, of actually calculating such values, to any required degree of exactness. These latter remarks apply, more especially, to the doctrine of Plane Trigonometry, which is here treated of only in a cursory manner: and it will be found, that they have not been lost sight of, in the short summary, which is given of that subject. The great practical objects of Trigonometry, are undoubtedly best attained by computation: and when a proposition is to be investigated, which is afterwards to be employed in that way, the investigation itself, generally speaking, can best be conducted in the appropriate language, and by the peculiar processes of universal arithmetic. Whenever, therefore, actual calculation is the end ultimately proposed, recourse may be had without scruple, to the application of Algebra to Geometry: and, it seems, accordingly, as if no doubt could well be entertained, in the management of our subject, where Geometry should end, and where Algebra should begin. No examples of calculation have been exhibited |