PREFACE.

T

Ir is remarkable, that, since the days of Theodosius and Menelaus, scarcely have any books been written, professedly, on Spherical Geometry: and, that those authors who have entered upon it, as a collateral subject, have given its rudiments always very imperfectly, and often very inaccurately. More than one cause may, indeed, be assigned, in order to account for the neglect, into which this branch of Geometry seems to have fallen. But whether it has been deservedly so neglected, may well be doubted.

And, first, as a matter of mere speculation, interesting only to our curiosity, the comparison of Plane, with Spherical Geometry, is of itself a most fertile and ample field, for the exercise of mathematical research. Of the three great bases of reasoning, in the former branch, namely, the equality of the three angles, of every triangle, to two right angles, the properties of parallel lines, and the pro

portionality of the sides about the equal angles of equiangular triangles, the first, it is soon discovered, has no existence at all in Spherics; the second, it is manifest, admits only of a partial application there, in the way rather of analogy, than of strict correspondence; and the third is found not to obtain in the case of any two triangles, on the same sphere, that are of different magnitudes. It might, therefore, on a first view of the question, seem unlikely that these two great provinces of Geometry should have many points of absolute contact, or near approximation. We are instantly prompted to enquire, under what circumstances spherical triangles are equal to one another ; whether the relative position of the greater side, and the greater angle, be the same in these, as it is in plane triangles; whether, since the sides of equiangular triangles, on equal spheres, are equal, each to each, the sides of equiangular triangles, on unequal spheres, be not still proportionals, although they are not equal to one another: whether spherical triangles, on the same sphere, and on the same base, or on equal bases, and between any two parallel circles whatever of the sphere, be equal; and, if not, whether there is not one particular pair of parallel circles, between which this equality may subsist: whether, again, a great circle can be drawn, cutting two given great circles, so

as to make the alternate spherical angles equal; and lastly, whether there be any common principles in what relates to the contact of circles, in Plane, and in Spherical, Geometry.

The result of such enquiries is, that, in many cases, the same general enunciation equally applies to the propositions of the one and of the other class; that between many theorems, of the one and of the other, which cannot entirely correspond, there exists a very close analogy: and that, even the very dissimilarities of these two branches of mathematical science are no less worthy of notice, than their many examples of coincidence.

All this, however, is pure theory. It will also, readily be allowed, that the principal use of Spherical Geometry is to be looked for in the doctrine of Spherical Trigonometry: and it must, further, be admitted, that it is possible to deduce the solution of spherical triangles from one general theorem, which in order to be understood, requires very little knowledge of what is called Spherical Geometry. Any extensive study of that subject might, therefore, appear to be a needless labour. But it may be urged, on the other hand, that trigonometrical propositions can, in some instances, more easily be deduced from very simple constructions, than from the general theorem itself; that all,

which relates to the species of the sides and angles of spherical triangles, is best demonstrated by the geometrical method; and, generally, that, unless the student be well versed in the Geometry of the Sphere, he will not be able thoroughly to understand Spherical Trigonometry; much less will he be properly qualified to enter upon the task of learning Astronomy.

The greater number of authors, accordingly, who have written upon either of the two last mentioned subjects, have felt the necessity of supplying certain subsidiary geometrical truths. But this incidental kind of supply has commonly been insufficient; and some degree of confusion has, perhaps, been created, by the intermixture of propositions belonging to two different departments of the mathematics.

The principal design, therefore, of the following Treatise, is to separate Spherical Geometry and Spherical Trigonometry by a marked line of boundary; to investigate more fully than is usual, the former subject, and thus to make it a step, which may lead to the more perfect comprehension

of the latter.

Of the method which has been pursued in the

execution of this design, some account remains now to be given.

The reader is supposed, in the first place, to be well acquainted with Euclid's Elements, or at least to have that book at hand. So that, in the following pages, after certain fundamental principles have been established, whenever, by the direct application of them, the proof of a spherical proposition becomes identified, in all its steps, with that of a corresponding proposition in Euclid, it has been thought sufficient to refer to Euclid's demonstration; in order to avoid all needless prolixity.

The partition, and the arrangement, of the materials have been regulated entirely by considerations of convenience, without any regard to precedent, even where it was to be found.

The definitions, belonging to the subject are introduced as they are wanted; instead of being prefixt, as they more commonly are, to the Part, or Section, in which they are used. Much might be said in favour of a general adoption of this method of proceeding. But, in Spherics, many of the definitions, which it is convenient to lay down, are founded on theorems: it would, therefore, have been preposterous to have announced them, before the propositions, upon which they are dependent,