4. If a leg be less than its opposite angle, their sum will be less than 180°; and if it be greater than its opposite angle, their sum will be greater than 180°. 5. The difference of the two oblique angles is less than 90°; and their sum is greater than 90°, and less than 270°. 6. The three sides are either all equal to, or less than, 90°, or two of them are greater than 90°, and the other less (y). The six cases of right-angled spherical triangles, be fore mentioned, may be ranged as follows: When a leg and its opposite angle are given, to find the rest. 1. To find the other leg. As rad tan giv. leg: cot opp. or giv. sin other leg. Which leg may be either an arc less than 90°, or its supplement. followed without any other restriction, they would frequently lead to an impossible triangle. The same observation may also be applied to the two corresponding properties of quadrantal spherical triangles. (y) A right-angled spherical triangle may have either, 1. One right 4, and two acute or two obtuse 4s; and one acute or one obtuse 4; may be right 4. used at present, is a performance of great labour and merit (0). It may also be further observed, that in consequence of the decimal division of the circle, now generally used by the French mathematicians, a number of persons have been employed, for several years past, at the Bureau de Cadastre, at Paris, under the direction of Prony, in computing new trigonometrical tables of this kind, to a far greater extent than any that have hitherto been devised; but though the work appears to have been nearly completed a considerable time since, it has not yet been offered to the public: which is much to be regretted. For, though the bulk and price of these tables would necessarily preclude them from coming into general use, there are many points of delicate calculation in which they might be advantageously consulted; and our common tables could be corrected from them, or new ones published under an abridged forın. It is therefore to be hoped, that this great monument of calculation will soon make its appearance, under the auspices of a government which declares itself to be l'ami des Arts et des Sciences (p). To this brief account of the works of some of the early writers on this subject, and the tables which, at (0) Dr. Wallis informs us, in the 2d vol. of his mathematical works, that an antilogarithmic canon was begun by Harriot, the algebraist (who died in 1621), and finished by Warner, the editor of his works, about the year 1640; but which was lost for want of encouragement to print it. (p) For a detailed account of the contents of this great work, and the manner in which it was computed, see the Report of De lambre, Mémoires de l'Institut, vol. v. different times, have been composed for facilitating its practical operations, it may also be proper to subjoin a slight sketch of the improvements which it has undergone in passing through the hands of the later analysts, who, by means of a more commodious algorithm, and the resources of a ready and comprehensive calculus, unknown to their predecessors, have enlarged the boundaries of the science, and simplified its rules and processes. These advantages, and the consequent discoveries which attended them, have chiefly arisen from the new views of the subject that had been opened to mathematicians by the theorems first given by Vieta, for the chords of the sums, differences, and multiples of arcs and their supplements; which though left without demonstration, and, in the latter case, probably formed by induction from the law of the terms and their coefficients, have, nevertheless, been the germ of most of the numerous and elegant formula which have since enriched this branch of the subject. We are also, in this respect, no less indebted to Napier, not only for his admirable discovery of logarithms, but for the new and excellent analogies which he introduced into that part of the science relating to the solution of spherical triangles, which still go by his name; as likewise for his other well-known rules, called the Five Circular Parts; which, though too artificial and restricted to be generally employed in the present advanced state of the science, are sufficient proofs of the skill and address with which he investigated every branch of a subject so intimately connected with the invention that has gained him such just celebrity. The works of Briggs, already mentioned, also greatly contributed to the advancement of this branch of the science, both by the assistance which they afforded to the practical calculator, in many intricate and difficult computations, and by the numerous improvements and discoveries of a higher kind, with which they abound. The method, in particular, which he appears to have first used in raising logarithms from their differences, and his skilful application of analytical principles to several subjects of difficult investigation, entitle him to rank with the first mathematicians of the time in which he lived. The logarithmic and other curves, likewise, which first began to be introduced about this time, greatly facilitated the conception of those numbers, by exhibiting some of their most remarkable properties in a more perspicuous way than could be done by the abstract methods of investigation employed by Napier and others. And though the doctrine itself has no necessary connection with these or any other geometrical figures, it was from this source that the new and advantageous mode of expressing logarithms by series was first derived. This happy improvement, which was introduced into the science about the year 1668, by Mercator and James Gregory, who were led to the discovery of some of the most simple forms of these series by contemplating the nature of the hyperbola, was soon afterwards extended to the trigonometrical part of the subject, or the arithmetic of sines, which Newton, Leibnitz, the Bernoullis, and others, enriched with similar formulæ ; and by this means assimilated the principles of logarithms and trigonometry with those of the new calculi, of which they were the inventors and improvers. The exponential formulæ, also, for the sines and cosines of arcs, which were first given by Demoivre, have greatly contributed to the progress of the analytical branch of this subject, by abridging its operations, and shortening the labour of investigation; and though some writers have represented expressions of this kind as founded upon principles which are repugnant to all our ideas of magnitude or quantity, yet their commodious form, and the ease and certainty with which they can be applied in many intricate inquiries, will always cause them to be regarded by the skilful analyst as an important acquisition to the science. Many other improvements, of more or less importance, have since been made, both in the practical and theoretical branches of this subject, by later writers; but of these, none have proved of such general advantage to the science as the substitution of the analytical mode of notation in the place of the geometrical; which useful change was first introduced by Euler; who, besides this simplification of the former methods, has developed and extended, in his numerous works, almost every part of the trigonometrical analysis; which, under his masterly hand, assumed the form of a new science. With respect to the projection of the sphere, which is also a branch of science connected with this subject, little more is known of the early part of its history than what can be collected from the writings of Ptolemy, who in his treatise on the planisphere, as well as in his geography, has left us a number of propositions relating to the stereographic method, as it is now generally called, of representing the surface of the sphere upon |