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the elementary principles of the science by the assistance of algebraical characters, and algebraical formulæ alone, without the aid of Geometry, he would most assuredly deceive both himself and his pupils.
It appears then, if the above observations be correct, that Trigonometry cannot be learnt expeditiously and completely, either from principles purely geometrical, or purely algebraical, without sacrificing utility to the uniformity of system.
By way of illustration, let us compare some parts of the Tri. gonometry at the end of Dr. Simson's Euclid, with similar parts of the Trigonometry at the end of Legendre's Geometry.
FIRST, Where the three sides of a spherical triangle are given, to find an angle.
In Prop. xxvii of Dr. Simson's work, the process is conducted from principles purely Geometrical, and the demonstration is long and tedious; whereas in Legerdre's Geometry article LXXVIII, page 389, (where the demonstration is similar to that given at page 215 of the ensuing treatise) it is short and perspicuous.
SECONDLY, in article LXXVII, page 387 of Legendre's Geometry, where he has shewn that the sines of the sides of a spherical triangle are directly proportional to the sines of their opposite angles; the demonstration, which is purely algebraical, is complicated, unnatural, and almost incomprehensible to a young student; whilst that given by Dr. Simson in Prop. XXIII, from Geometrical principles, is simple and obvious *. See likewise page 194 of the ensuing work, where the rule is deduced from Baron Napier.
These remarks are not introduced with a design to criticise the works of either of these eminent authors, but to shew the insufficiency of the Geometrical and of the Algebraical mode of demonstration independent of each other.
* It may be proper to remark that Legendre has given a Geometrical demonstration, at page 385 of his treatise, which is equally as plain as that given by Dr. Simson.
The following work may perhaps, at first view, appear to be too much extended for an introductory treatise, but a little attention to the subject will soon correct this supposition. The student is here supplied with THREE DISTINCT TREATISES, which are all essential towards his future progress in the science. Firstly, A treatise on Logarithms and the necessary tables. Secondly, A treatise on practical Trigonometry; and thirdly, a theoretical treatise on the subject. A work which is deficient in any one of these three dependent parts, will oblige the student to have recourse to different authors, and the want of uniformity of reference, and of regularity of composition, will considerably retard his progress at the commencement of his studies.
“ An Introduction,” says Lord Bacon)"ought to have two properties; the onė, that of a perspicuous and clear method; the other, that of an universal latitude and comprehension ; where the student may have a little pre-notion of every thing, like a model towards a great building.”
Convinced of the propriety of these remarks, the Author has employed much time and attention, in the arrangement and execution of the subject; bearing constantly in mind the purpose for which he was writing, viz. the Instruction of young Students.
The Work is divided into FOUR BOOKS.
Book II. Comprises the principles and practice of Plane Trigonometry, illustrated by a variety of useful examples in the mensuration of distances, heights, &c.
The substance of this book is the same as that of Book I. of the former edition, with the exception of some additions, and an alteration in the arrangement of the several chapters. The easy parts are here placed together, and the demonstrations of the rules are separated from their practical applications; so that the student who wishes to acquire the practical parts only,
may omit the demonstrations and proceed regularly with those parts which are the sole object of his pursuit; whilst the theoretical reader may, if he thinks proper, omit the practical parts, and continue the theory without interruption.
The greater part of the fourth Chapter has been added to this edition, and will be found to contain information which will be useful on several occasions.
The fifth Chapter is nearly the same as Chap. I. Book I. of the preceding edition, except that the formulæ have been somewhat extended. The substance of the two general Scholia from pages 6th and 11th of the second edition of Mr. Emerson's Trigonometry (which were noticed in the 11th page of the first edition) have been introduced, and commodiously arranged according to the notation of Mauduit, Legendre, &c. This chapter likewise contains a variety of formulæ, exclusive of those already mentioned, together with series for finding the sines, cosines, &c. of arcs, and the construction of a table of sines by the continual bisection of an arc. This method of construction was made use of on account of its simple and obvious principles, and is similar to that given in the Trigonometry of Nikiten and Souvoroff, inspectors of the Imperial Academy at Cronstadt, and to the method used by the late Dr. Horsley, Bishop of St. Asaph, at page 137 of his Treatise. The remaining part of the chapter contains general formulæ for the solutions of the different cases of Plane Trigonometry.
Book III. Comprehends the Theory and Practice of Spherical Trigonometry, and the Stereographic Projection of the Sphere.
The first Chapter contains various properties of Spherical Angles, Arcs, and Triangles, demonstrated in a plain and easy manner.
The second Chapter is wholly occupied with the Stereographical Projection of the Sphere, which was placed at the end of Spherical Trigonometry in the first edition; it has been brought forward, in this edition, in order that the constructions
of the different practical cases of spherical triangles may appear with the calculations. The only alteration here is in the arrangement.
The third Chupter contains the investigation of general rules for calculating the sides and angles of right-angled Spherical Triangles. The xxiiid proposition, page 163, which is partly derived from the general figure page 147, is very extensive in its application. The original construction of this figure is ascribed, by Dr. Horsley, to Copernicus the celebrated astronomer. The demonstration of Baron Napier's rule is derived from the xxIid proposition. This rule is the most simple and comprehensive that ever was invented, for solving the different cases of right-angled spherical triangles, but has generally been explained in a confused and unintelligible manner. Valuable as this rule is, it has been censured
« Too tificial and restricted to be generally employed in the present advanced state of the science.” * It is, notwithstanding this aspersion, susceptible of general application, and the species of the different cases may be as accurately ascertained from its equations, as from any rules, or formulæ, hitherto invented; in consequence of which, it will, undoubtedly, continue to be used, by those who know how to appreciate its value, when the works of the author who has presumed to censure it shall be forgotten. The fourth Chapter contains rules for the solutions of
quadrantal spherical triangles, and the fifth Chapter shews the manner of applying Baron Napier's rule to the solutions of the different cases of oblique-angled spherical triangles, from which several useful rules are derived.
The sixth Chapter is, in substance, the same as the IVth Chapter of Book II. in the first edition, but the mode of demonstration has been varied. In this edition the method used by Lagrange and Legendre has been adopted, as being more simple than that used in the first edition. This chapter contains the whole of the formulæ given by Legendre, from
page * Fram?
page 386 to 399 of his Geometry, besides several others. These formulæ, however, are not copied from Legendre, each succeeding step being regularly deduced from the preceding, or from different propositions, to which references are given. These particulars, so essential to a learner, are entirely disregarded by Legendre: for illustration, compare article LXXXVIII, page 398 of Legendre's Geometry, with X. 221. and Y. 222. of the succeeding work, &c.
The formule where auxiliary angles are introduced (X. 221. 1. 222, &c.) are, in effect, analogous to those which may be deduced from Baron Napier's rule, by drawing a perpendicular from the vertical angle of a spherical triangle upon the base, as in proposition xxv, page 191, &c.
The seventh and eighth Chapters include the substance of Book III. of the former edition, though the matter has undergone a considerable change; the original examples were all adapted to the year 1796, but in this edition those which depend on the Nautical Almanac are carried forward to the year 1813. This necessary alteration has been attended with considerable labour, and, it is presumed, will be of much advantage to the young student. The Nautical Almanac for that year can easily be procured. The method of finding the correct distance between thė sun and the moon, or between the moon and a star, by having the apparent distance, and the apparent altitudes of these objects given, is shorter than any that has hitherto appeared, where the common tables only are used, besides having the advantage of being perfectly correct; for it is not liable to those inaccuracies which frequently arise from taking the logarithmical, or natural, sines and cosines out of the tables; as the observed distance is never very small, nor ever near 180 degrees.
The Nautical Almanac was first published in the year 1767; from which time, no treatise on Trigonometrý had appeared, containing astronomical examples adapted to that work, until the first edition of the following treatise ; wherein the author,