ART. VI. A Treatise on Plane and Spherical Trigonometry. By Robert Woodhouse, A.M. F.R.S. Fellow of Caius Colige, Cambridge. London, 1809.

TRIG

RIGONOMETRY is one of the branches of the mathematics, which has received the greatest number of successive improvements, and has advanced the farthest beyond the boundaries within which it was originally confined. It dates its origin from the time of Hipparchus, and may boast, that its foundations were laid by the same person who first undertook to number the stars, and, in the language of Pliny, to leave the heavers for an inheritance so posterity. We do not know, from the writings of this astronomer, or those of his cotemporaries, by what circumstances he was led to apply number to measure the sides and the angles of triangles; but, in the history of a science, of which the objects are all necessarily connected with one another, the want of direct testimony may often be safely supplied by theoretical conjecture. This, we believe, is true, in the present instance. Geometeis were, no doubt, at first, satisfied, in the solution of problems, to determine the things sought from the things given by geometrical construction; that is, by mere graphical operations, or by drawing their figures, as we do, with compasses and a scale of equal ports. This would be sufficiently exact for common use, and for all the ordinary purposes of mensuration, whether of lines or surfaces. Cases, however, would sometimes occur, where the errors of such a method were too great to be overlooked, and where the results were palpably inconsistent with one another. When, for example, there were given in a triangle, the base and the adjacent angles, to determine the sides; and when, at the same time, it happened that the sum of the given angles was nearly equal to two right an gles, a very small error in laying down the angles, would necessarily be accompanied with a very great error in the determination of the sides. It is reasonable to suppose, that such inaccuracies. would first be perceived in the solution of astronomical and geographical problems, where it would often happen that the base was small in respect of the distance to be determined; and there, of consequence, some method of solution, more accurate than the construction of a diagram, would become extremely desirable. Hipparchus, to whom both the geography and astronomy of ancient Greece are greatly indebted, if he was not the first who was sensible of the defects in the constructions of practical geometry, appears to have been the first who had ingenuity and extent of view sufficient to find out a remedy for them. This remedy was the introduction of calculation founded on the famous property

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of the right-angled triangle, said to be discovered by Pythagoras, and now forming the 47th of Euclid's Elements. It is this theorem which is the connecting principle of arithmetic and ge ometry, or which renders the relation between the position of lines and their magnitude expresssible by number. The merit of perceiving this at so early a period as that of which we now speak, even if not done in all its generality, was certainly great; and it enabled Hipparchus to lay the foundation of trigonometrical calculation. The combination of the theorem just mentioned, with certain properties of the circle, particularly with the methods of inscribing regular polygons in that curve, led to general methods of computation, and to the construction of tables of the chords of circular arches, by which the computations were much facilitated. The work which Hipparchus composed on chords is lost; and we know nothing of the steps by which trigonometric calculation advanced, till we find it, some centuries later, in the Spherics of Mcnelaus, and in the Almagest of Ptolemy. The spherical trigonometry of the later was all contained in two theorems concerne ing the intersections of four great circles of a sphere; which have been well given, by the lite Dr Horseley, in his volume of Elementary Treatises. The idea of a spherical triangle had hardly as yet occurred; and the name of trigonometry was unknown. When the repeated irruptions of the Baroarians of the North forced the sciences and the arts of Europe to take refuge in the East, the mathematics found a most favourable reception; and, after a while, returned to their native country, with increased strength and multiplied resources. Trigonometrical calculation came improved by the addition of several new theorems, and, what was still more material, by the substitution of the sines for the chords of arches.

In the midst of that increased activity, which the perception of truth and reality could not but give to the mind, when it awoke from the slumbers and visions of so many ages, a branch of knowledge essential to the study of astronomy, was not likely to be forgotten. Purbach introduced a more commodious division of the radius, and added several other improvements. Regiomontanus, the inventor of decimal fractions, and one of the most,original, as well as the most laborious mathematicians of his time, introduced the use of decimal fractions into the tables and calculations of trigonometry he added, besides, a great number of new geometric theorems, and deduced from them nearly the same rules which are still in use.

In proportion as the fciences advanced, greater accuracy of calculation was required; and the difficulty of thofe calculations, as

well

well as the time confumed in them, increafed in the fame proportion. What all mathematicians were now withing for, the genius of NEPER enabled him to difcover; and the invention of logarithms introduced into the calculations of trigonometry a degree of fimplicity and ease, which no man had been fo fanguine as to expect. Neper made lik wife other great improvements on trigonometry. The theorems which were the foundation of the rules in that science, were not all fuch as to derive from logarithms an equal degree of advantage. To many of the cafes of trigonometry, therefore, though logarithms could be applied, they did not fo much. facilitate and abridge the labour of calculation, as if the rules had been of a different form. NEPER, as if his creative genius had always had the power of difcovering juft what he wanted, or fuch truths as were exactly accommodated to the occafion, found out two theorems which answered precifely to his views, and afforded rules perfectly accommodated to logarithmic calculation. Trigonometry, in the ftate to which it was now brought, continued, with hardly any change, except perhaps a better arrangement of its rules, and a more concife demonftration of its principles, till about the middle of the last century.

A few years before that period, Euler introduced the Arithmetic of the Sines, or, as it may properly be termed, the application of algebra to trigonometry,--a new branch of analyfis which has a peculiar algorithm, and is wonderfully adapted to inveftigation. Before this, trigonometric calculation was only employed to find out an unknown quantity, of fuch a fort, that there was no occafion to reafon about it till it was found; as is the cafe in mere arithmetical queftions refolved by the rule of three or the extraction of roots. But, in the folution of many mathematical queftions, it is neceflary to reason about a quantity while it is yet unknown; and the method of finding it, often is the refult of fuch reafonings. This is properly the bufinefs of analyfis, as dilinguithed from mere numerical computation; and it is what algebra performs, but what arithmetic cannot do, nor trigonometry, till improved, in the way just mentioned, by the application of algebra. Thus improved, it has a peculiar notation, and peculiar rules, both for the addition and the multiplication of the fines and cofines of circular arches, and of all functions of thofe quantities. By this means, the art of geometric investigation is enriched with a new branch, to which we may properly give the name of the Trigonometrical Analysis. In this new form, the fcience has been cultivated in France and Germany ever fince the change made in it by the improvements of Euler. In England it has been confined, till within these few years, to its first and original occupation. The methods of the foreign geometers, how

have come gradually into notice; our trigonometrical treatifes within the last ten years have generally contained fome of the fundamental theorems and operations in the arithmetic of the fines, and have followed the notation of Euler. None of them, however, appear to have done this in fo complete a manner as the treatise which is now before us.

Mr Woodhoufe, who is already known to the mathematical world by writings in which there was more room for originality and invention than there can be in the prefent, has long culti vated the profoundelt parts of the mathematical sciences, and has done much to turn the attention of his countrymen to fub jects that have been far more ftudied on the continent than in this ifland. His treatife on Trigonometry is deltined, we conceive, for the fame purpofe. He fays, that although he once believed that much of the matter contained in it was new, yet now he thinks that it contains nothing of which he could not point out the fubftance in other works. We, for our part, do not think that the author here does juftice to himself; but of this we are certain, that we have now before us a very concife, luminous and analytical view of an important fcience, which has never been fo fully treated of by any writer of our own country.

Mr Woodhouse embraces, in this treatife, not merely the elements of trigonometry, but many of the higher parts, and their moft difficult applications. But when we fpeak of elements, we are reminded that we have lately been accufed of defining that term, as it refpects geometry, in a very unfkilful and inconfiftent manner it may therefore be right, before we proceed any farther, to inquire into the grounds of this charge.

In order to prevent all error about what is elementary in geometry, and what is not, in a former Number we propofed this criterion, that every property of lines, of the first and fecond order, which, when tranflated into the language of algebra, involves nothing higher than a quadratic equation, providing, at the fame time, that it be a propofition of very general application, is to be accounted elementary.' This definition certainly comprehends in it more than fome authors of great authority are difpofed to include in the elements of geometry; and, if we would accommodate our definition to the fenfe of D'Alembert, infead of lines of the firft and fecond order,' we must fay, firaight lines and circles,' the two fimpleft lines of thofe orders. It is neediefs to give, in this place, our reafons for extending our definition farther than the geometer above named has chofen to do; but a remark has been made on that definition, to which it is more material that we fhould advert. It is alleged, that our criterion

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has not only the fault juft mentioned, but another directly oppofite, that of excluding from the number of elementary truths, certain propofitions which have always been ranked among them, and which therefore ought to have been included within the boundary which we profeffed to trace. Such,' fays the critic, are the propofitions relating to the contents of fimilar folids, which, when refolved, according to the most natural and obvious method, into algebraic expreffion, involve cubic equations. Some of them are capable perhaps of more circuitous folutions, by which cubic equations may be avoided: but I believe I may fafely challenge the Reviewer to reduce the proposition to a quadratic equation, in which it is ftated that "fimilar fold parallelopipeds are to one another in the triplicate ratio of their homologous "fides." Now, to cut fhort all difpute, and to decide at once concerning the merits of this defiance, let us take the propofition here given, and tranflate it directly into the language of algebra. Because the object of the theorem is to exprefs the ratio of two folids by means of the fides of the folids, we muft confider the folids as the unknown, and their fides as the known quantities. Call the folids, therefore, x and y, and their homologous or correfponding fides, a and b, then the propofition af firms, that x is to y as a3 to b3; therefore

X a3

=

y 63

But, this is strictly and literally a SIMPLE EQUATION: because the quantities of which the ratio is to be found, are each of them of one dimenfion, and are not multiplied into one ano ther; and the order of an equation, as every body knows, is not denominated from the powers of the known, but from those of the unknown quantities, as it is on these laft that the diffieulty of the queftion depends. Euclid's conftruction comes to the fame thing, for he directs to take c; fo that a:b::b:c; and afterwards d, fo that a:b::c:d; and then he fhows that

a

d

-- ; and thus, conformably to the notions of the most strict geometry, he expreffes the ratio of the folids by the ratio of two lines, found by the rule of proportion. The error of the critic is therefore quite manifeft; and fo grofs, as to appear unaccountable on any fuppofition, except that of extreme ignorance of the principles both of geometry and algebra. If this is not a true theory, it is at least a very plausible hypothefis.

After all, we confefs, that we have not answered literally to the defiance; we have not reduced the propofition in queftion to a quadratic equation; this indeed we hold, as well as the critic though, we hope, on very different grounds) to be impoflible

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