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ART. VI. A Treatise on Plane and Spherical Trigonometry. By
Robert Woodhouse, A.M. F.R.S. Yellow of Caius College, Cambridge. London, 1809.
TRIGONOMETRY is one of the branches of the mathematics, of 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 Jaid by the same person who first undertook to mumber the stars, and, in the language of Plinv, to leave the heavens for an inheritance $0 posterity. We do not know, from the writings of this astronomer, or those of his cotemporaries, by what circumstances he was led o apply number to measure the sides and the angles of triangles ; but, in the history of a science, of which the objects are all 1.ecessarily 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. Geometers were, no doubt, at first, satisfied, in the solution of problems, to determine the things soug’it 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 parts. This would be sufficiently exact for common use, and for all the ordinary purposes of mensuration, whether of lines or sur. faces. Coses, however, would sometimes occur, where the errors of such a method were too great to be overlooked, and where the results were paipably inconsis:ent with one another. When, for example, there were given in a triangle, the base and the adjacent an. gles, 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 geoa graphical problems, where it would often happen that the base was small in respect of the distance to be determined; and there, of consequence, soine meihod of solution, more accurate than the construction of a diagrain, woull become extremely desirable. Hipparchus, to whom both the geograp'ıy 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 ingenuily and extent of view sufficient to find out a remedy for ihem. This remedy was the introductio. of calcula:ion founded on the famous property
of the right-angled triangle, said to be discovered by Pythagoras, and now forming the 47th of Euclid's Elements. It is this theo. rem which is the connecting principle of arithmetic and ge, emetry, 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 Hipp.rchus 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 reguiar 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 Menelaus, and in the Almagest of Ptolemy. The spherical trigonometry of the later was all contained in two theorems concern. ing the intersections of four great circles of a sphere; which have been well given, by the lite Di 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 Barbarians 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 inproved 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 sciences advanced, greater accuracy of calçulation was required ; and the difficulty of those calculations, as
well as the time consumed in them, increased in the fame propora tion. What all ma hematicians were now wishing for, the genius of Neper enabled him to discover; and the invention of logarithms introduced into the calculations of trigonometry a degree of fimplicity and ease, which no man had been so fanguine as to expect. Neper made lik wise other great improvements on trigonometry. The theorems which were the foundation of the rules in that science, were not all such as to derive from logarithms an equal degree of advantage. To many of the cases of trigonometry, therefore, though logarithms could be applied, they did not so 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 discovering just what he wanted, or such truths as were exactly accommodated to the occasion, found out two theorems which answered precisely to his views, and afa forded rules perfectly accommodated to logarithmic calculation. Trigonometry, in the state to which it was now brought, continued, with hardly any change, except perhaps a better arrangement of its rules, and a more concise demonstration 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 analysis which has a peculiar algorithm, and is wonderfully adapted to investigation. Before this, trigonometric calculation was only employed to find out an unknown quantity, of such a sort, that there was no occafion to reason about it till it was found; as is the case in mere arithmetical queftions resolved by the rule of three or the extraction of roots. But, in the folution of many mathematical ques. tions, it is necessary to reason about a quantity while it is yet un, known; and the method of finding it, often is the result of such seasonings. This is properly the bufiness of analysis, as diftinguilhed 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 cosmes of circular arches, and of all functions of those 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 Trigonomeirical Ara’ysis. In this new form, the science has been cultivated in France and Germany cver since 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 furcign geometers, however, have come gradually into notice; our trigonometrical treatises within the last ten years have generally contained some of the fun. damental 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 so complete a manner as the treatise which is now before us.
Mr Woodhouse, 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 present, has long cultivated the profoundelt parts of the mathematical sciences, and has done much to turn the attention of his countrymen to subo jects that have been far more Itudied on the continent than in this island. His treatise on Trigonometry is destined, we conceive, for the same purpose. He says, 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 substance in other works. We, for our part, do not think that the author here does justice to himself; but of this we are certain, that we have now before us a very concise, lu. minous and analytical view of an important fcience, which has never been so fully treated of by any writer of our own country.
Mr Woodhouse embraces, in this treatise, not merely the elements of trigonometry, but many of the higher parts, and their most difficult applications. But when we speak of elements, we are reminded that we have lately been accused of defining that term, as it respects geometry, in a very unskilful and incongítent 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 proposed this criterion, that every property of lines, of the firit and second order, which, when translated into the language of algebra, involves nothing higher than a quadratic equation, providing, at the same time, that it be a proposition of very general application, is to be accounted elementary.' This definition certainly comprehends in it more than some authors of great authority are dis. posed to include in the elements of geometry; and, if we would accommodate our definition to the sense of D'Alembert, instead of " lines of the first and second order,' we must say, ' firaight lines and circles,' the two simplest lines of those orders. ki is needless to give, in this place, our reasons for extending our definition farther than the geometer above named has chulun to do: but a remark has been made on that definition, to which it is more material that we should advert. It is alleged, that our criteriou
has has not only the fault juft mentioned, but another directly opposite, that of excluding from the number of elementary truths, certain propositions which have always been ranked among them, and which therefore ought to have been included within the boundary which we profefied to trace. Such,' says the critic, are the pro• positions relating to the contents of similar solids, which, when • resolved, according to the most natural and obvious method, into « algebraic expression, involve cubic equations. Some of them • are capable perhaps of more circuitous folutions, by which cu"bic equations may be avoided : but I believe I may safely chal
lenge the Reviewer to rednce the proposition to a quadratic en
quation, in which it is stated that “ fimilar folid parallelopipeds " are to one another in the triplicate ratio of their homologous “ sides.” Now, to cut short all dispute, and to decide at once concerning the merits of this defiance, let us take the proposition here given, and translate it directly into the language of alge. bra. Because the object of the theorem is to express the ratio of two solids by means of the sides of the solids, we must consider the solids as the unknown, and their fides as the known quantities. Call the folids, therefore, and y, and their homologous or corresponding sides, a and b; then the proposition aí. firms, that x is to y as a3 to 63; therefore =
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 dimension, 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 difficulty of the question depends. Euclid's construction comes to the same thing, for he directs to take c; so that a:0::6:0; and afterwards d, so that a:0::c:d; and then he shows that
=>; and thus, conformably to the notions of the most strict geometry, he expreflies the ratio of the solids by the ratio of two lines, found by the rule of proportion. The error of the critic is therefore quite manifeft; and so gross, as to appear unaccountable on any supposition, 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,
Atier all, we confess, that we have not answered literally to the defiance; we have not reduced the proposition in question to a quadratic equation ; this indeed we hold, as well as the critic though, ire hope, on very different grounds) to be impoflible >