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we have reduced the propofition to a fimple equation, which is much more to our purpose: We have done more than we were challenged to do; but it may not, for all that, be out of the power of quibble and conceit to difpute the victory.
To return to the fubject of trigonometry, we must obferve, that every treatife on that fcience, whether elementary or not, naturally divides itfelf into three branches; the first of which explains in general the arithmetic of the fines, and the conftruction of the tables; the fecond treats of the folution of plane triangles; and the third, of spherical triangles. It is, as we conceive, a great excellence, in treating of thefe, to proceed as analytically as poffible, or with no more fynthetical demonftration than is abfolutely neceffary. One fundamental propofition, at lealt, in each of the above three branches, must be taken from the elements of geometry, and it fhould then be the object to deduce from thefe, by the trigonometric analyfts alone, all the remaining propofitions, whether theorems, or folutions of problems. In the first and fecond parts, this is very fuccefsfully accomplished in the Treatife that is now before us. From two problems in which it is required to exprefs the cofines and fines of the angles of a plane triangle in terms of its fides, and which our author refolves analytically, he deduces, in the fame way, the expreffion for the fine of the fum of two arches, in terms of the fines and cofines of the fimple arches themselves; an expreffion which is known to be the foundation of the whole arithmetic of fines.
From thefe problems are alfo deduced the rules for the folution of the various cafes of plane triangles. But, though this deduction, as well as the former, is made with great brevity and elearnefs, we think that the folution of the cafes of plane triangles, as it is the main bufinefs of the practical ftudent, fhould have been more feparated from the other parts, and with a dittinction more fuited to its importance. In another refpect, we think that there is in this part an error in the arringement; as the auther returns, after giving the folutions just mentioned, to treat again of the fines and cofines of multiple arches, and of the various ufes to which this calculus may be applied. This, which is more general and more difficult, fhould not have been connected with the folution of triangles. More refting places, and a fuller explanation of the tranfitions made from one fubject to another, would have been improvements that could have colt the author very little trouble, and might have faved confiderable perplexity to the young flulent. In other refpects, the inveftiga. Mons are conducted very kilfully, with much brevity, cleruefs
and accuracy, and are confiderably different from thofe which are ufually given.
The values of the fine of the fum or difference of two arches are deduced from the folution of the two problems concerning the plane triangle, as mentioned above. They are moft commonly derived from a geometrical theorem relating to the fines of three arches in arithmetic progreffion: In this way they are deduced by Thomas Simfon, Le Gendre, La Croix, and feveral others. The method of Mr Woodhoufe is more completely analytical, though perhaps it may not be fo easy to a learner. We cannot help thinking, that the moft elementary way of deducing them would be from Ptolemy's property of the quadrilateral infcribed in a circle, which he himself, for a fimilar purpose, introduced into the Almageft, and which has been added by Dr Simfon to the 6th of Euclid. The tranfition from it, and efpecially from that cafe of it where two of the adjacent fides of the quadrilateral are equal, to the theorems in queftion, is extremely eafy, and would be very fatisfactory to one who was juft paffing from the elements of geometry to thofe of trigonometry.
In this divifion of the work our author has given many of the most difficult applications of the trigonometric calculus to the folution of equations, the doctrine of feries, and the investigation of theorems; all in a very concile form, very clear for researches of fo much difficulty, and without the introduction of fluxions or of quantities infinitely fmall. From a curious property of the cofines of multiple arcs, which is first found, as he remarks, in the Mifcellanea Analytica of De Moivre, he deduces feveral curious propofitions of Waring, Vieta, and De Moivre himielf. To thefe we must add the property of the circle generally known by the name of Cotes's Theorem, which was found among the papers of that celebrated geometer, though without any thing that could lead to the analysis or the demonftration. So imperfect indeed were the indications of it, that it was not without confiderable difficulty that Dr Robert Smith difcovered their meaning; fo that he could fay, with laudible exultation, Rev cavi tandem ab interitu Theorema Pulcherrimum.' The title of Theorema Pulcherrimum is indeed well applied to a propofition which, for extent and fimplicity, has perhaps no parallel among the properties of geometrical figures. We may, however, be permitted to remark, that though nothing can exceed the beauty of this property of the circle, its utility as a means of advancing farther in the higher geometry, is not fo great as Dr Smith would have us to beeve, and as he himfeif no doubt imagined. It gives no facility to the integration of fluxionary expreflions beyond what is obtained by the analytical procefs of refolving certain quantities into their trinontial
trinomial divifors. Indeed it would in most cafes embarrass the proceeding, to introduce, in the room of the algebraic formule the conftructions by the circle which might be derived from the Cotefian theorem. It is the natural confequence of the improvement of the mathematical fciences, to give the greatet weight to that which is most truly analytical, and to render the conclufions of algebra as much as poffible independent of the affistance, or, as it would sometimes prove, the embarrafiment, of geometric diagrams.
The investigations above enumerated, afford excellent examples of the use of what we have called the Trigonometrie Analysis, the principles and the application of which may be both very well learnt from the work under review. And here we must observe, that whoever would make a very extensive addition to the field in which this analysis may be exercised, and one in which much novelty may be expected, will do well to look into those properties of the circle which are given without the demonstrations, in the General Theorems of the late Dr MATHEW STEWART. They form an assemblage of truths, hardly less general, or less simple, than the theorem of COTES, just mentioned; and are certainly among the most beautiful propositions known in the whole compass of the abstract sciences. They will be found peculiarly calculated to call forth the resources of the Trigonometrical Analysis; and the difficulties they will present even to those who come armed with that powerful instrument, will be felt as a high eulegium on a Genius, which, without such assistance, and employing only the antient geometry, was equal to such arduous investigations.
The preceding discussions are followed by the construction of the trigonometric canon, or the tables of sines and targents, &c; to which are subjoined, some farther application of trigonometric formula to various kinds of approximation, and to some problems that belong to physical astronomy. The only improvement we have to propose on all this, is one already referred to, a change of arrangement, by which the things that are easy would be made more completely to go before those that are more difficult. From the fundamental propositions for resolving the sines of the sums and difference of arches, into products of the sines or cosines of the simple arches, we would deduce immediately the construction of the canon. This, we think, should be followed first by plane, then by spherical trigonometry; and, in the last place, should come the applica tions of trigonometrical analysis to the solution of problems, the invention of series, demonstration of theorems, &c. The whole of trigonometry would thus be comprehended in four sections, or books, and the learner would proceed gradually from the easier to the more difficult parts.
VOL. XVII. No. 53.
The subject to which Mr Woodhouse next proceeds, is spherical trigonometry; and he has treated it with the same brevity, clearness and extent, that we have remarked in the preceding parts. In explaining the elements of spherical trigonometry, there are two objects to be attained. The first is to demonstrate certain properties of the circles of the sphere, and certain affections of the triangles formed by their intersections; the second is to investigate formulas, or rules, by which, when any three parts of a spherical triangle are given, the remaining three may be found. The first of these does not require any analytical reasoning; the propositions are usually simple corollaries from the geometrical propositions concerning the intersection of planes; and the synthetical demonstration occurs more readily than any other. Here, therefore, we can hardly expect one author to have any great superiority over another; and yet we think Mr Woodhouse has been singularly fortunate in the demonstration of some of these propositions. In one of them, the thing to be proved is, that in an Isosceles spherical triangle, the angles at the base are equal. Now, this cannot be demonstrated by treading in the steps of Euclid, and deriving it, as he does in the case of plane triangles, from this general proposition, that triangles which have two sides of the one, and the angle between them equal to two sides of the other, and the angle between them, each to each, are every way equal. For Euclid's demonstration proceeds on the supposition that the triangles in question can be so laid on one another, as perfectly to coincide; which is not true of spherical triangles; for such triangles may have the three conditions just enumerated, and, in fact, be equal; but, in consequence of not being similarly situated, it may be impossible to make them coincide. Though this consideration is obvious enough, it has not always been attended to, even by good writers; and inconclusive reasonings have in that way been introduced. Thus, in Vince's Elements of Trigonometry, a work otherwise of very considerable merit, the proposition, that if two spherical triangles have two sides of one equal to two sides of another, and the included angles also equal, the two triangles will be equal, is said to be proved just as the fourth proposition of the first of Euclid is proved of plane triangles; though a very little consideration must convince any one, that the principle of superposition, on which Euclid's demonstration depends, is not applicable, generally, to spherical triangles.
Mr Woodhouse's reasoning is subject to no such objections; he has proved, that the angles at the base of an Isosceles spherical triangle are equal, by drawing tangents to the arches that contain the angles to be compared. By producing the two tangents to the equal sides, till they meet the radius drawn through the intersec
tion of those sides, which they do in the same point, and then joining that point and the point in which the tangents, at the op posite extremities of the base intersect one another, it is easy to prove that the two plane triangles, thus formed; are equal, and that the angles opposite to their common base are equal, and consequently also the spherical angles, which are the same with them. This is a demonstration very remarkable for its simplicity; and the more, that so many authors have passed over the same ground without discovering it. We cannot take it upon us to affirm that it is new; but we can say that we have not met with it in any book which we have had occasion to consult.
In the investigation of the rules for resolving the cases of spherical triangles, the object at which later writers have generally aimed, is to deduce the whole, analytically, from one theorem. The geometer, we believe, who set the first example of this, was M. Bertrand of Geneva, in his Elements published in 1778, where he deduced all the rules of spherical trigonometry from one proposition expressing the relation between the three sides of a spherical triangle and one of the angles. EULER, with that superiority in simplification and comprehensiveness which characterized all his productions, did the same thing, in the acts of the St Petersburgh Academy for 1779. This latter memoir has been followed by La Croix, in the elementary treatise on Trigonometry, which is now taught in the National Institute of Paris. This last treatise is remarkable for its elegant and systematic form, and, we are sure, will be perused with pleasure by every one whose love of the accurate sciences has taught him to discover beauty in the structure of an algebraic formula. The same kind of excellence, that of deducing a great variety of propositions from one principle, has been very successfully aimed at by our author. The problems, from the solution of which he derives all the rules, are these: To express the cosine and the sine of an angle of a spherical triangle in terms of the sines and cosines of the sides. There is a double advantage in this manner of proceeding; the whole process is rendered analytical, so that synthetical demonstration is entirely avoided; and there is a close analogy preserved between the investigations of the rules for the resolution of spherical, and of -plane triangles.
It is curious to remark how much has been gained, in all the parts of algebra, by improvements in the mere notation, such, too, as often must have appeared slight, and of no value, but as abridgements of the labour of writing. The introduction of t and y, or some other symbol, to denote a thing unknown, converted arithmetical reckoning into analytical investigation. The use of similar symbols to express known quantities, brought algebra,