to the mathematical studies at Cambridge, it will probably be expected that we should specify some of our objections to his treatise. 1st, Then, we think it defective in point of arrangement. Things are often brought together which have no mutual connection; and others thrown wide asunder which ought to have been treated consecutively. Some particular enquiry or example, which seems suddenly to strike the author, is as suddenly thrust in, interrupting the regular progress of his problems, and in some cases making him forget that order altogether. Thus, after numbers 1, 2, 3, &c. on to 17; we have problems, 1, 2, 3, 4, in the latter of which it is required to express the tangent of the sum and difference of two arcs in terms of the tangents of the simple arcs.' The author then chooses to make a disgression of 21 pages, in which he intermingles theory and practice; and at last loses himself so completely as to resume the course of his problems with the sixth. After problem 10, we are favoured with another rambling disquisition; at the end of which the author gives the examples of the utility of trigonometrical formula which we quoted above, relinquishes his enumeration of either articles or problems, and begins a series of propositions. All this is excessively disadvantageous to the student. Had Euclid's Elements been founded on such a model, we may venture to affirm that they would have perished with the author, instead of being read with benefit and admiration at the end of 2000 years. 2dly. Mr. Woodhouse's demonstrations are in more than one instance obscure, and even inelegant, while his practical examples are deficient in perspicuity. In shewing the mutual relations of sines, tangents, secants, &c. his method is very operose: and, in his manner of deducing all the formulæ useful in trigonometry from the fundamental expressions for the cosine of an angle, though he has evidently and avowedly profited by the researches of Euler and others, his method evinces far less smiplicity and clearness than it is susceptible of. When A, B, and C, are the angles of a plane triangle, and a, b, c, the sides respectively opposite those angles, it will be manifest, from a mere inspection of the figure, that a=b. cos. C+c. cos. B. c=a. cos. B+b. cos. A. Multiplying the first of these equations by a, the second by b, the third by c, subtracting successively each of the products thus derived from the sum of the other two, and dividing by 26 c, 2ac, 2 a b, respectively; we shall have cos. A= b2+c2-a2 26c 2 ab These are the same as the expressions given at page 14; but they have the advantage of being derived more immediately and independently from pure trigonometrical principles. If, in these, we put, instead of cosine A, cosine B, &c. their equivalents 1-sin. A, 1-sin. B, &c. we immediately deduce theorems for sines of angles of plane triangles in terms of the sides; then arrive at the most useful results, 2 sin. B And then, by a series of very simple operations, all the rules for plane triangles, as well as the most useful theorems relative to the sines and cosines of the sums and differences of arcs or angles, might be deduced perspicuously in less than six pages. This Mr. Woodhouse has attempted; but his process is not so luminous and orderly as that pursued by several of the models before him. So again, in the spherical trigonometry, where the deduction is from the equations cos. a=cos. b. cos.c+sin. b. sin. c cos. A ; each of which comprises implicitly the expression sin. b. sin. A=sin. a. sin. B, which lies at the foundation of spherical trigonometry; though he starts from the same theorems as Lacroix and Puissant, his process is less elegant even than that of the latter. 3dly. Another defect in this work is, that the results are not always tabulated, or brought together into one place. Mr. Woodhouse must have known, that Tho. Simpson condenses all the rules for plane and spherical trigonometry, which he had previously investigated, into two short tables: that Lacroix, in like manner, throws his most useful results relative to trigonometricial formulæ upon two pages; that Cagnoli exhibits, in the comprehensive tables at the end of his trigonometry, every thing valuable in the doctrine of plane and spherical trigonometry, or in the solutions of equations by sines and tangents: and that Delambre, in his excellent preface to Borda's logarithms, has done the same in a still smaller compass. To condense the results of multifarious investigations, and exhibit them for use in one place, is at all times beneficial; but it must be peculiarly so in a work like this before us. 4thly. The author has entirely omitted some very curious and are concerned. interesting formula, where more than two arcs Let, for example, a, b, and c, be any three angles whatever; then is sin. a. sin. (br)+sin. b. sin. (c-a)sin. c. sin. (ab)o Let a, b, c, d, ., be any series of angles, then the two following equations always obtain : Sin. (a+b). sin. (ab)+sin. (b+c). sin. (b-c)+ &c. . . . . + sin. (a). sin. (pa) = o Cos. (a+b), sin. (a-b)+cos. (b+c). sin. (bc)+ &c. . . . . +cos. (a). sin. (a) = o We have also, sin. a+sin. 3a+sin. 5a+ &c. = 0 cos. a+cos. 3m+cos.5a+ &c. = • And, if the whole circumference be divided into any number whatever n of parts, each of which is a, and if c represent an arc 2a, the two following equations always obtain, whether a and c be considered as positive or negative. Sin. (a+c) + sin. (a + 2c) + sin. (a + 3c) + +cos. (a+nc) o. These formulæ are more easy to investigate than many which Mr.. Woodhouse has given: and they find a very ready and advan tageous application in the earliest researches relative to polygonometry, and other branches of mathematical inquiry, as yet but little pursued in this country: and for no other reason that we are aware of, than because the preliminary theorems have not found their way into any of our elementary books. 5thly. Mr. Woodhouse suffers his prejudices (shall we call them?) to operate somewhat too frequently. We have seen that Ludlam's is the only English book of Trigonometry which he condescends to mention, though it is, in fact, inferior to every one of the treatises noticed at the commencement of this article; having nothing to distinguish it but the author's perspicuous mode of tracing the mutations of the signs of sines, cosines, &c. when found in different quadrants. Why then is this mentioned, while no notice is taken of Vince or Bonnycastle? Why, again are Sherwin's Logarithmic Tables so strangely preferred to Hutton's? And lastly, why is the discovery of properties attributed to Dr. Waring, which were known a century before he was born, and which he only slightly modified so as to apply them to a particular purpose? since Mr. Woodhouse is, after all, constrained to acknowledge that Vieta is not to be entirely excluded from the honour due to the invention of the theoIt would be strange indeed if he were; and it is equally rem. 6 strange that Briggs's analogous theorems, full as curious as Waring's, should be totally overlooked. 6thly. Mr. Woodhouse is not so explicit as we could wish in his ascription of discoveries and investigations to former authors. He says, it is true, that though he once believed much of his matter was new, yet now he thinks that it contains nothing, of which he could not point out the substance in other works.' He might, indeed, do this, and more. He might not only point out the substance, for example, of his demonstration of Legendre's theorem for solving spherical as plane triangles, in the Geometry of Legendre, and in the Géodésie of Puissant; but as he must have known that all three had the same source, he might have excluded this expressly from the matter which he once believed to be new. He might also point out' something more than the substance' of his investigations of Demoivre's and Cotes's theorems, in Lagrange's work, entitled 'Leçons sur Calcul des Fonctions,' where it obviously originated. Direct avowals, on these occasions, would surely have caused no real diminution in Mr. Woodhouse's reputation for talents; while the omission of them may subject the author in some degree to the suspicion of disingenuousness, of which we do not and cannot think him guilty. We must not conclude without noticing the affectation of this writer's style. He does not, indeed, like the author of the Mathematical Treatise noticed in our last number, run about in pursuit of tropes and metaphors, till he is giddy; but he cuts his English too much on Latin, and he seems anxious to discard the definite article from his vocabulary. Thus, we have aliter mode of computing sine of small arc,'-' aliter method of solving the fifth case,' construction of trigonometricial canon,'' if equation be,''the angles of triangle,'' sides of supplemental triangle,'—' area of lunæ, angle intermediate of,' &c. These are liberties with the language which no talents can justify, and which no weight of character, we trust, will ever render popular. But we must now take leave of Mr. Woodhouse. He is evidently a man of extensive reading and investigation. If he would cultivate order and simplicity, borrow more scrupulously from his predecessors, and manifest a more just, if not a more favourable regard for the writers of his own country, we should not despair of meeting him hereafter in the very foremost class of writers on the Elements of Mathematical Science. ART. IX. The State Papers and Letters of Sir Ralph Sadler, Knight Banneret, edited by Arthur Clifford, Esq. To which is added, A Memoir of the Life of Sir Ralph Sadler, with Historical Notes by Walter Scott, Esq. 4to. 2 vol. pp. 1442. Edinburgh, Constable and Co.; London, Cadell and Davies. 1810. HE British library has become rich in such collections as this we except the Cabala, Digges's Complete Ambassador, and the Reliqua Wottonianæ, little worth naming had appeared before that period. In subsequent publications various plans have been adopted to render them subservient to their proper uses. The noble series, however, selected by Haynes and Murdin from the Cecil papers, Winwood's memorials, Forbes's state papers, and the Hardwicke and Strafford papers, are almost mere transcripts, unappropriated by any aid from their respective editors to their due stations in history; and with indexes, if any, almost useless. The laborious and accurate Strype has introduced into his various historical works a multiplicity of extracts from most valuable original correspondence, and has weakened their interest by breaking in on their integrity. From documents of this nature nothing can be spared. Dr. Birch, in his View of the Reign of Elizabeth, contrived, with infinite labour, to weave into his narrative, generally without abridgment, the numerous letters of the Bacon family on which that work is chiefly founded. This mode of publication is perhaps preferable to all others; but to perform it well requires a degree of zeal, industry and patience, which few writers except Birch have possessed, and a nicety of composition in which he was deficient. The method adopted by Macpherson, and Dalrymple, in their disclosure of the very valuable Stuart and Hanover papers, is a faint and irregular imitation of Birch's plan. Sir John Fenn's Paston Letters, a collection rendered very curious by its early date, but of little value in any other point of consideration, affords the first instance of a regular series of elucidation by marginal notes, of all that is worthy, and, by the way, of much that is unworthy, of notice in the originals. The Talbot and other papers, exhibiting a variety of curious matter, particularly with regard to Mary Queen of Scots during her imprisonment, were published of late years by Mr. Lodge, Lancaster herald, under the title of Illustrations of British History, Biography, and Manners: to these also is attached a series of notes, on a scale far more extensive and various, and abounding particularly with biographical information, chiefly derived from, and in a great measure peculiar to, the curious library of that college of which he is a member. |