bem + L, and m + L, and therefore could be assigned by merely prefixing the proper characteristic; but then, in order to know the num N . bers corresponding to 3" x N and we must multiply and divide N by 3" ". 3, and powers of 3 : we cannot multiply and divide by simply altering the place of the point or comma that separates integers from decimals, 60 that, in fact, not knowing by inspection such numbers as 3 N, 9 N, 27 N, and 3m x N, we should be obliged to insert the logarithms of all num. .bers in the tables. A single instance will elucidate this statement: with a base = 3, the logarithm of 2.7341 equals .915519, then the logarithms * of the numbers . 8.2023, . 24.6069, 73.8207, 221.4621 .' are 1.915519, 2.915619, 3.915519, 4.915519. . .: for the numbers 8.2023, 24.6069, &c. are produced by multiplying 2.7341 by 8, 32, 3, 34, respectively, but they are known only by actual multiplication, and consequently it would not be sufficient to insert in ta. bles of logarithms constructed to a base = $, the logarithm of 2.7341 only, but those of 8.2023, 24.6069, 73.8207, 221 4621, &c. must be also inserted : and it is plain, that the logarithms of 27.841, 278.41, 3734.1, 27341, .02741 must be also inserted : if these latter logarithms are not inserted, the computist would be obliged to undergo the labour of forming them, by adding to the logarithm of 2.7341, respectively, the jogarithms of 10, 100, 1000, &c. computed to a base = 3. . .e ri This is not the sole practical inconvenience that would arise from using a system of logarithms with a base not equal to 10: we might, indeed, as it has been explained, by elight arithmetical operations, directly find the logarithms of aumbers from cables of no greater extent than those which are in use, but the reverse operation of finding the number from the logarithm, could not at all conveniently or briefly be performed; for the logarithm proposed might be nearly equal to a logarithm which the tables did not contain: 1 hese considerations will, perhaps, be sufficient to shew the very great improvement that necessarily ensued on Briggs's alteration of the Logarithmic Base. The real value of that alteration does not seem to have been duly appreciated by writers on this subject." · Mr. Woodhouse is an author whom it is really difficult to characterize, He has some excellences, but he has many defects ; and the latter sadly preponderate. A rea der of this book will meet with specimens of knowledge and of ignorance, elegance and slovenliness, taste and awkwardness, adroitness and clumsiness, perspicuity and obscurity; pedantry and vulgarity, affectation and simplicity, vaniiy and modesty ; and thus of almost every quality and of its opposite; The consequence of this is, that the student must necessarily lose all confidence in his author; a very unfortunate circumstance to a mathematical reader. He who reads a work on any branch of geometry or of analysis, expects to find every proposition affirmed afterwards demon. strated ; yet be does not follow an author pleasantly, if he der; and unnatwork ber follow him cautiously, under the influence of doubt, No person who ever read and comprehended the demonstrations of the first 10 propositions of Euclid's first book, would feel the least doubt as to the truth of the eleventh proposition : he would believe it, on the authority of his author, antece. dently to demonstration : the perusa] of the demonstration would communicate delight, would strengthen conviction, would confirm belief, and would excite still greater confidence and belief in all that was to follow: the labour of study becoming wonderfully lightened by the advantages and fascinations of the method. Something like this is always effected by the best modern as well as ancient authors ; as must be evident to every judicious reader of Simpson, Mac. laurin, Bossut, or Lacroix. But no such advantages are offered to the readers of the work before us. Here the arrangement is forced and unnatural; there is no appearance of the lucid order; and even the punctuation is often so · much neglected as to render the author's meaning doubtful. He commences with an enumeration of articles or paragraphs, which is carried as far as art. 18. He then gives * problems relating to sines, cosines, and tangents of angles, and of sums and differences of angles, Then the order of the problems is broken, to make room for more than 20 pages of solutions of the cases of plane triangles: then we have. Prob. 6. (the author has not favoured us with Prob. 5) on the sine and cosine of twice an 'arc, &c. and then Probs. 7, 8, and 9, relating to expressions or multiple arcs, chords of arcs, Cotesian theorem, &c. Then appears another problem 9, in which tlie numerical value of the sine of 1 minute is found, to radius urity. Then farewell problems! as tre are to be treated with about 16 pages of miscellaneous discussions, which are followed by four “ instances of the útility of trigonometrical formulæ. Our author now takes leave of articles, problems, and instances; and presents us with 17 propositions relative to spherical geometry and trigonometry. Then he gives observations, remarks, and rules respecting ambiguous cases, the affections of sides and ana gles, the several combinations of cases, in spherical trigono. metry; after which, having forgotten that he had before exbibited seventeen propositions, he gives another Prop. 17, which be divides înto 6 cases, with aliter method of solving the fifth case,' aliter method of solving the sixth case, and so on. This brings us to the 141st page, after which the contents of the treatise, (with the exception of the break occa. sioned by the introduction of the word appendix, at p. 155.) Vol. VI. sk bid defiance to every thing like order, and are put down with little more mutual relationship and dependence, than if the materials had been taken promiscuously from a variety of heterogeneous disquisitions in the author's port-folio, and sent to the printer just as they came to hand. With a book so immethodically arranged before us, no wonder if our subsequens remarks should be insulated and miscellaneous. Such a deviation from strictness of method, as we have just mentioned, must of necessity lead the way to many awkwardnesses, deficiencies, and obscurities : and, in truth, there is a pretty large stock of them; we can only mention a few. Among the awkwardnesses, we sometimes find the same or connected subjects treated in different parts of the book. Thus, instead of investigating formulæ generally for sines, cosines, &c. supposing radius = 1, and then shewing how much they become simplified by making radius = l, he proceeds thecontrary' way, and shews both at pages 12 and 22 how to introduce radius = r, into formulæ deduced on the supposition of radius = 1. Thus, again, our author gives formulæ of ve. rification in different parts of bis work; the consequence of which is, that some of the most useful are omitted, and especially the comprehensive formula i sitt. (54°+A) I sin. (540-A) — sin. (18°+A) — sin. (180-A) = sin. (90°--A). .. Indeed, the omissions in different parts of the treatise, occasioned by this want of method, cannot easily be enumerated. . Another kind of awkwardness, is that of analytical expreso sion. Thus, Mr. Woodhouse uniformnly puts (sin. A), (cos. A), instead of sin. 2A, cos. 2A: whether from an affectation of singularity or not, we cannot say; his method has certainly no 11 advantage. Again, Mr. W. pnts Tor , &c. instead of V2, V3, &c.; and that in some such places, as seem to indicate that he has actually performed the tedious operations in division required by his expressions, and that he is perfectly unconscious that -=N N. Yet we are unwilling to be 'n na lieve, that one so adroit in the use of symbols should be unacquainted with the management of surds. · Another species of awkwardness is that, by which easy things are made difficult, and plain things obscure. This is remarkably the case in our author's method of finding expressions for the sines of the angles of a plane triangle in terms of the sides and the area; he employs about a page and a half in shewing, what might have been done far better in four lines, that in this triangle whose apgles are A, B, C, of the ear he se beenes are 2 area sin. He likewise pursues - He likewise pursues the most circuitous way possible in finding expressions for secants and cosecants; when he might at once accomplish the business, by saying, rad 2 from similar triangles, cos: rad :: rad : sec. = , and sin: rad :: rad: cosec = ". . But our author's happy knack at sin, unintelligibility is never more ingeniously exemplified, than in the interstice between Prob. 4 and Prob. 6, where he gives the solutions of the several cases of plain triangles. We once thought we could without difficulty work any example in right-aligled plane triangles; but after reading what Mr. W. says about them we began to feel ourselves puzzled, and shall be long before we again speak confidently on a subject which we find can be made to appear so abstruse. In this bewildered state, however, we pass on to oblique angled triangles, where our embarrassments thicken : there are “ given (A, B, a). Req. (b, c)”: we print verbation et literatim, that our readers may understand the matter more clearly. Here Mr. W. exhibits a solution which we presume to think incomplete, for there is still “ Req. (c).” The next point which this gentleman renders more obscure by elucidation, is the ambiguous case of oblique plane triangles, where two sides and an angle opposite to one of them are given. It is attempted to make the matter more clear and plain, by calling the same angle sometimes A, at others P, at others 'Q; and another angle sometimes B, sometimes N. Maugre all this immense care to illustrate an instance of ambiguity by rendering it more ambiguous, our author has not assigned the actual limits between which the ambiguity subsists. "If B be the fixed, and A and C the variable angles, then, when the length of b iş between ą and a sin. B, there may be two triangles to the same data ; when b exceeds a there can be but one triangle; and when b is less than a sin, B, the example is improperly proposed, and the construction impossible. For like reasons it happens that an essential step in the inves. tigation at p. 51 is illegitimate and unwarrantable, and the deductions at p. 23 defective. , The spherical trigonometry of this author is, we think, more open to çensure 'than the plane trigonometry. His definitions are loose; as when he says, ' a spherical triana' gle is called rectangular, isosceles, equilateral, in the same cases that a plane triangle is." Yet a spherical triangle may be at once rectangular and equilateral, which we apprehend, with the utmost deference to the contrary suggestion of our learned brother, a plane triangle never can. We are also inclined to conjecture that an isosceles plane triangle can never have two right angles, though an isosceles spherical triangle certainly can. Some of our author's demonstrations, also, in spherical trigonometry are equally vague and unsatisfactory; we may specify that of Prob. 12.: nor are we much satisfied with that which relates to Napier's rules of the circular parts, The rules for right angled spherical triangles are defective, in so far as we have no formulæ by which to obtain results in terms of t sides or angles, or of 450 + 1 sides or angles: these are at least as useful as the variety of theorems to which our second extract from Mr. Woodhouse's book refers. In lieu of these, however, we are presented with Napier's Rules for circular parts, to which, in our estimation, Mr. Woodhouse attaches a most extravagant value, Were he as conversant with the practice of trigonometry as with the theory, he would doubtless have spoken in a lower tone. After all, he forgets to mention à particular connected with these rules, which alone renders it next to inpossible to forget them ; we mean, that the vowels in the terms sin., cos., tan., are the samo as those in the first syllables of middle, opposite, and adjacent parts; for bence it becomes peculiarly easy to remember that sin. mid. = rect... tan. ad. = rect. cos. op, By the way, what could be the reason, that Mr. Woodhouse omitted Walter Fisher's extension of these rules, conveyed by the Cabalistic Sao, satom, tao sarsalm ? We might now proceed to make some remarks upon the clumsiness of Mr. Woodhouse's diagrams, and his extreme de ficiency in practical examples, especially in the plane trigonometry : but the magnitude to which this article is swelling renders it necessary to desist. We must, however, re. cord a few specimens of his style, which is certainly more peculiar than the style of any other person we recollect, even among the contributors to the Monthly and Critical Reviews. After that the decimal point has been', &c. p. 2.. «The demonstration of this method is not conciser, &c. &c. but the rule and connected computation is conciser', p. 29, . By same form', p. 32. • Angle intermediate of', p. 36, • The use of the aliter solution,' p. 42. Query, which of the meanings of aliter can be substituted to preserve this sentence from being pure nonsense ? Construction of tri. gonometrical canon', p: 62. • Descend down the same steps', |