Maclaurin, the other two mathematicians are inferior to the learned foreigner. "The methods of Maclaurin and Simpson (for Emerson's is plainly taken from that of the former,) extend to cases, in which more than one property is involved; but they are inapplicacle to the three cases, and the connected problems enumerated in p. 30. "Maclaurin's formula of solution is this: if x and z are functions of r, then if xds zdy be a minimum or maximum, xdy zds. This result is included amongst Euler's. For since xds expresses one property, and dx = dx, or since x is a function of x, we have by form 111, the quantity corresponding to P (see p. 41,) = d (x.), and for zdy expressing the other property, by form 11, dy dx ds dz the quantity, corresponding to P. da; consequently the re dy ds dx sulting equation is d (x-2)= a. dz, and xdy a zds, the same result in fact as Maclaurin's. "Simpson's method is equally restricted with Maclaurin's; it rests too on the assumption of the principle, that the property of minimum or maximum, true for the whole curve, is true also for any portion of it. The want of generality, therefore, in this principle, would vitiate the method in its application to the excepted cases. "The methods just described solve not problems of greater depth and intricacy than those of the Bernoullis; although it must be remarked, they are invested with greater analytical neatness and compactness. They are not however more perspicuous; and even if they did possess greater extent and clearness, it would not suit the purpose of the present tract longer to insist on them, since they conduct us not towards that formula and algorithm, with which the researches on this subject have been closed." (P. 48.) We must, however, be excused for thinking, that a little enlargement of the plan would have made this, at least in the estimation of Englishmen, a more perfect treatise. We would have recommended the addition of two chapters, each in fact independent of the plan, which the author has prescribed to himself. Of these the first should contain a distinct enunciation of the methods* proposed in the books of fluxions which have * The following short account may give some idea of the methods adopted since the time of Maclaurin. Emerson presents us with two rules. The first is deduced from the ordinates of a curve in arithmetic progression. The determination of the equation depends upon the position of the intermediate ordinate. The principle upon which his demonstration is founded is, that the maximum or minimum, which belongs to the whole curve, must belong to the part intercepted between the ordiuates. Thus if been published in this country, and a detail of the reasons why they are defective: the second should give the elements of the science, on geometrical principles. Many persons who shrink from the pursuit of abstruse enquiries, would read and understand the geometrical process, and make themselves masters of the elements. A very elegant and perspicuous chapter on this subject has recently appeared in the third volume of " A Course of Mathematics," by Dr. Hutton, who has reduced into system the chief propositions of L'huillier, Le Gendre, and Horsley, together with some additional propositions, which those geometers had not deduced. We should have recommended a geometrical chapter on the elements with the more earnestness, because with some writers it seems a settled principle, that geometry is never to be admitted, where analysis can by any contrivance supply its place. If the length of this article appears to be disproportioned to the magnitude of the work under consideration, this circumstance one given quantity be a maximum L = = A+B+C+D+E+ &c. and another which is required to = a + b + c + d + e &c. and all the quantities be supposed constant except two, which correspond, we have C + D constant, and c÷d a maximum; hence C + D = 0; and ċ + ¿ 0; and the parts C and D, c and d, being expressed in terms of the same variable quantities, we can from the solution of the equation determine the nature of the curve. His second rule applies to cases which are somewhat more complex. He supposes Az - Bx to be a maximum or minimum, and proves Ar Br, A and B being functions of 1 or 2z. This process merely reciprocates the functions A and B. The demonstration of Lyons is similar to Emerson's; the same remark applies to Mr. Vince's. = Simpson's theorem supposes that when fymu is equal to a given value, then Sy'. y is a maximum or minimum; and his conclusion is that 27 is a constant quantity. Or more generally, if R and S u2 + y2 must be a constant quantity. This expression is to be applied to particular cases, and the equation of the curve to be deduced by substitution. For the cases in which a new condition is introduced, and to which this q are constant quantities. The principle of uniformity is introduced by most writers in the solution of cases after the manner of John Bernoulli. VOL. III. NO. V. E will at least serve to prove to Mr. Woodhouse, that we appre ciate with due respect the labour and ingenuity which he has displayed in the compilation of his treatise; and will also, we hope, preclude the necessity of additional recommendation to the lovers of analytics. If there be any part of the volume which we could wish to expunge, it is the notice of the quarrel between the two Bernoullis; and we wish it merely for the credit of science. When liberal minds are engaged in mathematical disquisitions, and abstract truth is the only object of research, it might be supposed that acrimonious feelings would never intrude; or if the detection of occasional error did sometimes alarm the vigilance of self-esteem, that the pain would be slight and the recollection of it would soon pass away. Of the irritation which could induce John Bernoulli to treat with marked disrespect the memory of a brother, who had been dead for sixteen years, and against whom he had no reasonable charge, we trust there are few examples. The fact however must be recorded as an additional proof, where proofs in abundance exist already, that no attainments in science can supersede the necessity of religious principle; that philosophical speculation is unable to purify the mind, and that we must look to christianity alone for the conquest of the passions and the reformation of the heart. If it could be made out, but it certainly never can be proved, that the love of abstract verities is fatal to those charities and affections which bind man to man, and form the harmony of social life: if every mathematician in short were of the temper of John Bernoulli, we should consider attainments in science as purchased indeed at an extravagant rate; and the mildest observation which we could bring ourselves to pass upon such learning would be in the words of the poet, "When I weigh this seeming wisdom well, COWPER. ART. III. The West Indians defended against the Accusations of their Calumniators; or, Facts versus Prejudices. By a Gentleman. Meyler, Bath. Mawman; Robinson; Hardy; Lon don. 1811. 2. The present ruinous State of the West India Islands submitted to the People of the British Empire, with a few Remarks upon the Imposition and Oppressions, under which the Merchants and Planters of those Islands have long suffered. By a Native of Jamaica. London: Sherwood and Co. 1811. 3. An Essay on the good Effects which may be derived in the British West Indies, in Consequence of the Abolition of the African Slave Trade; including an Inquiry into the present insular Policy of those Colonies. By Stephen Gaisford, Esq. London: Baldwin; Hatchard. 1811. 4. Notices respecting Jamaica in 1808, 1809, and 1810. By Gilbert Mathison, Esq. Stockdale. 1811. 5. Practical Rules for the Management and medical Treatment of Negro Slaves in the Sugar Colonies. By a professional Planter. London: Vernor and Co.; Hatchard. 1811. 6. A Letter to the Governors, Legislators, and Proprietors of Plantations in the British West India Islands. By the Right Reverend Beilby Porteus, D.D. Bishop of London. Cadell; Rivington. 1808. EVEN in these extraordinary times it would probably excite some surprise, were the farmers of Great Britain to present a petition to the legislature, setting forth, that they laboured under great "imposition" and "oppression," inasmuch as it was found impossible for them to make large profits by their farms, consistently with their immunity from all personal superintendence; that their fields would not plough themselves, or their corn and cattle spontaneously proceed to the market or to the shambles, while their owners were enjoying the busy pleasures of the metropolis, or the vacant flutter of a watering place. Nor, supposing them to have succeeded by unfeeling, oppressive, or illegitimate contrivances, in rendering these opposite objects compatible, would their complaint appear more worthy of attention, should they fancy themselves injured by a legislative prohibition of these same contrivances. As well might the smuggler complain of the reduction of the duties on tea and spirits, or the band of midnight robbers and assassins, of those precautions of police which we trust will shortly deprive that numerous portion of the community of the vested interest which they have assumed to themselves in the acquisitions of industry. That the complaints preferred in the two first of the abovementioned pamphlets bear somewhat of the character to which we have alluded, we think will evidently appear from a brief summary of the contents of the third and fourth; while the two last afford experimental and convincing proofs, that the remedy for all the evils complained of has long been in the hands of the complainants; nay, that they have been solicited by motives of interest, of honour, of humanity, and of religion, by every sentiment in short which can influence the heart and the conduct of men, to hasten the application of it. Those who have been deaf to the solicitation, are very naturally suffering for their obduracy; and it is because we think that the skirmishing at the outposts, announced in these pamphlets, portends a grand attack upon the main citadel of the measure itself, that we have thought it incumbent upon us to investigate the merits of a subject which, amply as it was discussed for twelve years preceding the year 1807, is now reappearing under an aspect somewhat new, and supported by arguments which were merged in the supreme importance of the original question. The pamphlets before us are all (excepting the last), from the pens of persons of professional knowledge in West Indian affairs; their authority is therefore equal, and we think that a fairer mode of discussing the question can scarcely be adopted, than by first stating the evils which some of these gentlemen have felt, and after opposing to them the advantages experienced by others, to point out the practical results which have afforded complete satisfaction to the minds of the remainder. Thus may we hope to arrive at conclusions, which, however adverse they may be to the mistaken interests and the bad passions of some of the parties concerned, cannot with any shew of reason incur the charge of visionary humanity or morbid sensibility. We have somewhere read that the characteristic qualities of an English gentleman are courtesy and courage. But the "gentle (Mr. Edward White)* whose lucubrations it is our duty first to notice, seems disposed to atone for any little deficiency, in the former quality by a double portion of the latter. For in the very teeth of the recorded enormities of Messrs. Hodge and another planter whom we abstain from naming, to which the newspapers of the day gave enough of publicity to exonerate us from the painful task of detailing them, he actually sets out with the professed object of justifying the slave trade by proving "that the condition of the negroes in the West Indies is preferable to their state in their own country." With this laudable * See the title to the first pamphlet at the head of this article. |