tion ;-a monument truly are perennius, and only to be obliterated by the fuperior ingenuity of others, in the fame walk of science. The more fully to evince the merit of this extraordinary genius, Lord Buchan proceeds to give an account of the state in which Napier found arithmetic, and of the benefits which the art received by his difcoveries. The firft of his mechanical devices was the Rhabdologia, or the art of computing by figured rods. Thefe are fo well known by the name of Napier's bones (being probably originally made of ivory or bone), as not to require the particular description which Lord Buchan gives of them; though, perhaps, a full account of them was neceffary, in a work profeffedly containing the hiftory of Napier's invensons. The multiplicationis prontuarium is another of Napier's mechanical contrivances for leflening the operations of arithmetic. Any description of this machine, without the delineations, would be unintelligible, as woul alto the method which Napier practifed, and called arithmetica localis, of calculating by counters peculiarly placed on the fquares of a cheis board, or fimilar table. Lord Buchan gives a clear idea of the form and use of these arithmetical machines, and the reafons on which the different operations on them are founded. The hint of the Rods, and of the Promptuary, which is only an improvement of the Rods, feems to have been taken from the Abacus Pythagoricus; and Napier's acquaintance with chefs, probably gave rise to his Arithmetica localis. The Promptuary, at leaft for multiplication, is greatly fuperior to the other two; for partial products of two numbers, each confifting of ten places of figures, may, by a little practice, be exhibited on that machine in the space of one minute, and no numbers are required to be written out, except the total product. Had logarithms remained undifcovered, these machines would, in all probability, have been in common use among calculators: at prefent they are only regarded as mathematical curiofities. In the next fection, the author gives Napier's Theory of the Logarithms, which conceives them to be generated by the motion of a point having an accelerated or retarded velocity. After amply explaining this theory, Lord Buchan fhews its refemblance to, or rather identity with the doctrine of Aluxions, as delivered by Newton. He fays, under the article Habitudines Logarithmorum, Napier thus expreffes the relation between two natural numbers and the velocities of the increments or decrements of their logarithms, "Ut finus major ad minorem ita velocitas Incrementi aut Decrementi apud majorem." What difference is there between this language and that of the great New ton ton now in use,x:y :: Log.x: Log. y. ?' We have tranfcribed this paffage because we think the quotation from the Canon mirificus is erroneous: not having that work at hand, we correct the paffage thus from memory; ut finus major ad minorem; ita velocitas Incrementi aut Decrementi apud minorem, ad velocitatem incrementi aut decrementi apud majorem. The remainder of the fection is employed in fhewing that Napier was the inventor of logarithms, and in refuting the opinions of those who attribute their invention to earlier mathematicians. Lord Buchan proceeds to give Napier's method of conftructing his logarithmetical tables, and then fhews that the common logarithms were firft devifed by Napier, and prepared for publication by Briggs. The difadvantages of Napier's firft logarithms were fufficiently apparent; but whether Napier or Briggs firft fuggefted the new fpecies of logarithms, is a queftion which the learned have not perfectly decided. By extracts from several books, it appears that the common logarithms occurred to Napier before they occurred to Briggs. Lord Buchan difmiffes the enquiry with obferving that Napier and Briggs had a reciprocal efteem for each other, and there is not the imalleft evidence of there having exifted in the breaft of either, the leaft particle of jealoufy;that after the invention of logarithms, the difcovery of the beft fpecies of them was no difficult affair ;-and that the invention of the new fpecies of logarithms is far from being equal to fome other of Briggs' invention.' The next fection treats of the improvements that have been made on logarithms after the death of their inventor. Next after Napier and Briggs, Gunter has the beft claim to the gra titude of the Public. He firft applied the logarithms to fcales, which are to this day in common ufe in the Navy, and in the Excife. Mercator, more than 50 years after Napier's death, invented an infinite feries expreffive of Napier's logarithms, but Gregory of St. Vincents had, 20 years before this period, thewn that the affymptotic areas of the hyperbola were logarithms. It is somewhat aftonishing that this identity between the hyperbolic areas and logarithms was not fooner obferved; for had Napier placed his two lines (one of which generated numbers by the equable motion of a point, and the other logarithms by an accelerated motion) at right angles to each other, he must have found that the curve of the hyperbola would have been defcribed. This circumftance occafioned the denomination of hyperbolic, which was given to Napier's logarithms, and which has been, and now is, ufually adopted by moft mathematical writers. The abfurdity, for we cannot give it a better term, of calling Napier's logarithms hyperbolical must be apparent, when it is confidered that all logarithms are hyperbolical; the only differ ence ence between different fpecies of logarithms being the inclination of the affymptots of the hyperbola to each other. Thus Napier's logarithms correfpond with an hyperbola whofe affymptots are at right angles, when the fine of the angle is unity, which is the modulus of that fyftem of logarithms. Briggs's, or the common logarithms, correfpond with an hyperbola whofe affymptots are inclined at an angle of 25° 44+ whofe fine is .43429, &c. which is the modulus of Briggs's logarithms. All logarithms are therefore hyperbolical; and it feems that the epithet hyperbolical was given to Napier's unjustly, and probably with a view to fupprefs the inventor's name. We must obferve by the way, that all through this publication, the words area and areas are mifprinted arca and arcas. The remaining part of this fection defcribes the different tables that have been published, and the preference is given to the tables portatives of Monf. Jombert, published at Paris in 1783. Why Lord Buchan prefers Jombert's tables, printed in France, to Hutton's, printed in England in 1785, is fomewhat extraordinary, when his Lordship points out an error in the French edition, but none in the English. It muft, however, be acknowleged that the French tables are much more diftinctly and elegantly printed than the English. This we fay from having feen both books, and not from the fpecimen which Lord Buchan's printer has given of Jombert's tables, where there is an error by placing 9019 in a wrong line. The 7th fection describes the use of logarithms; and the 8th, which clofes the work, enumerates the important improvements which Napier made in trigonometry. An appendix is given, containing, Ift, the analytical theory of logarithms; 2d, A table of Napier's logarithms of all natu ral numbers from 1 to 101, to 27 places of figures; we can pronounce this table correct from having examined many of the logarithms. 3d, A collection of trigonometrical theorems. 4th, A defcription of the hyperbolic curve as connected with logarithms; and, 5th, The principal properties of the logarithmic curve. From the recital of the contents of this performance, it appears to have been a work of no fmall labour on the part of Lord Buchan as well as of his affociate, Dr. Minto; to whom his Lordship acknowleges himself indebted, especially in the mathematical department. Napier's life, we are informed, is to be fucceeded by other lives, in which Lord Buchan is at prefent engaged, on condition that this fpecimen meets with the approbation of the learned world. His Lordship's zeal is great, and undoubtedly demands the gratitude of the Public. When noblemen not only patronize literature, but themfelves take an active part in its cultiva tion, the greateft expectation may be formed that its true interefts will be more generally promoted. We cannot close this article without mentioning a defect which Lord Buchan may eafily avoid in his future publications. His book is carelessly printed. The errors, however, are fuch as any mathematician may correct, and must be attributed to the inattention of those who undertook to conduct the work through the prefs. R......m. ART. XI. A Poem on the Bill lately passed for regulating the Slave Trade. By Helen Maria Williams. 4to. pp. 24. 1s. 6d. fewed. Cadell. TH 1788. HE accounts lately given to the Public refpecting the Slave Trade, were horrid enough to call into vigorous exercife the amiable fenfibility of the female breaft. By the ladies, this fubject has been contemplated through the pure medium of virtuous pity, unmixed with thofe political, commercial, and felfifh confiderations which operated in feeling the hearts of fome men against the pleadings of humanity: to find THEM, therefore, writing on it, by no means excited wonder. Though among the laft, Mifs Williams is not the leaft deferving of notice. In eafy, harmonious verfe, fhe pours forth the fentiments of an amiable mind; nor do we recollect, among the poems which have lately attracted our attention, to have perufed one with more pleasure than that which now lies before us. She thus addreffes her country, on the subject of her poem: That fooths defpair, is fram'd by Thee! Gilds the thick hade with fofter day.' The laft lines of this extract lead us to obferve that our poetess is peculiarly happy in the choice and application of her fimilies: The The traders in flaves are described as beings • Whose harden'd fouls no more retain Their fubftance change, when lock'd in frost, Who view unmov'd, the look, that tells The pang that in the bofom dwells.' The picture that follows of the wretched negro juft landed in the West Indies, and fold, is extremely natural: • When borne at length to Western Lands, In mute affliction, fee him try If one bleft glance of mercy there, One half-form'd tear may check despair!'- His fway the harden'd bofom leads To Cruelty's remorfelefs deeds; Darts to its goal with baleful force, Nor heeds that ruin marks its courfe.' Our approbation of this poem has induced us to allow it more room than fuch small publications ufually occupy; but we choose our poetry as our fruit, by the fpirit and flavour, not by the fize. We prefer a peach to a pumpkin. It may not be thought unfriendly to warn this ingenious lady below against as frequens admision of the bites, which is not a beauty in poetry for inftance, Again, • Deform Creation with the gloom Of crimes' How far the fpirit can endure Several more inftances of this imperfection might be produced, but the above may fuffice to convey the hint. 6 Page 10, 1. 147, fhould not the opening bloom' of a' ray,' be likewife reconfidered? Moo-y. the unharmonious over ART. over-flow of one line inh another; which renders the Composition to prosaic: |