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tion ;-a monument truly are perennius, and only to be obliterated by the superior ingenuity of others, in the same 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 discoveries.
The first of his mechanical devices was the Rhabdologia, or the art of computing by figured rods. These are fo well known by the name of Napier's bones (being probably originally made of ivory or bone), as not to require the particular description which Lord Buchan gives of them; though, perhaps, a fult account of them was necessary, is a work prott fledly containing the history of Napier's invensins.
The multiplicationis prontuarium is another of Napier's mechanical contrivances for lesening the operations of arithmetic, Any description of this machine, without the delineations, would be unintelligible, as would alto the method which Napier practised, and called arithmetica localis, of calculating by counters peculiarly placed on the squares of a cheis board, or fimular table.
Lord Buchan gives a clear idea of the form and use of these arithmetical machines, and the reasons on which the different operations on them are founded, The hint of the Rods, and of the Prompruary, which is only an improvement of the Rods, seems to have been taken from the Abacus Pythagoricus; and Napier's acquaintance with chess, probably gave rise to his Arithmetica localis. The Promptuary, at least for multiplication, is greatly superior to the other iwo; for partial products of two numbers, each consisting 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 undiscovered, there machines would, in all probability, have been in common use among calculators : at present they are only regarded as mathematical curiofities.
In the next section, 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 shews its relemblance to, or rather identity with the doctrine of Auxions, as de. livered by Newton. He says, under the article Habitudines Logarithmorum, Napier thus expresses 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 now in use, x:y:: Log.x: Tog. y.?' We have transcribed this passage because we think the quotation from the Canon mirificus is erroneous : not having that work at hand, we corre&t the passage thus from memory; ut finus major ad minorem; ita velocitas Incrementi aut Decremenii apud minorem, ad relocitatem incrementi aut decrementi apud majorem.
The remainder of the section is employed in Dewing that Napier was the inventor of logarithms, and in refuring the opi. nions of those who attribute their invention to earlier mathematicians.
Lord Buchan proceeds to give Napier's method of construct. ing his logarithmetical tables, and then shews that the common logarithms were first devised by Napier, and prepared for public cation by Briggs. The disadvantages of Napier's firft Jogarithms were sufficiently apparent; but whether Napier or Briggs first suggested the new species of logarithms, is a question which the learned have not perfe&tly decided. By extracts from several books, it appears that the common logarithms occurred to Napier before they occurred to Briggs. Lord Buchan dismisses the enquiry with observing that Napier and Briggs had a reciprocal esteem for each other, and there is not the imallest evidence of there having existed in the breast of either, the least particle of jealousy ;-hat after the invention of lozarithms, the discovery of the best species of them was no difficult affair ;-and that the invention of the new species of logarithms is far from being equal to some other of Briggs'invention.'
The next section treats of the improvements that have been made on logarithms after the death of their inventor. Next after Napier and Briggs, Gunter has the best claim to the gratitude of the Public. He tirlt applied the logarithms to scales, which are to this day in common use in the Navy, and in the Excise. Mercator, more than 50 years after Napier's death, invented an infinite series expressive ot Napier's logarithms, but Gregory of St. Vincents had, 20 years before this pericd, shewn that the assymptotic areas of she hyperbola were logarithms. It is somewhat astonishing that this identity between the hyperbolic areas and logarithms was not sooner observed; 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 acceJerated motion) at right angles to each other, he must have found that the curve of the hyperbola would have been described. This circumstance occasioned the denomination of hyperbolic, which was given to Napier's logarithms, and which has been, and now is, usually adopted by moft mathematical writers. The absurdity, for we cannot give it a better term, of calling Napier's logarithms hyperbolical must be apparent, when it is considered that all logarithms are hyperbolical; the only difference between different species of logarithms being the inclinationi of the affymptots of the hyperbola to each other. Thus Napier's logarithms correspond with an hyperbola whose assymptots are at right angles, when the fine of the angle is unity, which is the modulus of that system of logarithms. Briggs's, or the common logarithms, correspond with an hyperbola whose affymptots are inclined at an angle of 25° 44' + whose fine is .43429, &c. which is the modulus of Briggs's logarithms. All logarithms are therefore hyperbolical; and it seems that the epithet hyperbolical was given to Napier's unjustly, and probably with a view to suppress the inventor's name. We must observe by the way, that all through this publication, the words area and areas are misprinted arca and arcas.
The remaining part of this section describes the different tables that have been published, and the preference is given to the tables portatives of Mons. 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 somewhat extraordinary, when his Lordlhip points out an error in the French edi. tion, but none in the English. It must, however, be acknowleged that the French tables are much more diftinétly and elegantly printed than the English. This we say from having seen both books, and not from the specimen which Lord Buchan's printer has given of Jombere's tables, where there is an error by placing 9019 in a wrong line.
The 7th section describes the use of logarithms; and the 8th, which closes 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 natural numbers from 1 to 101, to 27 places of figures ; we can pronounce this table correct from baving examined many of the logarithms. 30, A collection of trigonometrical theorems. 4th, A description 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 small labour on the part of Lord Buchan as well as of his affociate, Dr. Minto; to whom bis Lordship acknowleges himself indebted, especially in the mathematical department.
Napier's life, we are informed, is to be succeeded by other Jives, in which Lord Buchan is at present engaged, on condition that this specimen meets with the approbation of the learned world. His Lordship’s zeal is great, and undoubtedly demands the gratitude of the Public. When noblemien not only patronize literature, but themselves take an active part in its cultiva
tion, the greatest expectation may be formed that its true interests will be more generally promoted.
We cannot close this article without mentioning a defect which Lord Buchan may easily avoid in his future publications. His book is carelessly printed. The errors, however, are such as any mathematician may correct, and must be attributed to the inattention of those who undertook to conduct the work through the press.
is. 6d. sewed.
Slave Trade, were horrid enough to call into vigorous ex-
• BRITAIN! the noble, bleft decree
Oh, first of EUROPE's polith'd lands,
Gilds the thick Ahadę wice softer day.'
The traders in flaves are described as beings
• Whose harden'd souls no more retain
that in the bosom dwells.'
• When borne at length to Western Lands,
One half-form'd tear may check despair!'-
• His sway the harden'd bosom leads
Nor heeds that ruin inarks its course.'-
It may not be thought unfriendly to warn this ingenious lady below against otrogonition of the time, which is not a
wymie poetry for instance,
• Deform Creation with the gloom
• How far the spirit can endure
Calamity' Several more instances of this imperfection might be produced, but the above may fuffice to convey the hint.
Page 10, 1. 147, Mould not the opening bloon' of a 'ray,' be likewise reconfidered?
the unharmonious oner-flow of one line inh another; which render the Couporihin ho prorace: