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The substance of all the differential equations given by Cagnoli, La Lande, Delambre, &c. is included in the XIIIth Chapter; these differential equations may readily be deduced from Simpson's Fluxions.

The fourteenth Chapter consists of Miscellaneous Proposi. tions, which could not easily be classed under different heads.

The first comprehends the French division of the circle. The sexagesimal division of the circle, which is ascribed to the Egyptians, has often been objected to by mathematicians, as by Oughtred, Wallis, &c. who recommend a decimal, or centesimal division. Several tables have been constructed centesimally, particularly one by Dr. John Newton. Dr. Charles Hutton, in a memoir published in the Philosophical Transactions for 1783, proposed to divide the quadrant into equal decimal parts of the radius ; by which means the degrees, or the divisions of the arc, would be the real lengths of the arcs, in terms of the radius.

The French have lately adopted the centesimal division of the circle in their calculations, and tables of greater extent than those by Dr. Newton have been published, as the tables of Hobert and Ideler, printed at Berlin, in the year 1799, the tables of Borda, printed at Paris, an. IX, with a preface by Delambre, &c.

The advantages of this new division of the circle, should it be generally adopted in practical calculations, are few and trifling, when compared with the confusion and perplexity it would occasion. It is true that degrees, &c. would be more readily turned into minutes or seconds, et vice versá, and some other advantages of minor importance would be obtained, were the new division to be universally adopted ; at the same time all our valuable tables would be rendered useless; the many well-established trigonometrical and astronomical works, which from time to time have been published, would be little better than waste paper; the most valuable mathematical instruments, which have been constructed by celebrated artists, must be considered as lumber in the different observatories of Eu, rope; the latitudes and longitudes of places must be changed, which change would render all the different works on Geography useless ; or otherwise the Astronomers, and those in

the habit of making trigonometrical calculations, must be perpetually turning the old division of the circle into the new, or the new into the old.

The best instruments in the observatories of Paris, Palermo, and of the different capitals of Europe, have been constructed by British artists, viz. by Ramsden, Troughton, &c. and are graduated according to the sexagesimal division of the circle; it is plain, therefore, that all observations made with these instruments must be reduced to the centesimal division of the circle before they can be used in calculation. Again, The logarithmical tables of sines, tangents, &c. which were originally constructed by the British mathematicians, have passed through so many hands, and have been so often examined, that they may be depended upon as correct; whilst the new tables would require great caution in using them.

In devising short-methods of constructing the new tables, foreign mathematicians have, however, discovered several formulæ, which may be useful in examining or extending the old tables, and in some astronomical calculations. These formula comprehend the most beneficial parts of their labours, as their new division of the circle will doubtless, ere long, share the same fate as their new calendar; unless the works, which may hereafter be framed according to that model, should possess extraordinary and continued merit.

From an attentive investigation of this subject, it will appear, that, if any merit can be attached to the new division of the circle, the invention cannot, with any degree of justice, be ascribed to the French mathematicians, and that their anxious desire for its universal adoption is the natural result of the French revolution, the general tendency of which has invariably been to depreciate the merit of every subject which has not the glory of the French nation or the praise of Frenchmen for its basis.

The improvements made by the French in the various branches of mathematics, though highly extolled, will be found, in many cases, to be more specious than real; the modern analysis so universally adopted (and which in some instances is certainly illogical, if not unscientifical) has contributed greatlý to vitiate the taste and explode that solid and accurate method

of reasoning which is so conspicuous in the writings of the ancients. A flimsy mode of demonstration, grounded on a dextermous management of algebraical characters, has frequently been substituted for perspicuity and logical exactness. By making these remarks, the author does not mean to insinuate that the French have made no improvements in the mathematical sciences, he is convinced that they have made several, and that in a variety of instances their method of demonstration is neat and perspicuous; but his opinion is, that the value of those improvements has been greately over-rated.

After considering the French division of the circle, the remainder of the fourteenth Chapter is occupied with problems which have been frequently applied in extensive trigonometrical surveys.

Book IV. treats of the Theory of Navigation, a subject intimately connected with Trigonometry, and immediately derived from the sphere.

The work CONCLUDES with a table of the logarithms of all numbers from an unit to ten thousand; a table of natural sines; a table of logarithmical sines and tangents to every degree and minute of the quadrant; with some other tables useful in the solutions of the different astronomical problems. These tables occupy only fify pages, and will be found a valuable acquisition both to the teacher and the learner, as they will save the expence of a large set of tables, and answer evey purpose of instruction with equal advantage.



After the foregoing developement of the nature and design of the work, and of the improvements which this edition contains, I feel myself compelled, however reluctantly, to notice the injustice done me by a contemporary author.

Mr. Bonnycastle, a gentleman well known from his various mathematical works, published a treatise on Plane and Sphe

rical Trigonometry in the year 1806, into which he copied, without the least acknowledgment, a very considerable portion of my work.

In consequence of this I waited upon his bookseller, Mr. Johnson, and expressed my displeasure with the transaction. He assured me verbally (and by letters which are now in my possession) that no steps whatever should be taken, by advertisements or otherwise, to dispose of Mr. Bonnycastle's publication until he (Mr. Johnson the proprietor of the work) had indemnified me for the injury which I might sustain.

I stated that I should be satisfied by his agreeing to cancel such parts of Mr. Bonnycastle's book as I could prove to be copied from mine. This demand, so just and reasonable in itself, did not appear to meet with that approbation which might have been expected, and several letters passed between Mr. Johnston and myself on the subject; the last (excepting one) was dated the 30th of November 1806, which repeated the assurance that the work should not be advertised either directly or indirectly.

Notwithstanding these repeated assurances, a partial review of Mr. Bonnycastle's work appeared in the Philosophical Magazine for November 1806, evidently drawn up with the design of counteracting the effects of any unfavourable impression that might have gone abroad respecting it, and of paralizing the efforts I might make to enforce my demand. The parts copied could not be cancelled without destroying nearly half of the work.

In the same Magazine for January 1807, I made some remarks upon this review, wherein I stated that “ Exclusive of detached matter (which is very considerable) there are upwards of seventy pages (of Mr. Bonnycastle's treatise) in which there are scarcely ten lines in any one page which are not directly copied from my work; and in many pages not a single line nor figure but what is copied.” The truth of which I illustrated by pointing out some errors in my book, which Mr. Bonnycastle had copied most minutely. My remarks produced a reply, in February, which clearly convinced me that the feelings of

unison, that the slighest irritation of the nerves of the one, immediately caused those of the other to vibrate.

There was not a sentence, or a word, in my observations, at which this reviewer (unconnected with Mr. Bonnycastle) could possibly be offended, yet his reply was not only intemperate but illiberal. I made some observations upon this reply, in a letter to the Editor of the Philosophical Magazine (dated 9th March 1807); part of which letter was inserted in the succeeding Magazine, but several of the most essential passages were suppressed.

The reviewers says, “The charges which Mr. Keith has adduced against Mr. Bonnycastle it is not my iminediate business to refute.” I would ask, for what purpose then did he make his reply? He makes a most ridiculous comment on the words copying, and collecting from the Nautical Almanac. Notwithstanding which, my observations were perfectly correct, for the tables of the sun's right ascension and declination inserted in my Book, were collected from the different pages of the Nautical Almanac, with the alteration of the last figures, and the arrangement was my own; .whereas Mr. Bonnycastle's tables were exactly copied from my work. Now, why did Mr. Bonnycastle copy these tables from my treatise ? The answer is obvious, they were copied for the express purpose of solving the various astronomical problems which he had taken from my work, for they were of no other use whatever.

The reviewer next makes some remarks on two diagrams noticed in my first letter to the Editor of the Magazine; and here, it appeaas, he has carefully measured the figures to the greatest nicety, and exults at the (supposed) advantage which he has obtained over me. In consequence of which, like a conqueror of true courage and humanity, he assumes a milder tone in the next sentence, and says, “There is in general only one way of constructing these figures; so that the cry of plagiarism on such an occasion is perfectly ridiculous, as I have no doubt Mr. Keith very well knows."

I feel highly flattered that this learned reviewer will allow me to know any thing, and in return for this mark of his condescension, I beg leave to refer him to the 120th page of the second

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