Among the Dictionaries of Arts and Sciences which have been publifhed, of late years, in various parts of Europe, it is matter of forprife that Philofophy and Mathematics fhould have been fo far overlooked, as not to be thought worthy of a feparate treatife in this form. Thefe Sciences conftitute a large portion of the present stock of human knowledge, and have been ufually confidered as poffeffing á degree of importance to which few others are entitled; and yet we have hitherto had no diftinct Lexicon, in which their conftituent parts and technical terms have been explained, with that amplitude and precifion, which the great improvement of the moderns, as well as the rifing dignity of the fubject, leem to demand.

"The only works of this kind in the English language, deferving of notice, are Harris's Lexicon Technicum, and Stone's Nathematical Dictionary; the former of which, though a valuable performance at the time [when] it was written, is now become too dry and obfolete to be referred to with pleafure or fatisfaction: and the latter, confiting only of one volume in 8vo. must be regarded merely as an unfinished ketch, or brief compendium, extremely limited in its plan, and neceffarily deficient in ufeful information.

"It became, therefore, the only refource of the reader, in many cafes where explanation was wanted, to have recourse to Chambers's Dictionary, in four large volumes folio, or to the Encyclopædia Britannica, now in eighteen large volumes 4to. or the fill more flupendous performance of the French Encyclopedifts; and even here his expectations might be frequently difappointed. Thefe great and ufeful works, aiming at a general comprehenfion of the whole circle of the Sciences, are fometimes very deficient in their defcriptions of particular branches; it being almoft impoffible, in fuch extensive undertakings, to appreciate, with exactnefs, the due value of every article: they are, betides, fo voluminous and heterogenous in their nature, as to render a frequent reference to them extremely inconvenient; and even if this were not the cafe, their high price puts them out of the reach of the generality of readers.

"With a view to obviare thefe defects, the public are here prefented with a Dictionary of a moderate fize and price, which is devoted folely to Philofophical and Mathematical fubjects. It is a work for which materials have been collecting through a courfe of many years, and is the refult of great labour and reading. Not only moft of the Encyclopædias already extant, and the various publications of the learned Societies throughout Europe, have been carefully confulted, but also all the original works, of any reputation, which have hitherto appeared upon these fubjects, from the earliest writers. down to the prefent times.

From the latter of thefe refources, in particular, a confiderable portion of information has been obtained, which the curious reader will find, in many cafes, to be highly interefting and important. The Hiftory of Algebra, for inftance, which is detailed at confiderable length in the first volume, under the head of that article, will afford fufficient evidence to fhew in what a fuperficial and partial way the inquiry has been hitherto inveftigated, even by profelled writers on

M

SRIT, CRIT. VOL. XI. FEB. 1798,

the fubject; the principal of whom are M. Montucla, our country man the celebrated Dr. Wallis, and the Abbé de Gua, a late French author, who has pretended to correct the Doctor's errors and mifre prefentations."

The fentiments we have expreffed in the opening of this article, clearly evince, we apprchend, our coincidence with fe veral of thefe obfervations. The refpectable labours of Harris and Stone, and the biaffed narrations of others, certainly fall confiderably fhort of the improvements and liberality. of the prefent age. The extent of fcience mocks the limits of time and place; it is not commenfurate with the life of an individual; and to induftry and ability, not to party, we are to look for its advancement..

As the contents of the work now before us may very properly be referred to three heads, or arranged in three parts, the biographical, hiftorical, and scientific, our remarks on it will naturally follow that divifion, and be difpofed in the order in which the parts are here mentioned. It is evident that the greateft part of thefe divifions can only be a compilation fromformer writers, and, therefore, a confiderate reader will not expect extenfive novelty; but, from the eftablished character of the author, he will expect a careful, judicious, and scientific felection of materials, a clear arrangement of them, and a faithful and perfpicuous reprefentation of facts.

In the Biographical part, Dr. Hutton has, in general, compiled with much care. He prefents to our view memoirs of the lives of many ancient Mathematicians and Philofophers, as well as of thofe who have lived in latter times; and he has preferved fuch anecdotes as enliven narration, or contribute to the difplay of character. His departure from former accounts,. in retrenchment or alteration, is in feveral inftances laudable; but we think he would have afforded more fatisfaction to his readers, and done more justice to former writers, if he had more frequently mentioned the fources from which he drew his information.

While we give this author due praife for dwelling with evident pleasure upon the characters of thofe who eminently diftinguished themfelves by the extenfion of science, we cannot help regretting that he has made no mention of fome who ought to have been noticed with respect, for their exertions. Of the life of Dr. Harris, the learned author of the Lexicon Technicum, he has given us no memoir, nor of Machin, Pemberton, or Sterling. These are names which must defcend to pofterity, and be gratefully remembered by men of science; that they are paffed over in filence in the work before us, we tonfider therefore as a remarkable deficiency..

In the Hiftorical part of the work, Dr. Hutton's induftry and perfeverance appear evident, and particularly in his history of Algebra. In the beginning of that article he makes the following obfervations, and gives an account of the manner in which he executed this part of his defign.

"There have arifen great controverfies, and sharp difputes, among authors, concerning the hiftory of the progrefs and improvements of Algebra; arifing partly from the partiality and prejudices which are natural to all nations, and partly from the want of a clofer examination of the works of the older authors on this fubject. From thefe caufes it has happened, that the improvements made by the writers of one nation, have been afcribed to thofe of another; and the discoveries of an earlier author, to fome one of later date. Add to this alfo, that the peculiar methods of many authors have been defcribed fo little in detail, that our information derived from fuch hiftories, is but very imperfect, and amounting only to fome general and vague ideas of the true ftate of the arts. To remedy this inconvenience, therefore, and to reform this article, I have taken the pains carefully to read over, in fucceffion, all the older authors on this fubject which I have been able to meet with, and to write down distinctly a particular account and description of their feveral compofitions, as to their contents, notation, improvements, and peculiarities; from the comparifon of all which, I acquired an idea more precife and accurate than it was poffible to obtain from other hiftories, and in a great many inftances very different from them. The full detail of thefe defcriptions would employ a volume of itfelf, and would be far too extenfive for this place: 1 muft therefore limit this article to a very brief abridgment of my notes, remarking only the most material circumftances in each author; from which a general idea of the chain of improvements may be perceived, from the first rude beginnings, down to the more perfect state; from which it will appear, that the difcoveries and improvements made by any one fingle author, are fcarcely ever either very great or numerous; but that, on the contrary, the improvements are almost always very flow and gradual from former writers, fucceffively made, not by great leaps, and after long intervals of time, but by gradations, which, viewed in fucceflion, become almoft imperceptible.'

Every reader, who has attended to the hiftory of the arts and fciences, will readily perceive that the foregoing obfervation may be applied to almost every branch of them. The hiftorian's labour does not arife from the multiplicity of the difcoveries, which he meets with in the courfe of his enquiries, but from the care neceffary to diftinguifh, in many cafes, between improvement and invention. To do this frequently requires a very commanding view of the fubject, and the exertion of keen difcernment. These the prefent author difplays in his hiftory of Algebra, and prefents to his readers the peculiarities of each work, of any character, publifhed on the fubjet, till about the middle of the laft century. Publications

M 2

after

after this period he mentions in general terms, which is certainly all that could be reafonably expected. Beyond this point, the appropriate features of difcovery, with a few exceptions (and thefe are here recorded) are too taint to be conveyed by defcription.

In the difficult task of afcertaining difcoveries, it can hardly be expected that mistakes thall be wholly avoided; and we have obferved one that is remarkable, refpecting the date of the invention of the Binomial Theorem. In vol. i, p. 208, under that word, this Theorem is faid to have been first difcovered by Sir Ifaac Newton in the year 1669; in vol. ik, p. 732, we are informed, that it was difcovered by him about the year 1666. But it appears, by Sir Ifaac's letter to Mr. Oldenburgh, dated October 24, 1676, that he had discovered this Theorem before he obtained the quadratures of the circle and hyperbola in feries*; and by a paffage in Jones's preface to the Analyfis per quantitatum feries, published in 1711, it appears that thele quadratures were difcovered in 1665. Mr. Jones's words are, Ex Newtoni fchedis quibufdam a me vifis intellexi, quod is quadraturam circuli, hyperbola, et aliarum quarundam curvarum per feries infinitas ex Wallifii noftri Arithmeticâ Infinitorum, per interpolationem ferierum ejus, prims deduxit, idque Anno 1605t. This, therefore, is the date of the invention of this celebrated Theorem.

Having been told in the preface, that "the whole of this work was written before it was put to prefs," and that the reader would find it of an equal and uniform nature and construction throughout," and having read Dr. H.'s hiftory of Algebra with much fatisfaction, we were in hopes of reaping fimilar information and pleafure in a perufal of the hiftory of other articles. In this expectation, however, we have beenfrequently difappointed. In feveral inftances the narrations defective, and in many we meet with that only which was well known before, and has been often repeated.

are very

The Scientific part of the work before us contains a confiderable quantity of valuable original matter; and, in most of the articles, that which has not the recommendation of novelty, is, at leaft, clearly arranged, and perfpicuoufly defcribed. We have perufed with pleature Dr. H.'s account of the ancient Analyfis, with an example of it from Pappus; and we have

* See the Commercium Epiftolicum, Edit. 1722, No. LV, p. 143, 144, 145.

See the Preface above-mentioned, p. 3, from the end,

derived

derived much fatisfaction from his comparifon of the ancient Analyfis with the modern, in which the value of each is juftly appreciated. Much praife is alfo due to him for the manner in which he prefents to his readers fuch rules for folving Algebraic equations, as have from time to time be n invented. Among thefe, we find one for reducing biquadratic equations to cubics, invented by Lewis Ferrari, which, till now, feems to have been but little known. Dr. H. appears to us to have been equally fuccefsful in explaining the principles, and in illuftrating the notation of Fluxions.

Under the article Gunnery, we meet with fresh proofs of Dr. H.'s laudable perfeverance in the caufe of fcience. It is well known, we prefume, in the philofophical world, that he published a volume of Tracts in 1786, where n is detailed an extenfive courfe of new experiments in artillery, which were carried on at Woolwich, in the years 1783,'1784, and 1785.

"Since the publication of thofe Tracts," fays the author in the work before us, we have profecuted the experiments fill farther, from year to year, gradually extending our aim to more objects, and enlarging the guns and machinery, till we have arrived at experiments with the fix pounder guns, and pendulums of 1800 weight. One of the new objects of enquiry, was the refinance the atmoiphere makes to military projectiles; to obtain wi ich, the guns have been placed at many different diftances from the pendulum, againft which they are fired, to get the velocity loft in paffing through thofe fpaces of air; by which, and the ufe of the whirling machine, defcribed near the end of the first volume of Robins's Tracts, for the flower motions, I have inveftigated the refiftance of the air to given balls moving with all degrees of velocity, from o up to 2000 feet per fecond: as well as the refiftance for many degrees of velocity, to planes and figures of other fhapes, and inclined to their path in all varieties of angles; from which I have deduced general laws and formulas for all fuch motions. All thefe experiments agree in evincing the very enormous refiftance the air makes to the fwift motions of military projectiles, amounting in fome cafes to 20 or 30 times the weight of the ball itfelf; on which account the common rules for projectiles, deduced from the parabolic theory, are of little or no ufe in real practice; for, from theie experiments it is clearly proved, that the track defcribed by the flight even of the heaviest fhot, is neither a parabola, nor yet approaching any thing near it, except when they are projected with very fmall velocities; in fo much that fome balls, which in the air range only to the diftance of one mile, would in vacuo, when projected with the fame velocity, range about 10 or 20 times as far."

Similar experiments were made by Dr. H. refiftance of the air to bodies in motion.

to afcertain the A full account of

thefe, he fays, would make a book of itself, and must be re