< The principles of common algebra and geometry, fays Mr. Landen, having been thought infufficient to enable the Analyft to purfue his fpeculations in certain branches of fcience; new principles, very different from those made ufe of, have, through a fuppofed neceffity, been introduced into analytics. The Fluxionifts following Sir Ifaac Newton, introduce an imaginary motion, and recur to the generation of quantities by a fuppofed flowing, or continual increase of their parts. Mr. Leibnitz and his Followers, to avoid the fuppofition of motion, confider quantities as compofed of infinitefimal elements; and reject certain parts of the infinitely small increments of quantities, as infinitely less than the other parts. In the Refidual Analysis (admitting no principles but fuch as were anciently received in algebra and geometry) we neither have recourfe to infinitefimals nor the principles of motion; but confider magnitudes as already formed, without any regard to their genefis, except in particular cafes, where the manner of their being generated may be proper fubject of enquiry: and as this analyfis is not lefs (if not more) ufeful than the fluxionary, or differential calculus, it will confequently appear, that the analytic art, founded and carried on upon fuch principles as were anciently received therein, (without the aid of any foreign ones relating to an imaginary motion, or infinitefimals) is far more extenfive than Mathematicians have hitherto reckoned it.'

Our Author adds, That although fuch borrowed principles may enable us to give very concife folutions, yet, perhaps, we must not expect to bring the analytic art to its utmost perfection, otherwife than by proceeding on its own proper principles. -I am, therefore, induced to think, that not only in the refolution of problems purely algebraical, but likewife in geometry and natural philofophy, when an analytical process is requifite, and what is called common algebra is infufficient, the Refidual Analyfis, which is founded (as I conceive) on the genuine principles of Analytics, is, for the most part, more properly applicable than the Auxionary analysis, which is founded on new principles, borrowed as above-mentioned. But, however, very far from being pofitive in this matter, I freely fubmit it to the judgment of the public.'

The above comparifon between the principles on which the Refidual Analyfis, the doctrine of Fluxions, and the Differential Calculus, are founded, will fhew the reafons which induced the Author to purfue this difcovery; but whether the method given by this learned Gentleman for obtaining the fpecial value of a given quantity, and which is to serve as a substitute for the Auxionary method, will not be much more troublesome in its application

application to the folution of problems de maximis et minimis, drawing tangents, &c. than that already known, is a point that well deferves confideration.

Mr. Landen, in the beginning of the treatife before us, fhews that the Refidual Analyfis depends upon the investigation of the following equation;

In Cor. 1. to Example 11, our Author fays, that seeing

The method by which Mr. Landen obtains the fpecial value

of a propofed quantity, is, by dividing v-w ·w by v-w, as we have already obferved, and then by making vw in the quotient arifing from that divifion, which then becomes the va

lue fought. But if we expand

appear that the feries produced can only hold true when w is less than v; for when vw, both the divifor and dividend will vanish, or become equal to o. Let us, for instance, sup

pofe m2, and r = 1; alfo v=w= 4. Then

= o. Confequently

Mr. Landen's quotient o+w, or 2v, will, in this cafe, be

equal

equal to o. But, according to the Refidual Analyfis, we shall have 2 for the above quotient, the special value of v2.

If we now take the fluxion of v2, we shall have 2vú † ¿¿. But the quantity being infinitely fmall with refpect to 2v, is rejected, and 20 taken for the fluxion of v. Let this Auxion be divided by, and the quotient 2v, will be the fame with Mr. Landen's fpecial value of v2. In this example we have taken 2v instead of 2v+; that is, we have rejected the infinitely small quantity, and by that means obtained the finite quantity 2v. Whereas by Mr. Landen's method we obtain the fame finite quantity 2 from an expreffion abfolutely equal to nothing. Had this able Mathematician told us, that the expreffion, or error, in excefs, arifing from the divifion of

- w, would, when v was made equal to w,

m

be the very fame with the fluxion of divided by, he had been undoubtedly right, and at the fame time removed what, we imagine, will be a principal objection against his Residual Analyfis.

We would not, however, be understood to mean, that the Refidual Analyfis will be of no ufe in mathematical enquiries: on the contrary, we are perfuaded, that in many cafes it will prove of very fingular advantage; but we cannot think it, upon the whole, fuperior to the Fluxionary Calculus.

The Semi-Virgilian Hufbandry, deduced from various Experiments: Or, an Effay towards a new Courfe of National Farming, formed from the Defects, Loffes, and Disappointments, of the Old and New Hufbandry, and put on the true Bias of Nature, in the Production of Vegetables, and in the Power of every Ploughman, with his own Ploughs, &c. to execute. With the Philofophy of Agriculture. Exhibiting, at large, the nutritive Principles derived from the Atmosphere, in a Rotation of Nature, from their being exhaled, to their Defcent into the Pores of the Soil, when duly - prepared, for the Purposes of Vegetables. By Mr. Randall, fome time fince Mafter of the Academy at Heath, neat Wakefield, Yorkshire. 8vo. 6s. Law.

AHIS treatife is dedicated to the Society for the Encouragement of useful Arts and Sciences*;" to whom the Au

TH

• We know of no fuch Society in Great Britain: Mr. Randall means the Society for the Encouragement of Arts, Manufactures and Commerce.

thor

thor flies for patronage, in his attempts to serve the public.'-It was wrote, as we learn from the preface, about eight years ago, and afterwards fent to the London Chronicle, in order, from time to time, to communicate the contents to the public, for the confideration of thofe Gentlemen who are defirous of improving agriculture:' but, after fome sheets had been published in that manner, the Bookfeller treated with the Author, in or der to print the whole in the present volume.

This being the cafe, he hopes for indulgence, on account of method and manner of expreffion; as, he owns, he gave himself no fort of trouble about fuch forms and correctness of flyle: and whatever tautologies may occur, he hopes they are not improperly placed; as the fame principles, repeated, may be useful. The whole was wrote for the fake of that part called the Semi-, Virgilian Husbandry, which, he prefumes, will be of confequence, when duly confidered, and carried properly into execution. All that precedes this part is, by way of contraft, to fhew Gentlemen the defects in conducting agriculture, on the principles and practice of both the Old and New Hufbandry.'-He adds, that he has himself experienced all the three forms; and, from the great facility in executing the Semi-Virgilian, and from the advantages thence arifing, he is willing to hope, that fome public-fpirited Gentleman will try, in an effectual manner, whether it bids fair for a national courfe of Hufbandry; and the more fo, as it is of such a nature, as to be in the power of every ploughman to execute, without being puzzled with difficulties in the inftruments.-He has, however, added a few inftruments, which he found extremely useful to his defign: but as they are not [abfolutely] neceflary, ploughmen may [ftill] ufe their own; his whole view having been to put those men as little out of their way as poffible.

The parts of knowlege [proper] for a Gentleman, he says, who would understand this treatife, are, the philofophy of earth (mould) in general,-the nutritive principles which promote vegetation, the philofophy of the atmosphere, and how the celeftial influences affect the foil. Thefe principles in general are here premifed, under the title of a Preface; in which we are made acquainted with the conftitution of the globe of the earth, in order to a proper diftinction of mould, effential to vegetation. We are next fhewn the nature of specific gravity, in order to conceive fomething of the general fubfidence of the earth, for the purpofes of vegetation; and how it came to pas that we enjoy the different firata of moulds, initead of having fluids diffufed over the face of the globe. Then follows an enquiry into the nature of mould, its various forts and qualities: and the Preface conclad wolkrtion, that all the [dif

3

ferent]

ferent] claffes of foils are [either naturally] proper for vegetal. tion, or may be made fo, by man's industry and skill.

In the Introduction, which is very fhort, we are told, that The philofophical part of agriculture, in order to rife as aci ence, fhould imitate natural philofophy, in making experiments; and that the Author, who here throws in his mite towards perfecting this truly ufeful and agreeable part of knowlege,hopes what he delivers will be as lovingly received as he intends

it.'

The general treatise, or effay, as it is ftyled, is divided into chapters; of which the following view may, perhaps, be fufficient for such of our Readers as are not profeffed admirers of agriculture; and to thofe who are, we would recommend the perufal of the work itfelf, in which they will meet with a great many useful, and fome very ingenious, obfervations, tho' too often almoft loft in a needlefs deluge of words. We readily fuppofe, that the Author intended his frequent amplifications as real illuftrations of his arguments; but, if we may be allowed to judge of other people's fenfations by our own, we cannot help being of opinion, that the book would be more universally read, and much better underftood, if the fubftance of it were reduced into half its prefent compafs. And if ever, a fecond edition fhould be called for, we would recommend: this point to the confideration of the Author.

Chap. I. explains the general nature of the Atmosphere, together with the various principles of which it confifts, fuch as aqueous vapours, fulphurous, nitrous, and faline particles, all of them of confequence in the growth of plants.

Chap. II. gives us the manner of recruiting worn-out land,. by duly preparing it for the reception of the treasures of the atmosphere. And this is propofed to be effected, 1. by redu cing the foil to a fitting degree of fineness and richness; and 2. by the deftruction of weeds. The firit point is chiefly to be obtained by making a proper ufe of the plough and harrows.; but as our Author's method is fomewhat fingular, we fhall give a fhort quotation, relating to a worn-out loam, in his own words.

The loam, immediately after harvest, is to be turned up, and as we shall fuppofe it will allow the ploughman to go very deep, this is a point to be obtained, at any rate, for a worn-out foil. In order to this, one plough is to go the ufual depth, and another plough to follow at the fame depth, in the fame furrow, which will throw the mould over him, and bury the ftubble." In this cafe, the field will lie under the advantage of being turn⚫ed upfide down, double-fpitted, as it were, more than a foot

decp,

1