take a purse, no more deferves fuccefs, than he who prefents a piftol.'

gar

It requires a good deal of art and temper for a man to write confiftently against the dictates of his own heart. Thus, notwithstanding our Author talks fo familiarly of us, the great, and affects to be thought to ftand in the rank of Patrons, we cannot help thinking, that in more places than one he has betrayed, in himself, the man he fo feverely condemns for drawing his quill to take a purfe. We are even fo firmly convinced of this, that we dare put the queftion home to his confcience, whether he never experienced the unhappy fituation he fo feelingly defcribes, in that of a Literary Underftrapper? His remarking him as coming down from his ret, to rummage the Bookfeller's fhop, for materials to work upon, and the knowlege he difplays of his minutest labours, give great reafon to fufpect he may himself have had concerns in the bad trade of book-making. Fronti nulla fides. We have heard of many a Writer, who, patronized only by his Bookfeller,' has, nevertheless, affected the Gentleman in print, and talked full as cavalierly as our Author himself. We have even known one hardy enough, publicly to ftigmatize men of the firft rank in literature, for their immoralities*, while confcious himfelf of labouring under the infamy of having, by the vileft and meaneft actions, forfeited all pretenfions to honour and honefty.

·

If fuch men as thefe, boafting a liberal education, and pretending to genius, practife, at the fame time, thofe arts which bring the Sharper to the cart's tail or the pillory; need our Author wonder, that learning partakes the contempt of its profeffors.' If characters of this ftamp are to be found among the learned, need any one be furprized that the Great prefer the society of Fidlers, Gamefters, and Buffoons ?

We are forry to obferve further, on this occafion, that it has been more frequently found, that the Patrons of Literature and the Polite Arts have been difgufted at the diffoluté manners of their profeffors, than that those arts have really wanted patronage. Nor is it at all ftrange, if men of the best fenfe and tafte fometimes refuse to countenance the greatest efforts of genius, when they cannot do it without appearing to protect bad men, and promoting the interefts of those who would repay their benevolence by infolence and ingratitude.

* Even our Author feems to have wandered from his fubject into calumny, when, fpeaking of the Marquis d'Argens, he tells us, He attempts to add the character of a Philofopher to the vices of a Debauchée.'

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A Dif

A Differtation on the Ufe of the Negative Sign in Algebra, containing the Demonftration of the Rules ufually given concerning it; and fhewing how Quadratic and Cubic Equations may be explained, without the Confideration of Negative Roots. To which is added, as an Appendix, Mr. Machin's Quadrature of the Circle. By Francis Mafleres, M. A. Fellow of ClareHall, Cambridge. 4to. 14s. in Boards. Tho. Payne.

T

HE profeffed defign of this Author is, to remove the difficultics that have arifen in fome of the less abftrufe parts of Algebra, from the too extenfive ufe of the Negative Sign; and to explain them, without confidering that Sign in any other light than as the mark of Subtraction of a lefs quantity from a greater. He informs the Reader farther, in his Preface, that the first part of this work, contains the Demonftrations of the feveral operations of Addition, Subtraction, Multiplication, and Divifion, applied to Compound Quantities; that the fecond contains the doctrine of Quadratic and Cubic Equations; towards the understanding of which, he fays, no previous knowlege of any part of the Mathematics is abfolutely neceffary; excepting only that of the common operations of Arithmetic, with the Reafons and Principles on which those operations depend.

This work, being intended for the ufe of beginners, its merit chiefly confists in an attempt to treat the fcience of Algebra with the fame propriety and accuracy of reafoning, that has ufually been thought neceffary in books of Geometry, which has been almoft univerfally neglected in those which treat of Algebra: and to which neglect he attributes the general complaints, of the obfcurity and perplexity of the Algebraic Science.

As to the use of the Negative Sign in Algebra; he fays, at the beginning of the fift chapter; The cleareft idea that can, as I apprehend, be formed of a Negative Quantity, is that of a quantity that is fubtracted from another greater • than itself. To denote this fubtraction, the fubtracted quantity has the Negative Sign prefixed to it. Hence it follows, that a fingle Quantity can never be marked with either of thefe figns, or confidered as either affirmative or negative; for if any fingle Quantity, is marked either with the fignor, without affuming fome other quantity, to which it is to be added, or from which it is to be fubtracted, the mark will have no meaning or fignification. Thus, if it be faid, that the fquare of -5 is equal to +25, such an affertion muft either fignify no more than

' that

that 5 times 5 is equal to 25, without any regard to the Signs, or it must be meer nonfense and unintelligible Jargon.'

This is all the Author fays concerning the Negative Sign but how does this agree with the title, which promifes a Differtation on the Ufe of the Negative Sign; and how does it remove the difficulties that have arifen from its too extenfive application? One would expect from the title, and the preface, that the greateft part of this work confifted chiefly of the explanation of this fign. Does then the reftraining the ufe of the Negative Sign to one particular cafe only, explain its ufe in all others? and does his Differtation confift in no more, than in a bare affertion, without the least proof, and contrary to all mathematical reasoning? If this is his opinion, we must beg leave to differ from him.

As Mr. Maferes is not the only Author who has, through a mistaken notion, ftarted many ftrange difficulties concerning this fign, and as fome have even gone fo far as to use it without the leaft objection, and afterwards raised difficulties which, without any fcruple, they have left their Readers to folve as they could; the Reader will not be difpleafed at the following explanation, wherein will be fhewn the abfolute neceflity of using the fign - and, in the application; and that the idea of this fign, in all cafes whatsoever, is as clear and diftinct as any we have of any other fymbols or figns which are used in Algebra.

That the Negative Sign before a fingle quantity is often very useful, appears amongst many examples, in Logarithms; for fince the Logarithm of Unity is o, thofe of all numbers above Unity are pofitive, and thofe of all numbers less than Unity are negative: Thus the Logarithms of any proper Fractions as,, &c. are negative: and will any one then difpute the usefulness of this fign? And that they are indifpenfibly neceffary, will likewife appear by the following example, from amongst a multitude that might be given. In the Divifion of a Circle, the Equation which folves the Problem, contains twice as many Roots as there are to be divifions; and thefe Roots exprefs the Sines and Co-fines anfwering to the points of divifion: all the Sines which fall above the Diameter drawn thro' the beginning of the divifions, are pofitive, and all thofe which fall under or below that Diameter are negative: all the Co-fines which fall between the beginning of the divifions and the center are pofitive, and all thofe that fall beyond the center negative. Now, as i would be impoffible to know where the points of divifion fall, with

Dd 4

out

out thefe pofitive and negative Sines and Co-fines: it is ma nifeft, beyond all contradiction, that the negative and pofitive figns are not only useful, but abfolutely neceffary.

Some Authors have objected against the Negative Sign annexed to a Single Quantity, as obfcure, and even as impoffi ble; but if they admit the pofitive one, it would be ridiculous to except against the negative. That -3, or —a, is full as clear as +3, +a, admits of no doubt; fince both these figns mean no more, than that the quantities to which they are annexed, are, the firft to be added, and the second to be fubtracted, without changing the values of the quantities: confequently, -3, or +3, are neither more nor less than 3 as to quantity.

The Author now under confideration was fenfible, that by admitting the fign+before a fingle quantity, the fign—must likewife be admitted; and therefore rejected them both, tho' with no better reasons than those alleged against the negative one only. For fince quantities admit of being increased or diminished, whether they are abftractedly confidered or not, the increase is marked with the fign +, and the decrease by the fign both the increase and decrease are real quantities, and are therefore as clearly to be understood as any others. And to increase a by b, we write a +b; to diminish a by b, we write ab; which, in the language of Algebra, is expreffed, to add + b, orb to a. Now where lies the obfcurity in the conception of +b, orb, before they are added? Does it mean any more in the common language, than that is to be added, and -b to be subtracted? It is faid, that there must be another quantity to which these are to be added or fubtracted; otherwise these marks will have no meaning or fignification. Let this be fo; is it not the fame with regard to all other figns? for what meaning or fignification is there in the fign X of Multiplication, or in the fign of Equality, before they are applied? or what meaning is there in any quantity reprefented by a letter a or b, before they are actually applied? What reafon have we, therefore, to object against the fignsor, before they are applied, more than against the figns X,, or against the letters a, b, &c.? Since then the latter characters are used without the leaft fcruple or objection, thofe who object against the figns + or - being prefixed to fingle quantities, ought to prove, that the one may be clearly and diftinctly conceived, but not the other. Which, we conceive, neither has been, nor can be done.

- Now,

Now, as we have fhewn, that a, anda, convey as +a, clear an idea to the mind as a without any fign; the one +a, as an increment, and the other—a, as a decrement; or in the vulgar language, the one to be added, and the other to be fubtracted, in all future operations of addition and fubtraction; it remains to fhew, that the application of these increments and decrements requires no other meaning or fignification than those marked in their definitions.

Since we have proved, that a, ora, convey the fame idea as the quantity a; their doubles, triples, quadruples convey the fame idea as the double, triple, quadruple, &c. of a; that is, +3a, +4a, +5a, are the same as to quantity as 3a, 4a, 5a: again, 3a, 5a, -8a, the fame as 3a, 54, 8a, it follows, that fingle quantities, with the fame fign+ or prefixed to it, may be added together: thus, 2a, 3a, + 4a, gives + 2a, + 34, +4a, or +9a: the fame as 2a, 3a, 4a: again, that when added together, give 9a, - a,. 44, 9a, or 144, the fame as a, 4a, 9a; with this only difference, that as the fign of decrement is prefixed to them, or that of fubtraction, the fame fign must be prefixed to the fum; fince the rule of Addition is no more than collecting the several parts of the fame quantity together, without changing its quality or meaning.

---

a, 4a,

Now, because to add a and b together, we write +ab, according to the rule of Addition given by all Authors, without any other clear or precife meaning of the value of these quantities, than that they must be of the fame kind, it remains to fhew, that whether a is greater, equal to, or less than b, this addition is rightly performed; contrary to the opinion of our Author, who will allow but the firft cafe.

1. When a is greater than b, the most strenuous objectors. to the Negative Sign, allow this operation to be right. 2. When a is equal to b, then abo. This cannot be denied, fince in common Arithmetic a number may be taken from an equal one, and the remainder is then 0. 3. But when a is less than b, the difference ab becomes negative, or a decrement. For in the cafe when a is greater than b, the difference becomes pofitive, or an increment; fo, of confequence, when a is less than b, the difference must be negative, or a decrement; that is, it must be contrary to the former, and is as real as to quantity; that they cannot both. be the fame is clear and evident to common fenfe. It must be obferved, that the lefs is always fubtracted from the great

er,