588 Professor Plai/fair on Mathematical and Physical Science. 



of rcturninp, was perhaps the most valu 

 adie pnrt ol the a:ncieiii maiheinatics, in- 

 asmuch as B method of discovering tiiiili 

 is more valuable ihnii the truths it has 

 already discovered. Unfortunately, how- 

 ever, the fragments containing this pre- 

 cious remnant had suifered more from 

 the injuries of time than almost any 

 other. 



SIGNS PLUS ANB MINUS. 



The use of the signs plus and wjnus lias 

 given rise to sr/me dispute. 'Ihese sijjns 

 were at first used the one to denote addi- 

 tion, tiie other subtraction, and for a 

 long time were applied to no other pur- 

 pose. But as, in the multiplication of a 

 quantity, consisting of parts connected 

 by those signs, into another quantity 

 similarly composed, it was always found, 

 and could be universally demonstrated, 

 that, in uniting the particular products 

 of which the total was made up, those of 

 which boili the factors had the sign minus 

 before them, must be added into one sum 

 Willi those ol which all the factors had the 

 sicn phis; while those of which one of 

 tlfe (actors had the sign plus, and the 

 other the sign minus, must be subtracted 

 from the same,— this general rule came 

 to be more simply expressed by saying, 

 that in multiphcation like signs gave 

 flus, and that unlike signs gave minus. 

 ■ Hence tlie signs plus and minus were 

 eonsidered, not as merely denoting the 

 relation of one quantity to another placed 

 before it, but, by a kind of Jiclion, they 

 were considered as denoting qualities 

 inlierent in the quantities to the names 

 of which thev were prefixed. This fic- 

 tion was found extremely useful, and it 

 was evident that no error could arise 

 from It. It was necessary to have a rule 

 for determining the sign belonging to a 

 product, from the signs of the (actors 

 composing that product, independently 

 of every other consideration; and this 

 was precisely the purpose for which the 

 above fiction w.is introduced. So neces- 

 sary is this rule in the generalizations of 

 algebra, tliat we meet with it in Dio- 

 phantus, notwithstanding the imperl'ec- 

 tion of the language he employed ; for he 

 8tate», that Ai»4i« '"'° Ae»4'K gives 



. "Tvao^K:, &c. The reduction, there- 

 fore, ot the operations on quantity to an 

 . arithmetical form, n«cessniily involves 



-this use of the signs phis ot minus; that 

 is their application to denote something 

 4iite absolute qualities in the objects they 

 collect together. The attempts to free 

 .algebra from this use of the signs have of 

 course failed, and roust ever do so, if we 

 would preserve to that science the extent 

 and facility bf its op^iation*. 



Kven the most scrupulous purist ia 

 mathematical language must admit, that , 

 no real error is ever introducr-.d by em- 

 ploying the signs in this most abstract 

 sense. If the equation a*-\-p.i''\-gx-\-r 

 =0, lie said to have one positive and two 

 negative roots, this is certainly as cx> 

 ceptionuble an application of the term 

 negative, as any that can be proposed; 

 yc(, in reality, it means nothing but tlii> 

 intelligible and simple truth, thata'-t-/)a» 

 -t-jj+r=(a — a){x-\-b){x + c); or that 

 the former of these quantities is produced 

 Ly the multiplication of the three bino- 

 mial factors, j: — a, x-\-b, x + c. W« 

 might say the same nearly as to imaginary 

 roots ; they show that the simple factors 

 cannot be found, but that the quadratic 

 factors may be found ; and tiiey also 

 point out the means of discovering them, 



IGNORANCE or TIIE ANCIENTS. 



Though the phenomena of the material 

 world could not but eaily excite the cu- 

 riosity of a being who, like man, receives 

 his strongest impressions from without, 

 yet an accurate knowledge of those phe- 

 nomena, and their laws, was not to be 

 speedily acquired. The mere extent ami 

 variety of the objects were, indeed, such 

 obstacles to that acquisition, as could not 

 be surmounted but in the course of many 

 ages. INIan could not at first perceive 

 from what point he must begin Ins inqui- 

 ries, in what direction he must carry them 

 nn, or by what rules he must be guided. 

 He was like a traveller going forth to ex> 

 plorc a vast and unknown wilderness, in 

 which a multitude of great and interest- 

 ing objects presented themselves on 

 every side, while there was no path fur 

 him to follow, no rule to direct his sur- 

 vey, and where the art of observing, and 

 the instruments of observation, miist 

 equally be the work of bis own inven- 

 tion. In these circumstances, the selec- 

 tion of the objects to be studied was the 

 effect of instinct rather than of reason, 

 or of the passions and emotions, more 

 than of the understanding. When things 

 new and unlike those which occurred iii 

 the course of every day's experience pre- 

 sented themselves, they excited wonder 

 or surprise, and created an anxiety to 

 discover some principle which might ain- 

 nect them with the appearances com- 

 monly observed. About these last, men 

 felt no desire to be farther informed ; 

 but, when the common order of things 

 was violated, and something new or sin- 

 gular was produced, they begi.n to exa- 

 mine into the f.icr, and attempti d to in- 

 quire into the cause. Nobody sought t» 

 know why a stone fell to the ground, 

 why smoke ascended, or wliy the stais 

 revolviii 



