197 



ALGEBRA. 



ALGEBRA. 



stand for any number we please, provided that it keeps the same 

 meaning throughout the question. Hence in what are called addition, 

 multiplication, &c., of algebraical quantities, we do not ask, ' What 

 umber does this multiplication give,' but ' what net of operations are 

 equivalent to, and, if we please, may supply the place of, this multipli- 

 cation ? ' For example, suppose it occurs in a question that one 

 number is to be added to, as well as subtracted from, another, and that 

 the two results are to be multiplied together. Let a and 6 stand for 

 the two numbers, of which let a be the greater. So long as we use 

 general symbols, that is, so long as we do not assign some particular 

 numbers, whfch a and b are to signify, we cannot perform the above 

 operations, but can only indicate them by the marks above mentioned ; 

 for example, a + b stands for the sum of a and b, a b for the dif- 

 ference, and 



(a + M x (a b) 



for the product of this sum and difference. So far we need nothing 

 more to tell us what to do, so soon as a and b shall have their values 

 assigned to them : for instance, if a be 7, and 6 be 3, a + b is 10, and a b 

 is 4, and the above product is 10 x 4, or 40. But, in the meanwhile, 

 we see in the above a sort of double operation : there is inside each 

 pair of brackets something to be done, while the results of the brackets 

 themselves are connected by a further process. It is asked then, what 

 simple processes will supply the place of the preceding, so that whatever 

 numbers o and b may stand for, the product of this sum and difference 

 may be obtained from them ? The answer to this is obtained by the 

 process of algebraic mi>ltit>li<-<iti<>tt, and proves to be aa bb, or b 

 multiplied by itself, and the result subtracted from a multiplied by 

 itself. In the preceding example, this is 7 x 7 3 x 3, or 49 9, or 

 40, as before. For details of variaus operations, see the general heads 

 already quoted, and BINOMIAL THKOHEM, DEVELOPMENT, SERIES. 



The earliest treatise on algebra of which we can fix the date within 

 two centuries is that of Diophantus, an Alexandrian Greek, who lived 

 in the sixth century. It is very unlike a modern treatise on algebra, 

 being almost destitute of general symbols, and consisting altogether of 

 a species of problems which have since received the name of D'wjihantine, 

 in which it is required to solve certain questions, the answers to which 

 shall be whole numbers only. It is so like the Hindoo algebra in its 

 character, that it is impossible to suppose the two wholly unconnected. 

 But as the Hindoo algebra is of a much higher cast than that of 

 Diophantus, we are obliged to suppose, either that Diophantus obtained 

 from the East a part of their knowledge, or that the Hindoos, setting 

 out with the Greek algebra only, made considerable improvements after 

 the sixth century. As the Hindoo algebra has been very much extolled 

 by some, and more than proportionally cried down by others, we quote 

 from Delambre, who is distinguished among the latter. " The Hindoos 

 had algebra of the first and second degrees ; they knew how to solve 

 indeterminate problems ; and they made these acquisitions themselves ; 

 they are also the authors of the system of arithmetic now universally 

 received by us." ' Histoire de 1' Astronomic Ancienne, vol. i., p. 556. 

 To these we might add many minor points, and also that, in the 

 solution of indeterminate equations of the second degree, they had 

 made as much progress as ever was made in Europe before the middle 

 of the eighteenth century. [VlGA GANITA, in Bloo. Div.] 



The Persians and Arabs confessedly derived their knowledge of the 

 subject from the Hindoos. We do not, however, find that they 

 proceeded as far as '.their masters : for the Arabic treatises, so far as 

 we know, contain only the solution of equations of the first and second 

 degree, and their application to various arithmetical questions, excluding 

 all mention of indeterminate equations. 



It was by means of the treatise of Mohammed Ben Musa, who lived 

 in the time of the Caliph Al Mamun, that the science was introduced 

 into Europe. A complete and able translation of this work, by Dr. 

 Rosen, with the original Arabic, was published in 1831, by the 

 Oriental Translation Fluid. 



Thus much of the science was introduced into Europe, or rather 

 into Italy only, at the tieginning of the thirteenth century, by Leonardo 

 Bonacci of Pisa. Algebra lay dormant in Italy, without receiving any 

 material improvement, till the middle of the sixteenth century, when 

 it was introduced into Germany, France, and England, nearly about 

 the same time by Stif elius, Peletarius, and Robert Recorde, respectively. 

 The Hindoos, instead of using the letters of the alphabet, designated 

 various unknown quantities by the names of different colours ; the 

 Persians and Arabs employed the word answering to ' thing ' in their 

 language for the unknown quantity, and the Italians adopted the word 

 ' cow ' for the same purpose : hence algebra came to be called the 

 Ker/ola de la Coaa in Italy, and the Cosiike Art in England. It is to 

 be observed, however, that in no country, up to the time of Vieta, 

 were letters used to signify anything but quantities sought ; those 

 u'lfin being always certain numbers, and never arbitrary representations 

 of numbers in general. Hence the simple word ' thing! or any abbre- 

 viation of it, was sufficient for their purpose. 



While algebra was being introduced into the various countries of 

 Europe, the Italians began to make the first steps towards its improve- 

 ment. The solution of an equation of the third degree was discovered 

 by CARDAN and TARTAQLIA ; that of the fourth by FERRARI ; while 

 various other discoveries were made by BOMBKLLI and MAUROLICO. 

 We must refer the reader to the several lives of these mathematicians 



in the BIOG. Div. of this work. VIETA, a Frenchman, who died in 1603, 

 made the grand improvement of using letters to stand for known as 

 well as unknown quantities, and with the additional power derived 

 from this improvement, laid the first steps of the general theory 

 of equations. In England, HARRIOT, who died in 1621, carried on and 

 extended the discoveries of Vieta ; and from the time of the two latter 

 we must date the modern form of the science. 



Our limits will not allow us even to name the crowd of discoverers 

 who have extended this branch of pure mathematics since the time of 

 Vieta. We must refer to the work of Hutton already cited, to Bonny- 

 castle's translation of Bossut's ' Histoire des Mathematiques,' or to the 

 original work itself : to the preface of the mathematical part of the 

 French Encyclopaedia ; or to the histories of Montucla and Cossali. 

 The first and second are the most likely to fall in the way of the 

 English reader. Libri's ' History of Italian Science ' has much informa- 

 tion on early Italian algebra ; and several very ancient treatises have 

 been recently printed, for the first time, by Prince Boncompagni. 



The only necessary preliminary to the study of algebra is a good 

 knowledge of the four rules of arithmetic, and of common and decimal 

 fractions. Without so much it is impossible to read any work with 

 profit ; and in the want of it we m\ist look for the reason why the 

 science appears repulsively dry to most persons. On this subject, we 

 refer the student to some remarks in page 59 of the treatise on the 

 ' Study of Mathematics ' published by the Society for the Diffusion of 

 Useful Knowledge. 



We shall now give some notion of the form which algebra has taken 

 in the last thirty years. Instead of attempting to make our work of 

 reference supply the place of a school-book, we prefer to give those 

 to whom the common elementary work was unnecessary, some o 

 the notions which are gradually enlarging the boundary of the science, 

 and assisting, beyond what was once imagined possible, in clearing 

 the difficulties which it presented to reflecting minds. 



1. In pure arithmetic the subject matter is simple number. Its symbols 

 are 0, implying the absence of all number ; 1 or unity ; 2, the abbre- 

 viation of 1 + 1 ; 3, that of 1 + 1 + 1, &c. Also the symbols H- x -=- 

 and those of powers and roots. All these symbols have specific mean- 

 ings, and the symbols of operation are connected with known operations. 

 There are also general symbols of number, a, b, c, &c., definite but not 

 specific : that is, each one stands for a different number, known or 

 Unknown, throughout any one train of thought, though it need not 

 represent the same number in all investigations. 



2. The fundamental direct symbols of operation, seen in a + b, ab or 



a x b, a b , are connected with inverse operations, seen in a I,- or 

 1 6 



a+b, i/a or a Each inverse operation offers certain difficulties ; 

 those of a-^-b are overcome by the introduction of the notion of 

 fractions ; those of Va by the considerations in LIMIT, applied to what 

 is called the approximate extraction of roots. But a b presents cases 

 of absolute impossibility, as in 3 7 ; and this impossibility is in- 

 capable of being removed as long as the symbols are the symbols nf 

 pure arithmetic. 



3. The rules by which operations of pure arithmetic are performed 

 are found to be capable of classification under five heads : no step in 

 any train of thought leads to anything but the use of one of these five 

 simple rules, or of complex rules obtained by their combination. Of 

 course these rules are understood as applied only when they give intel- 

 ligible results. 



I. Rule of signs. Whenever two of the signs + and both act upon 

 any symbol, the result is that like signs give + , unlike signs give . 



II. Convertibility of additions and subtractions, as in + i c = 

 c + b = c + a + b, SLC. 



III. Convertibility of multiplications and dMlUMH, as in 







IV. Distributireness of multiplications and divisions, as in (i 



, ,c + d c d 

 =i ao + ac aa, and - H 

 a a a' 



r d) 



V. Rules of ernonenti. These are a x = ' -=- = . 



bb b be be 



a c = (ar) , and ( ) = a . 



Every master of the use of these rules can perform all the steps of 

 an arithmetical operation, including all that is commonly called alge- 

 braical, but provided that no steps enter except those which are arith- 

 metically intelligible. Even the common rules of computation, so far 

 as they are not acts of mere memory or trial, consist in uses of the first 

 four rules. 



4. On examining problems into which the impossible subtraction 

 enters, whether in process or result, whether from inconsistency in tin- 

 problem or in the mode of solving it, the impossibility is always to be 

 traced to a diametrically wrong sense put upon the meaning of some 

 one quantity, as reading gain for loss, or ascent for descent, &c. And 

 since the impossible subtraction can always be reduced to the form 

 o, or a (thus 3 7 is 3 3 4, or 4), and since this is found 

 always to require, as a correction, that units should be taken of the 

 Bort diametrii-nlh/ oppotite to what was supposed when a iras being 



