IMA 



centre of the town is new built market-house and 

 h unities. An excellent free school WHS founded here 

 in 1.1.~>i), by Humphry Wuldron and Henry Greenfield, 

 and well endowed. The t-loth manufacture once flou- 

 rished here in n very great degree, but, though still 

 .Irr.ible. it has grr.it! . dtvivaM-d. llminster is 

 Mirnmmled with fine orchard*, and very extensive pros- 

 pects re commanded by the neighbouring eminences. 



The following is the abstract of the population for 

 the town and parish for 1811 : 



Number of houses 365 



\timberoffamilies 403 



Families employed in trade 231 



Do. in agriculture 121 



Males 1022 



Female* '.....,... 1138 



Total population . . . 2160 



See the BeaulU's nf England and Wales, vol. xiii. page 

 539. 



IMAGE. See OPTICS. 



IMAGINARY EXPRESSIONS or QUANTITIES, or im- 

 possible quantities in Algebra, are such as have the sym- 

 bol v/ 1 in their analytic expression. They are so 

 called, because the square root of a negative quantity 

 can have no real existence ; for whether a quantity be 

 positive or negative, its square is a positive quantity. 



The origin and nature of imaginary quantities have 

 been explained in our article ALOEBKA, 189 193. 

 They are there shewn to be of two distinct kinds, one 

 which is altogether impossible, and can denote no real 

 quantity ; and another, which denotes real quantities. 



The first class, when reduced to their most simple 

 expression, have this form a -(- t/ b-, or a + b*/ 1, 

 where a may in some cases be = o. These occur, 

 when a problem is to be resolved which from its nature 

 requires that the data be contained within certain li- 

 mits in respect of magnitude, while at the same time, 

 in the particular case proposed, they pass those limits. 

 For example, if it be required to construct a right 

 angled triangle, the hypothenuse of which shall be 

 eqti.il to a given line a, and one of the sides equal to 

 another given line 1>; from the nature of the case, a 

 must be greater than b ; and if, in the particular state 

 of the data, a be less than b, the thing required cannot 

 be done. 



The unknown side of the triangle expressed in sym- 

 bols, algebraically is ^/ (a 1 b') ; now, if b be greater 

 than a, the quantity a* 6 2 is negative, and the ex- 

 pression for the side of the triangle has the form 

 \ f >i'=r n y' 1, which is imaginary. The impos- 

 sibility of giving a significant numerical value to this 

 symbol, corresponds, in this instance, to the impossibi- 

 lity of placing between a given point, and a straight 

 line given by position, a line of a given length, that is 

 shorter than the perpendicular from the point on the 

 line, or, which is the same, of determining the intersec- 

 tion of a straight line, and a circle which lies wholly on 

 one side of the line. 



In GEOMETRICAL Problems, passing the first order, 

 the unknown quantities are determined either by the 

 intersection of a straight line with a curve, or else by 

 the intersection of two curves : Now, although it may 

 be possible that the conditions to be fulfilled in a pro- 

 blem may be all satisfied at once, yet in many cases 

 there will be limitations of the data ; for example, by 

 one condition a straight line may be required to be of 

 a given length ; and by another, that its extremities 



t IMA 



be on the circumference of a given circle. These can *.ej|'7 

 only be satisfied at once, whn the straight line, is less "antme,. 

 than the diameter. In like manner, one Condition re- 

 quiring that a straight line touch a circle, and another, 

 that it pass through a certain point, can both be satis- 

 fied only when the point is without the circle. When 

 the data of a problem are in this way limited, as often 

 as they cannot be all satisfied at once, the incongruity 

 is indicated geometrically by their being no intersec- 

 tion of the lines, which should meet and determine the 

 unknown quantities ; and algebraically, by the impos- 

 sible symbol \S 1 entering into their values, and in 

 such a way as not to admit of its being eliminated. 

 The presence of the symbol </ 1, in the algebraic 

 expression for a quantity, serves not only to shew the 

 impossibility of finding that quantity in the particular 

 state of the data, but it also indicates the boundary 

 which separates the possible from the impossible cases, 

 and thus determines the greatest and least values that 

 can be given to the different quantities concerned in 

 the problem. 



For example, let it be required to find a fraction 

 which, together with its reciprocal, shall be equal to a 

 given number, and also the limits within which the 

 problem is possible. 



Calling the fraction x, and the given number 2 a, 

 the condition to be satisfied will be expressed by this 

 equation : 



which produces the quadratic equation **' 2u= 1, 

 and, this resolved, gives 



From this expression, it appears that the problem is 

 impossible if a be a fraction, positive or negative, be- 

 tween the limits of +1 and 1, because then a 1 will 

 be less than 1 , and a 2 1 a negative quantity, and 

 */(a- 1) an impossible quantity. However, if a be a 

 positive quantity not less than + 1 , or a negative quan- 

 tity not greater than 1, (here we reckon ' 2 to be 

 less than 1, and 3 less than 2, and so on), the 

 problem will always be possible, and, excepting the ca- 

 ses 0=4-1, and = 1, x will have two values, which 

 will be reciprocals of each other, because their product 

 is unity. 



It also appears that the least positive value of the 



expression x -f is +2, and its greatest negative value 



X 



2, reckoning, as before, that negative quantity to be 

 greatest, which, independently of the sign, is expressed 

 by the smallest number. 



Hence, we learn that no real value of* can be found 



that shall make the expression \ (x -\ ') equal to a 



proper fraction, either positive or negative, but that, 

 this expression may represent any positive or negative 

 quantity whatever that is not between the limits of 

 -f. 1 and 1 , 



From this example, it appears that the theory of im- 

 possible quantities may sometimes be applied with 

 great advantage to a very elegant and interesting class 

 of problems, namely, such as require the determination 

 of the greatest and least values of a variable quantity. 

 The general method of proceeding w to suppose, that 

 the quantity to be a maximum or minimum, is equal to 

 a given quantity; and then to inquire what is the 



