IMAGE
, in Optics, is the spectre or appearance of an object, made either by reflection or refraction.
In all plane mirrors, the Image is of the same magnitude as the object; and it appears as far behind the mirror as the object is before it. In convex mirrors, the Image appears less than the object; and farther distant from the centre of the convexity, than from the point of reflection. Mr. Molyneux gives the following rule for finding the diameter of an Image, projected in the distinct base of a convex mirror, viz, As the distance of the object from the mirror, is to the distance from the Image to the glass; so is the diameter of the object, to the diameter of the Image. See Lens, mirror, reflection, and Refraction.
IMAGINARY Quantities, or Impossible Quantities, in Algebra, are the even roots of negative quantities; which expressions are Imaginary, or impossible, or opposed to real quantities; as √- aa, or √4- a4, &c. For, as every even power of any quantity whatever, whether positive or negative, is necessarily positive, or having the sign +, because + by +, or - by - give equally +; from hence it follows that every even power, as the square for instance, which is negative, or having the sign -, has no possible root; and therefore the even roots of such powers or quantities are said to be impossible or Imaginary. The mixt expressions arising from Imaginary quantities joined to real ones, are also Imaginary; as a - √- aa, or b + √- aa.
The roots of negative quantities were, perhaps, first treated of in Cardan's Algebra. As to the uneven roots of such quantities, he shews that they are negative, and he assigns them: but the even roots of them he rejects, observing that they are nothing as to common use, being neither one thing nor another; that is, they are merely Imaginary or impossible. And since his time, it has gradually become a part of Algebra to treat of the roots of negative quantities. Albert Girard, in his Invention Nouvelle en l'Algebre, p. 42, gives names to the three sorts of roots of equations, calling them, greater than nothing, less than nothing, and <*>nvelopée, as √- 3: but this was soon after called Imaginary or impossible, as appears by Wallis's Algebra, p. 264, &c; where he observes that the square root of a negative quantity, is a mean proportional between a positive and a negative quantity; as √- bc is the mean proportional between + b and - c, or between - b and + c; and this he exemplifies by geometrical constructions. See also p. 313.
The arithmetic of these Imaginary quantities has not yet been generally agreed upon; viz, as to the operations of multiplication, division, and involution; some authors giving the results with +, and others on the contrary with the negative sign -. Thus, Euler, in his Algebra, p. 106 &c, makes the square of √- 3 to be - 3, of √- 1 to be - 1, &c; and yet he makes the product of two impossibles, when they are unequal, to be possible and real: as | ; and or 2. But how can the equality or inequality of the factors cause any difference in the signs of the products? If be , how can , which is the square of √- 3, be - 3? Again, he makes . Also in division, he makes to be = √+ 4 or 2; and ; also that 1 or ; consequently, multiplying the quotient root √- 1 by the divisor √- 1, must give the dividend √+ 1; and yet, by squaring, he makes the square of √- 1, or the product , equal to - 1.
But Emerson makes the product of Imaginaries to be Imaginary; and for this reason, that “otherwise a real product would be raised from impossible factors, which is absurd. Thus, and &c. Also and &c.” And thus most of the writers on this part of Algebra, are pretty equally divided, some making the product of impossibles real, and others Imaginary.
In the Philos. Trans. for 1778, p. 318 &c, Mr. Playfair has given an ingenious dissertation “On the Arithmetic of Impossible Quantities.” But this relates chiefly to the applications and uses of them, and not to the algorithm of them, or rules for their products, quotients, squares, &c. From some operations however here performed, we learn that he makes the product of √- 1 by √- 1, or the square of √- 1, to be - 1; and yet in another place he makes the product of √- 1 and to be Mr. Playfair concludes, “that Imaginary expressions are never of use in investigations but when the subject is a property common to the measures both of ratios and of angles; but they never lead to any consequence which might not be drawn from the affinity between those measures; and that they are indeed no more than a particular method of tracing that affinity. The deductions into which they enter are thus reduced to an argument from analogy, but the fo<*>ce of them is not diminished on that account. The laws to which this analogy is subject; the cases in which it is perfect, in which it suffers certain alterations, and in which it is wholly interrupted, are capable of being precisely ascertained. Supported on so sure a foundation, the arithmetic of impossible quantities will always remain an useful instrument in the discovery of truth, and may be of service when a more rigid analysis can hardly be applied. For this reason, many researches concerning it, which in themselves might be deemed absurd, are nevertheless not destitute of utility. M. Bernoulli has found, for example, that if r be the radius of a circle, the circumference is . Considered as a quadrature of the circle, this Imaginary theorem is wholly insignificant, and would deservedly pass for an abuse of calculation; at the same time we learn from it, that if in any equation the quantity should occur, it may be made to disappear, by the substitution of a circular arch, and a property, common to both the circle and hyperbola, may be obtained. The same is to be observed of the rules which have been invented for the transformation and reduction of impossible quantities*; they facilitate the operations of this imaginary arithmetic, and thereby lead to the knowledge of the most beautiful and extensive analogy which the doctrine of quantity has yet exhibited.
* The rules chiefly referred to, are those for reducing the impossible roots of an equation to the form .”
Imaginary Roots, of an equation, are those roots or values of the unknown quantity in an equation, which contain some Imaginary quantity. So the roots of the equation , are the two Imaginary quantities + √- a a and - √- a a, or + a √- 1 and - a √- 1; also the two roots of the equation , are the Imaginary quantities ; and the three roots of the equation , or , are 1 and and , the first real, and the two latter Imaginary. Sometimes too the real root of an equation may be expressed by Imaginary quantities; as in the irreducible case of cubic equations, when the root is expressed by Cardan's rule; and that happens whenever the equation has no Imaginary roots at all; but when it has two Imaginary roots, then the only real root is expressed by that rule in an Imaginary form. See my paper on Cubic Equations, in the Philos. Trans. for 1780, p. 406 &c.
Albert Girard first treated expressly on the impossible or Imaginary roots of equations, and shewed that every equation has as many roots, either real or Imaginary, as the index of the highest power denotes. Thus, the roots of the biquadratic equation , he shews are two real and two Imaginary, viz, 1, 1, , and ; and he renders the relation general, between all the roots and the coeffi<*>ients of the terms of the equation. See his Invention Nouvelle en l'Algebre, anno 1629, theor. 2, pa. 40 &c.
M. D'Alembert demonstrated, that every Imaginary root of any equation can always be reduced to the form , where e and f are real quantities. And hence it was also shewn, that if one root of an equation be , another root of it will always be <*> and hence it appears that the number of the Imaginary roots in any equation is always even, if any; i. e. either none, or else two, or four, or six, &c. Memoirs of the Academy of Berlin, 1746.
To discover how many impossible roots are contained in any proposed equation, Newton gave this rule, in his Algebra, viz, Constitute a series of fractions, whose denominators are the series of natural numbers 1, 2, 3, 4, 5, &c, continued to the number shewing the index or exponent of the highest term of the equations, and their numerators the same series of numbers in the contrary order: and divide each of these fractions by that next before it, and place the resulting quotients over the intermediate terms of the equation; then under each of the intermediate terms, if its square multiplied| by the fraction over it, be greater than the product of the terms on each side of it, place the sign +; but if not, the sign -; and under the first and last term place the sign +. Then will the equation have as many Imaginary roots as there are changes of the underwritten signs from + to -, and from - to +. So for the equation , the series of fractions is 3/1, 2/2, 1/3; then the second divided by the first gives 1/6 or 1/3, and the third divided by the second gives 1/3 also; hence these quotients placed over the intermediate terms, the whole will stand thus, . + + - + Now because the square of the 2d term multiplied by its superscribed fraction, is 16/3x4, which is greater than 4x4 the product of the two adjacent terms, therefore the sign + is set below the 2d term; and because the square of the 3d term multiplied by its overwritten fraction, is 1<*>/3x2, which is less than 24x2 the product of the terms on each side of it, therefore the sign - is placed under that term; also the sign + is fet under the first and last terms. Hence the two changes of the underwritten signs + + - +, the one from + to -, and the other from - to +, shew that the given equation has two impossible roots.
When two or more terms are wanting together, under the place of the 1st of the deficient terms write the sign -, under the 2d the sign +, under the 3d -, and so on, always varying the signs, except that under the last of the deficient terms must always be set the sign +, when the adjacent terms on both sides of the deficient terms have contrary <*>igns. As in the equation , + + - + - + which has four Imaginary roots.
The author remarks, that this rule will sometimes fail of discovering all the impossible roots of an equation, sor some equations may have more of such roots than can be found by this rule, tho' this seldom happens.
Mr. Maclaurin has given a demonstration of this rule of Newton's, together with one of his own, that will never fail. And the same has also been done by Mr. Campbell. See Philos. Trans. vol. 34, p. 104, and vol. 35, p. 515.
The real and imaginary roots of equations may be found from the method of fluxions, applied to the doctrine of maxima and minima, that is, to find such a value of x in an equation, expressing the nature of a curve, made equal to y, an abscissa which corresponds to the greatest and least ordinate. But when the equation is above 3 dimensions, the computation is very laborious. See Stirling's treatise on the lines of the 3d order, Schol. pr. 8, pa. 59, &c.
