effects, as well as several anomalies, seem to point at something of this nature; and as opinions formed agreeably to this view of the subject will account for most of the galvanic phenomena in a simple and plausible manner, without the aid of mysterious principles, the subject assumes an highly interesting character, by the increasing probability, that the phenomena of galvanism are most intimately connected with many other important branches of natural philosophy. ARTICLE VIII. Defence of the Opinion that all Numbers have Four Imaginary Cube Roots. By James Lockhart, Esq. SIR, (To Dr. Thomson.) I AM much obliged to Dr. Tiarks, and to your Correspondent N. R. D., for their attention to my late communication. The disagreement of these Gentlemen in respect of the value of the imaginary quantity gives me encouragement to hope that some doubt of the error which they suppose I have made will be excited. Dr. Tiarks affirms that the quantity is nothing but a different form of a well-known root of 64; whereas N. R. D. insists that it is a cube root of 8, and not of 64; and thus it would appear that the quantity is the square, and the square root of itself also. If impossible expressions, only a little complicated, universally lead to such difference of sentiment, it will be wise to abandon them altogether. Nevertheless, it now becomes me to endeavour to show that I have not made a hasty assertion, and that I was duly acquainted with the nature and construction of the quantity in question; and for this purpose I resort to the following remarks and demonstration. t In the general equation xbx = c, there are three roots, x the greater, the middlemost, and the least. The rule promulgated by Cardan gives all the three values, which, however, is denied by some eminent algebraists of the present day. I shall place the cube roots in their order under Cardan's binomials; and I believe that it is the first time of their being so exhibited. By means of these roots, and 28 varieties connected with them, the cube roots of all binomials may be obtained, if such roots admit os a finite expression, even when they are irrational, and without trial or assumption. The imaginary quantity which I introduced relates only to t, and to the second cube root in the column on the left hand, which cube root is thus demonstrated to be exact : adding bt - 13 other side. bt-13 2 to one side of the equation, and its equal to the extracting the cube roots No other value can be used in this case for the cube root of the binomial, which the algebraist may readily prove by adapting it to an irreducible equation where there is no ambiguity in respect of the square root. Such is the equation x3 63 x 162, where the binomial is 81 +2700, and the cube root fort is 3+ = To obtain the imaginary quantity which is the subject of consideration, I employed the reducible equation - 24 x 72, where a 6, t = 3 + √ — 3, v = 3 ✓ - 3; and by Cardan's rule the roots of the equation are thus expressed :√ 36 + √ 784 + √36 784 and by the previous demonstration, the cube root of the binomial on the left hand connected with t is the quantity I gave; namely, + √ − ( 13 - -2/3 √ — 8). 3+ √3 3 Algeaists universally give precedency in magnitude to the binomial on the left hand; and in this they follow the old masters. It would be strange indeed to call the first binomials, and the latter 64. The binomial on the left hand being, then, by common consent 3 and usage, equal to 36+ 28 or 64, it follows that my number is a true cube root of 64, and not of 8, which your correspondent N. R. D. affirms it to be, and that I have properly, and in conformity with the practice of algebraists, taken the positive square root of 784. I conceive, therefore, that I have now only to show that the quantity is different from the known forms of the cube roots of 64. Dr. Tiarks has divided x3 64 by x 4, and by means of the quotient he obtains 2√ - 12, which are the cube roots connected with the equation x3 48 x 128, where by Cardan's 3 rule the roots are represented by √ 64 + √ 4096 4096, and where, by the roots previously 2 ± √ 64-4096 exhibited depending on t and v, the cube roots become -12; but these are the cube roots of binomials in their vanishing state, in which state they have functions and connexions widely different from those deduced from binomials which are not evanescent. The means taken by Dr. Tiarks to prove my quantity to be equal to - 2 2 3 is by no means sufficient. This, as well as the correctness of my assertion, may be sufficiently evidenced by the nature of vanishing fractions; and on this evidence, and not on any ambiguity of expression, I entirely rest my opinion. If binomials are not in a vanishing state, one of the roots of the equation from which the binomials are deduced will, by a simple operation, become extinct; but all the roots will be preserved if the binomials are evanescent. Here the roots are preserved, because the binomials connected with the given equation vanish. But Here the value of unity is extinct, because the binomials cube roots in respect of the root unity are - and ÷ + √ - ; but if, under this conception, we should assimilate the sum of these roots to the root of the equation 3 x − x3 = 2, a greater mistake, in my opinion, could not be made. In the same manner may my quantity be divided by 4, and it will be a cube root of unity, but never can it be conceived to be a root of the equation 3 x x= 2; but if Dr. Tiarks's number be so divided, it will be, together with, a root of the equation 3 x = 2. The equations 7 x6 and 2 have a similar root unity; but it is seen that all equality is lost when they are converted into fractions, and this is precisely the case of our two numbers. It is the province of the lovers of the science to decide on the question. 3 May 9, 1815. SIR, I am, Sir, your obedient servant, JAMES LOCKHART. Another Communication on the same subject. (To Dr. Thomson.) May 3, 1815. As the subject proposed by Mr. Lockhart on the algorithm of imaginary quantities is one of considerable importance in a variety of analytical investigations, you will be induced probably to admit a few remarks on the two answers published in your last number. The first thing which appears singular is, that one of your correspondents has shown Mr. Lockhart's expression to be the cube root of 64, but under a different form to that usually given; and the other, that it is not the square root of 64, but of 8. The fact is, that Mr. L.'s expression, -- 3 + √ + (13 — 3√3), the same as all other quantities in which the sign of the square root enters, admits of two values; and as there is no previous condition, either of them may be employed; aud the quantity will be accordingly either the 64 or 8. R. N. D. is therefore too positive when he says, "it is not the cube root of 64, but of 8." He is also wrong in stating that by squaring aban ambiguity is introduced; for the ambiguity has place in the before the operation of squaring; in fact, the only case in which there is no ambiguity is when we know the origin of the quantity whose root is to be extracted, as is shown in one of the latter numbers of Nicholson's Journal, where the object was to explain why 13+ √ ÷ + ÷ √3, which is known to be equal to 187938, or 2 sin. 70°, is not (when squared by the usual process) equal to the square of the same number. The |