Historian’s Office,

Salt Lake City,

Utah Territory

Sept. 18^{th} 1876.

Editor of Analyst:

Dear Sir: Yours of the 6^{th} inst. was received on the 14^{th}. I am pleased that you have received the volume which I sent you. Only a very few copies of the edition were published in England. In this country it has not been circulated, and is generally unknown. Only a very few copies of the edition were published in England, In this country it has not been circulated, and is generally unknown. It is now out of print. Should there be a sufficient demand to warrant a new edition, I should be pleased to add some improvements, as well as new matter.

I have, with great pleasure, perused Dr. Nelson’s translation of the solution of the fifth degree equation, so far as published in the Analyst. It will, undoubtedly, be a great curiosity and intellectual feast to all lovers of analytical science. Permit me to call your attention to p. 145, third line from the bottom,; it reads “A or B and D”; should it not read A and B and D? Is the capital, IV., solvable, if A only = 0? Is it solvable, if B an D only = 0? Also on p. 144, §.4. twelfth line from the bottom, it reads, “both these products = √P “. Would not the real meaning be more perfectly expressed, if it read – each of these products = ?

The Equation of Differences <to which you refer,> was so named, in order to

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correspond with the names of previously discovered equations, which have been called “Equations of the squares, cubes, and with powers of the differences.” In an elementary work <like mine>, I should scarsely suppose, the name would be confounded with those, called “Differential Equations”, “Equations of Finite Difference,” to., which are terms usually connected with the Higher Calculus.

The Equation of Differences can <also be> very easily be deduced from my formula

(A). ± 2 ((q/3) -m )^{½}, ± (q/3 – m)^{½} ±√(3m) , ± ((q/3) – m)½ ± √(3m) .

Subtract the 2^{nd} from the 3^{rd} forms <using the upper signs> and we have

Q √(3m) = diff. = Z’’ ;

hence,

m =z”²/12 ………………………………………………..(I)

Subtract the 1^{st} from the 2^{nd}, we have

+ 3 ((q/3) -m)^{½} – √(3m) = Z’.

Subtract the 3^{rd} from the 1^{st}, and we have

– 3 ((q/3) -m)^{½} – √(3m) = Z’’’

By these three differences, is formed the general Equation of Differences, viz.,

Z³ – 3qZ + 24 (m^{3/2} – m^{½} ) = 0 ………………………..(d)

But

24√(3m) (m^{3/2} – (q/4) m^{½}) = ± √(4q³-27r²) = r’…………….(II.)

q in this last formula must be considered positive. Divide by 6√3 and we have

4 m^{3/2} – qm^{½} = ± √(4q³-27r²) / 6√3 = ± (- r²/4 +q³/27)^{½}. This is equal to Cardan’s

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radical with the signs of both terms changed; and as q is positive, the term itself must be real, for all cases of where the three roots are real. It is somewhat remarkable, that the product of the three differences, when their signs are changed, is equal divided by 6√3 , gives a quotient in <the form and> equal to Cardan’s <celebrated> radical, when the signs of both terms are changed. I am inclined to believe, that a similar law obtains for equations of the higher degrees. If so, a general solution of the equation of the n^{th} degree <may> be accomplished, so far as the radical part of the solution is concerned.

I have not yet succeeded, in developing the general forms of (A), directly from the general equation. They are the results of inductive reasoning, and carry with them all the data necessary for rigid demonstration. In the transformation of Equations, I find them to be invaluable. <In many cases,> they reveal new methods of demonstration, in many cases, far more simple and than those already known.

That Cardan’s formula is “equally general” with mine, is not for a moment doubted: but in the case of all the <roots> being real, you agree with me, that the forms of (A) are better adapted <than his> for finding the roots by trial.

When two roots are imaginary, the three forms of Cardan are reducible. In this case, the first form of (A) is the real root; the last two forms are the imaginary roots. This arises from the fact, that –m is a negative quantity, between the limits

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of 9/3 and 0, after which it becomes positive, and therefore, each of the last terms of the last two forms of (A), must necessarily become imaginary. This will be better illustrated by an example.

Required, by formula (A), the three roots of the general equation, when q = 0.

We have x³ = ± r; hence x = ±r^{⅓ }. The first form of (A)

gives x = ± 2 (-m)^{½} = ^{ } r^{⅓ };

hence -m = r^{3/2 }/4, and m = – r^{3/2 }/4 .

Substitute these values of m in the three forms of (A), and we have

± r^{1/3} , ± r^{1/3}/2 ± (r^{1/3}/2) (√-3) , ± r^{1/3}/2 ± (r^{1/3}/2) (√-3)

These values are the same as would be given by the three forms of Cardan, contained in your letter. So far as the cubic contains imaginary roots, it must be acknowleged, that Cardan’s formula, though tedious in its arithmetical operations, is superior to the forms of (A), unless the general value of m, in a reducible form, can be discovered.

I would prefer to delay the publication for of the my formula, I have given, until I have made some further researches. I very much regret that my daily avocations are such, that I can only devote, now and then a leisure hour, to this very interesting subject. Would some original problems be acceptable for the Analyst? Must the solutions always accompany the problems <to insure their insertion>? Yours very sincerely, Orson Pratt, Sen.

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[Transcribed by Nora Fowers, Dick Grigg, and Heather Hoyt; Feb. 2011]