(1) This memoir in its present form is of the nature of a trilogy; it is divided into three parts, of which each has its action complete within itself, but the same general cycle of ideas pervades all three, and weaves them into a sort of complex unity. In the first is established the validity of Newton’s rule for finding an inferior limit to the number of imaginary roots of algebraical equations as far as the fifth degree inclusive. In the second is obtained a rule for assigning a like limit applicable to equations of the form Σ(
ax
+
b
)
m
=0,
m
being any positive integer, and the coefficients
a
,
b
real. In the third are determined the absolute invariantive criteria for fixing unequivocally the character of the roots of an equation of the fifth degree, that is to say, for ascertaining the exact number of real and imaginary roots which it contains. This last part has been added since the original paper was presented to the Society. It has grown out of a foot-note appended to the second, itself an independent offshoot from the first part, hut may be studied in a great measure independently of what precedes, and constitutes, in the author’s opinion, by far the most valuable portion of the memoir, containing as it does a complete solution of one of the most interesting and fruitful algebraical questions which has ever yet engaged the attention of mathematicians (1). I propose in a subsequent addition to the memoir to resume and extend some of the investigations which incidentally arise in this part. The foot-notes are numbered and lettered for facility of reference, and will be found in many instances of equal value with the matter in the text, to which they serve as a kind of free running accompaniment and commentary. 2) In the ‘Arithmetica Universalis,’ in the first chapter on equations, Newton has given a rule for discovering an inferior limit to the number of imaginary roots in an equation of any degree, without proof or indication of the method by which he arrived at it, or the evidence upon which it rests(²). Maclaurin, in vol. xxxiv. p. 104, and vol. xxxvi. p. 59 of the Philosophical Transactions, Campbell (³) in vol. xxxviii. p. 515 of the same, and other authors of reputation have sought in vain for a demonstration of this marvellous and mysterious rule (
4
). Unwilling to rest my belief in it on mere empirical evidence, I have investigated and obtained a demonstration of its truth as far as the fifth degree inclusive, which, although presenting only a small instalment of the desired result, I am induced to offer for insertion in the Transactions in the hope of exciting renewed attention to a subject so intimately bound up with the fundamental principles of algebra.
A Fifteenth-Century Plan of the Cathedral of Seville