As corollaries to the general formulae he adds the formulae relating to the representation of a number as a sum of five squares and also of seven squares.
It is the first example in Italian literature of a national biography, the first attempt in any literature to trace the vicissitudes of a people's life in their logical sequence, deducing each successive phase from passions or necessities inherent in preceding circumstances, reasoning upon them from general principles, and inferring corollaries for the conduct of the future.
It meant the excision of an alien element which fed like a cancer on the body politic; it meant the recovery, at comparatively little cost, of the command of the principal rivers of Poland, the Vistula and the Niemen; it meant the obtaining of a seaboard with the corollaries of sea-power and world-wide commerce.
But two corollaries follow from our present investigation of his extant writings; the first, that it was the long continuance of the Peripatetic school which gradually caused the publication, and in some cases the forgery, of the separate writings; and the second, that his Peripatetic successors arranged and edited some of Aristotle's writings, and gradually arrived by the time of Andronicus, the eleventh from Aristotle, at an order of the whole body of writings forming the system.
It is, in fact, the confluence of the Malthusian ideas with the theories of Ricardo, especially with the corollaries which the latter deduced from the doctrine of rent (though these were not accepted by Malthus), that has led to the introduction of population as an element in the discussion of so many economic questions in modern times.
This proposition, which he called the mystic hexagram, he made the keystone of his theory; from it alone he deduced more than 400 corollaries, embracing, according to his own account, the conics of Apollonius, and other results innumerable.
Of it and three corollaries are given in appendix 4 of the second edition of Dalbys Balancing of Engines (London, 1906).
Modern speculations (mainly corollaries of Indo-Germanic theory) add little of value to the Greek accounts quoted above.
He held the portfolio of ways and communications in the first responsible Magyar administration (March 23, 1848) under Batthyany, but his increasing apprehension of a revolution, with its inevitable corollaries of civil war and a rupture with the dynasty, finally affected his mind, and on the 5th of September he was removed to an asylum.