icomachus,
Ptolemaeus, Paciolus, Boetius, have written much thereon. For men like
these never came near to discover one-hundredth part of the things
discovered by me. But with regard to this matter--as with divers others--I
leave judgment to be given by those who shall come after me. Nevertheless
I am constrained to call this work of mine a perfect one, seeing that it
well-nigh transcends the bounds of human perception."[111]

FOOTNOTES:

[84] It was published at Milan by Bernardo Caluschio, with a
dedication--dated 1537--to Francesco Gaddi, a descendant of the famous
family of Florence. This man was Prior of the Augustinian Canons in Milan,
and a great personage, but ill fortune seems to have overtaken him in his
latter days. Cardan writes (_Opera_, tom. i. p. 107):--"qui cum mihi
amicus esset dum floreret, Rexque cognomine ob potentiam appellaretur,
conjectus in carcerem, misere vitam ibi, ne dicam crudeliter, finivit: nam
per quindecim dies in profundissima gorgyne fuit, ut vivus sepeliretur."

[85] There is a reference to Osiander in _De Subtilitate_, p. 523. Cardan
gives a full account of his relations with Osiander and Petreius in
_Opera_, tom. i. p. 67.

[86] November 1536.

[87] Ferrari was one of Cardan's most distinguished pupils. "Ludovicus
Ferrarius Bononiensis qui Mathematicas et Mediolani et in patria sua
professus est, et singularis in illis eruditionis."--_De Vita Propria_,
ch. xxxv. p. 111. There is a short memoir of Ferrari in _Opera_, tom. ix.

[88] _Opera_, tom. i. p. 66.

[89] Fra Luca's book, _Summa de Arithmetica Geometria Proportioni e
Proportionalita_, extends as far as the solution of quadratic equations,
of which only the positive roots were used. At this time letters were
rarely used to express known quantities.

[90] The early writers on Algebra used _numerus_ for the absolute or known
term, _res_ or _cosa_ for the first power, _quadratum_ for the second, and
_cubus_ for the third. The signs + and - first appear in the work of
Stifelius, a German writer, who published a book of Arithmetic in 1544.
Robert Recorde in his _Whetstone of Wit_ seems first to have used the sign
of equality =. Vieta in France first applied letters as general symbols of
quantity, though the earlier algebraists used them occasionally, chiefly
as abbreviations. Aristotle also used them in the _Physics.--Libri. Hist.
des Sciences Mathematiques_. i. 104.

[91] _Opera_, tom. iv. p. 222.

[92] In the conclusion