Bombelli, Rafael 47

Just before publishing his work, Bolyai learned that

the great C

ARL

F

RIEDRICH

G

AUSS

(1777–1855) had

already anticipated much of this theory, even though he

had not published any material on the matter. Bolyai

decided to delay the release of his work. In 1832 he

printed the details of his new theory only as a 24-page

appendix to an essay his father was preparing. Later, in

1848, Bolyai discovered that Lobachevsky had pub-

lished a similar piece of work in 1829. Bolyai never

published the full version of his original treatise. His

short 24-page piece was practically forgotten until

Richard Blatzer discussed the work of both Bolyai and

Lobachevsky in his 1867 text Elemente der Mathematik

(Elements of mathematics). At that point, Bolyai’s piece

was recognized as the first clear account of the mathe-

matics of a new type of geometry. He is today regarded

as having independently founded the topic.

Bolyai died on January 27, 1860, in Marosvásárhely,

Hungary (now Tirgu-Mures, Romania). In 1945 the

University of Cluj honored Bolyai by including his name

in its title. It is today known as the Babes-Bolyai Univer-

sity of Cluj.

Bolzano, Bernard Placidus (1781–1848) Czech

Analysis, Philosophy, Theology Born on October 5,

1781, in Prague, Bernard Bolzano is remembered as the

first mathematician to offer a rigorous description of

what is meant by a

CONTINUOUS FUNCTION

. The related

theorem, the

INTERMEDIATE

-

VALUE THEOREM

, is some-

times named in his honor.

Bolzano studied philosophy and mathematics at

the University of Prague and earned a doctoral degree

in mathematics in 1804. He also completed three years

of theological study at the same time and was ordained

a Roman Catholic priest two days after receiving his

doctorate. Choosing to pursue a career in teaching,

Bolzano accepted a position as chair of philosophy and

religion at the university later that year.

In 1810 Bolzano began work on understanding the

foundations of mathematics and, in particular, the log-

ical foundations of the newly discovered

CALCULUS

. He

found the notion of an

INFINITESIMAL

troublesome and

attempted to provide a new basis for the subject free

from this concept. In his 1817 paper “Rein Analytis-

cher Beweis” (Pure analytical proof), Bolzano explored

the concept of a

LIMIT

—anticipating the foundational

approach of A

UGUSTIN

-L

OUIS

C

AUCHY

offered four

years later—and proved the famous intermediate value

theorem. Bolzano was also the first to provide an

example of a function that is continuous at every point

but differentiable at no point.

Bolzano also anticipated much of G

EORG

C

AN

-

TOR

’

S

work on the infinite. In his 1850 article “Para-

doxien des Unendlichen” (Paradoxes of the infinite),

published by a student two years after his death,

Bolzano examined the nature of infinite sets and the

paradoxes that arise from them. This piece contains the

first use of the word “set” in a mathematical context.

Bolzano died on December 18, 1848, in Prague,

Bohemia (now the Czech Republic). His work paved

the way for providing rigorous underpinnings to the

subject of calculus. In particular, Bolzano identified for

the first time in “Rein Analytischer Beweis” the “com-

pleteness property” of the real numbers.

Bombelli, Rafael (1526–1572) Italian Algebra Born

in Bologna, Italy, in 1526 (the day and month of his

birth date are not known), scholar Rafael Bombelli is

remembered for his highly influential 1572 book L’Alge-

bra (Algebra). In this work, Bombelli published rules for

the solution to the

QUADRATIC

,

CUBIC

, and

QUARTIC

EQUATIONS

, and was one of the first mathematicians to

accept

COMPLEX NUMBERS

as solutions to equations.

Bombelli began his career as an engineer specializ-

ing in hydraulics and worked on a number of projects

to turn salt marshes into usable land. Having read the

great Ars magna (The great art) by G

IROLAMO

C

AR

-

DANO

(1501–76), Bombelli decided to write an algebra

text that would make the methods developed there

accessible to a general audience and be of interest and

use to surveyors and engineers. He intended to write a

five-volume piece but only managed to publish three

volumes before his death in 1572.

Bombelli noted that Cardano’s method of solving

cubic equations often leads to solutions that, at first

glance, appear unenlightening. For instance, examina-

tion of the equation x3= 15x+ 4 leads to the solution:

Although scholars at the time rejected such quantities

(because of the appearance of the square root of a neg-

ative quantity) Bombelli argued that such results