4 
Journal of the Mitcheee Society. 
[May 
In order to make the matter clear to “the man in the 
» 
street,” it is necessary to speak, not so much as a mathema- 
tician as one who knows, let us say, no more of mathematics 
than is taught in the Freshman year in the college or university. 
We recall that Euclid uses three terms in laying the foundations 
for his geometry: Definitions (’o/aoi), Postulates ( airo^ara ), 
and Common Notions (icoivot em>nu). He defined his elements: 
point, line, etc.; he assumed that you can draw a straight 
line from one point to another; and he laid down as accepted 
such statements as “Things equal to the same thing are 
equal to each other,” etc. For Euclid’s Common Notions 
later geometers substituted the unfortunate term — unfortun- 
ate, as we shall subsequently see — Axioms. This word Axiom 
(Greek, aino/m) is used by Aristotle to mean “a truth so 
obvious as to be in no need of proof” — virtuall} T in the modern 
sense of a “self-evident truth.” Euclid used only five Postu- 
lates and thirteen Common Notions, none of which chal- 
lenged doubt save the celebrated “parallel-postulate.” Indeed, 
all were very simple except this fifth postulate,* which excited 
suspicion, not only on account of its cumbrous form, but be- 
cause it is used only once — to prove the inverse of a proposition 
already demonstrated — the seventeenth. “It requires,” says 
Staeckel, “a certain amoutit of courage to declare such a 
requirement, alongside the other simple axioms and postu- 
lates.” The Swiss mathematician, J. H. Lambert, f averred 
that Proklos, Euclid’s first commentator (410-485 A. D. ) 
argued that the parallel-postulate was demonstrable, because 
it was the inverse of the seventeenth proposition. 
Euclid’s twenty-seventh proposition: that straight lines 
ledge my general indebtedness to the writings of Professor Halstsd, to 
which I occasionally refer. 
*Also given in various editions of Euclid as a Common Notion — elev- 
enth, twelfth, or thirteenth. 
tLambert’s Theory of the Parallel Linen was not published until 1786 
twenty years after it was written and nine years after his death, by Bern- 
ouilli and Hindenberg in the Magazin fur die reine und angewandte Matlie - 
matilc. 
