AveustT 23, 1918] 
nucleuses for the student’s further develop- 
ment along this line. 
To emphasize the fact that some of the 
present historical notes are useless from this 
standpoint we may mention that many of our 
elementary geometries state that according to 
tradition Pythagoras was so jubilant over his 
discovery of the Pythagorean theorem that he 
sacrificed 100 oxen to the gods on the oc- 
easion. It is difficult to see why authors of 
text-books waste space on such a statement. 
It is probably not true that such a sacrifice 
was made by Pythagoras, and if it were true, it 
eould only lessen our respect for him. Just 
imagine meeting now’ a man in the act of 
sacrificing 100 oxen because he had made a 
mathematical discovery. Would you not con- 
clude:that he ought to be in an asylum for the 
insane? 
By the time the average student reaches 
college he may easily have read twenty-five 
historical notes in this various mathematical 
text-books. It is therefore of interest to list 
twenty-five topics which have been epoch 
making in the development of pure mathe- 
matics and are nucleuses of an extensive his- 
tory. It can scarcely be expected that all 
would agree entirely on what twenty-five 
points should be regarded as most important 
in the history of secondary mathematics but 
one may perhaps assume general agreement 
as regards the fact that each of the subjects 
of the following list is worthy of consider- 
ation in connection with this question. The 
order of these subjects is supposed to be 
chronological with respect to the beginnings 
of their history. (1) Numeration and nota- 
tion; (2) Value of +; (3) Irrational quantities 
and irrational numbers; (4) Science of ele- 
mentary geometry; (5) Science of elementary 
algebra; (6) Translations of treatises into a 
different language; (7) Science of trigonom- 
etry, or arithmetical geometry; (8) Algebraic 
solution of the general cubie and of the gen- 
eral biquadratic equation; (9) Use of log- 
arithms for numerical calculations; (10) Sci- 
ence of analytic geometry, or algebraic geom- 
etry; (11) Science of differential and integral 
ealeulus; (12) Scientific societies supporting 
SCIENCE 
183 
publications; (13) Special mathematical peri- 
odicals; (14) Ecole polytechnique; (15) Sci- 
ence of arithmetic, or the theory of numbers; 
(16) Reality of complex numbers; (17) found- 
ing of descriptive and projective geometry; 
(18) Theory of functions; (19) Non-euclidean 
geometry; (20) Theory of groups; (21) Johns 
Hopkins University; (22) Theory of aggre- 
gates; (23) International mathematical con- 
gresses; (24) Large modern mathematical en- 
eyclopedias; (25) International commission 
on the teaching of mathematics. 
The above list of twenty-five important 
topics in the history of secondary mathematics 
does not include any of the subjects of applied 
mathematics, which haye furnished strong 
motives for the development of pure mathe- 
matics, From the earliest times astronomy 
and surveying have furnished such motives 
especially for the development of spherical 
and plane trigonometry respectively. Among 
the subjects which furnished strong motives 
for some ‘of the later developments in pure 
mathematics we may mention celestial me- 
chanics, hydromechanics, and the theory of 
heat. 
Some may be surprised to find in the above 
list of important topics the names of two in- 
stitutions, but a little reflection will tend to 
make it clear that these institutions have been 
the centers of unusually strong mathematical 
influences. The early courses offered at these 
institutions are of great historical interest. 
It is true that the influence of the latter 
might be regarded to have been national 
rather than international, but the national 
activity which it fostered has been sufficiently 
extensive to merit general recognition. 
It is, however, not our purpose to justify 
the particular selection of topics noted above. 
Tf this list will tend to direct the attention of 
teachers and authors to the really seriouseand 
fundamental questions of mathematical his- 
tory, the purpose of its compilation will be 
fulfilled. Just as the material of the body of 
a text-book is selected with a view to furnish- 
ing matter of permanent usefulness rather 
than to arouse ephemeral interest, so it seems 
that the material for the historical notes 
