NABOKLE 
409 
THURSDAY, MARCH 3, 1904. 
THE HISTORY OF ELEMENTARY 
MATHEMATICS. 
(Geschichte der Elementar-Mathematik in  system- 
atischer Darstellung. By Dr. Johannes Tropfke. 
Zweiter Band. Pp. viii + 496. (Leipzig: Veit and 
Co., 1903.) 
le his first volume of this work, already reviewed in 
Nature, Dr. Tropfke confined himself to the 
lhistory of elementary arithmetic and algebra. He has 
now completed his work by giving with equal care, and 
the same wealth of references, the story of the progress 
of the other branches of elementary mathematics— 
geometry, logarithms, plane and spherical trigono- 
metry, series, stereometry, analytical geometry, and a 
few other topics of minor importance. 
To geometry 140 pages are devoted, besides the 
chapters on conics and analytical geometry ; this is not 
very much, and some paragraphs are so condensed that 
a large part of them consists of titles. But the 
arrangement is good, and on‘ several points very 
interesting details are given. Thus we have side by 
side, and in the original, the definitions of Euclid and 
those of Hero of Alexandria. The comparison of the 
two sets is instructive; thus Hero adds to the bare de- 
finition of a line (i.e. curve) a statement of its genesis 
by an ideal point continuously moving in space 
(yiyverat 5€ anpetov puevros avabev Karo evvoia Tt KaTa 
osvvexecav). Similarly he adds to Euclid’s definition 
of a straight line another definition of it as 
the shortest line joining two points. In_ trans- 
lating Euclid’s definition of a straight line, Dr. Tropfke 
wenders the very difficult phrase é& icov by ‘ gleich- 
miassig (in derselben Anordnung und Richtung),”’ 
where, of course, the parenthesis is the translator’s 
gloss, and probably puts more into the definition than 
Euclid intended. Perhaps ‘‘ symmetrically,’’ as we use 
the term, is the nearest equivalent. Whatever is pro- 
posed, it must be remembered that the current version 
in English editions of the ‘‘ Elements ”’ is a mistrans- 
lation. Euclid says nothing about the extreme points 
of the line; he says ‘‘ a straight line is one which lies 
<& ioov with respect to the points on it,’’ that is, to 
all the points on it. 
Two other entertaining sections are those on the 
construction of regular polygons and on the various 
approximations to 7. It is amazing that the author 
has made the statement, so often seen in print, that 
‘Gauss proved that the circle can be divided by rule 
and compass into n equal parts only when n is a prime 
of the form 2™+1. Probably it is a slip of the 
pen in this case, for only seven pages earlier the 
‘division into fifteen parts is referred to. Nevertheless, 
the wrong statement is definitely made, and it really 
seems as hopeless to try to get this vulgar error 
‘corrected as to expect authors to spell the name 
Bernoulli properly. After all, the practical man, with 
tools that range from a sixpenny protractor to a 
‘dividing engine, cares not for these abstractions. How 
different it was a few centuries ago! not better, of 
NO. 1792, VOL. 69] 
course, but certainly more amusing. The ‘‘ divine pro- 
portion ”’ or ‘‘ golden section ’’ impressed the ignorant, 
nay even learned men like Kepler, with a sense of 
mystery, and set them a-dreaming all kinds of fantastic 
symbolism. Even to the Greeks it was the section; and 
their philosophers, doubtless infected by the East, 
speculated about atoms and regular solids in a way 
that seems to us childish, but was serious enough to 
them. At any rate, the man who first found out an 
exact construction for a regular pentagon had reason 
to feel proud of his exploit; and the superstitions which 
have gathered about the pentagramma mirificum are 
grotesque echoes of his fame. 
Mathematicians now alive must sometimes feel it a 
rather mournful privilege to have read what is 
practically the last chapter in the chronicle of z=. 
The first that is known at present is in the Rhind 
papyrus (2000-1700 B.c.), where the approximation 
m™=256/81=3-1605 is given; how it was obtained is, 
unfortunately, quite uncertain. Dr. Tropfke gives in 
detail Archimedes’s very ingenious method, which he 
carried out far enough to prove that 3} >r>3}?. It is 
well known that 355/113 is a remarkably near approxi- 
mation, which is easily remembered. It appears that 
this is due to a German mathematician, Valentinus 
Otho, who is said to have obtained it from #2 (Archi- 
medes) and 337 (Ptolemy) by subtracting numerators 
and denominators. Shanks’s calculation to 707 places of 
decimals still holds the record. The symbol = for the 
ratio of circumference to diameter is first used in 
William Jones’s ‘‘ Synopsis palmariorum matheseos ”” 
(1706); Euler made it popular. 
Part iv. (pp. 141-186) is on logarithms. It gives a clear 
account of the methods of Biirgi and Napier, with speci- 
mens of their tables; of the later developments of logar- 
ithmic series; and of the most noteworthy logarithmic 
tables. By a curious irony of fate, the expeditious 
methods of calculation now familiar were not dis- 
covered until after the tables of Briggsian logarithms 
had been computed. The base of Biirgi’s logarithms 
is nearly e, and that of Napier’s nearly e-*; but neither 
of them was acquainted with the true theory of natural 
logarithms, and ‘‘ Napierian logarithms” is really a 
misnomer, when applied to natural logarithms. 
Noticeable in the sections on plane and spherical 
trigonometry are the specimens of early tables, includ- 
ing Ptolemy’s, and the account given of the treatise 
attributed to Nasir Eddin Tusi, of which a French trans- 
lation was published at Constantinople in 1891. Dr. 
Tropfke describes this as the first systematic treatise on 
plane trigonometry, considered as an independent sub- 
ject; moreover, it discusses oblique-angled triangles 
after the modern manner, instead of reducing the solu- 
tion of them to that of right-angled triangles. This 
part of the book brings out the services rendered to 
mathematics in the middle ages by Arab, or more pre- 
cisely Arabic-speaking, geometers. The inventors of 
the more important formule are also indicated. On p. 
198 there is a note on Hero’s Mezpixa, published so re- 
cently as last year; in this work occurs a term for the 
fourth power of a quantity, not previously known to 
have been used before the time of Diophantus. 
it 
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