Marcu 5, 1914] 
NATURE 3 
de J. Tannery. 
7. 
Tereenss 
Troisi¢me Edition. Pp. vi+ 
(Paris: A. Hermann et Fils, 1914.) Price 
(1) T N the old days a boy had to reason in 
| geometry “acording to the rules of the 
game.” Many a boy felt that he must put aside 
common sense for fear that it might contradict 
the “rules,” and that he must “play the game 
according to the rules.”” Hence came grotesque 
howlers, and the boy felt it unreasonable that 
grotesque results should be ridiculed, for they 
arose from strict adherence to rules. While in 
the present happy days we have altered that as 
regards geometry, the old method persists in 
algebra; the subject is treated in a purely abstract 
manner, it has no relation to anything in every- 
day life; the student can only commit to memory 
the rules of the game, and do his best to play the 
game. In this case the results also are generally 
abstract, so that appeal to common-sense would 
be impossible even if the boy allowed the validity 
of the appeal. 
Now, however, a brighter dawn is breaking for 
the ill-treated boy. Usherwood and Trimble con- 
nect the algebraic work with the concrete through- 
out. The boy no longer has to play a game with 
rules he does not understand. Each algebraic 
process arises out of concrete instances, which are 
themselves easy of comprehension, and give a 
common-sense meaning to the process. The book 
is thus in the van of the movement to humanise 
algebra as geometry has been humanised, and in 
another generation no headmaster will mourn the 
blighting influence which x+y had upon him. 
The authors approve of the use of contracted 
methods, as do most of the best teachers at the 
present moment. We venture, however, to ques- 
tion whether these authors and teachers do well in 
this matter. It is customary to work to a signi- 
ficant figure more than will be required in the 
result; this generally gives the result to the re- 
quired approximation, but not always. Are we to 
chance the accuracy, or are we to complicate the 
process further by an estimate of the trustworthi- 
ness of the result? Moreover, the estimation of 
the number of figures to be retained, even in the 
normal case, is a matter of no little skill; we have 
frequently known ‘professors and schoolmasters 
of good standing to be at fault. 
There appears to be no educational principle at 
stake, and the question is simply whether con- 
tracted methods conduce to speed and accuracy 
or not. Does the shortness of the contracted 
calculation compensate for the time spent in de- 
ciding how far to contract, and for the chance of 
error by excessive contraction? For the expert 
calculator, like the teacher of arithmetic or the 
NO. 2304. VOL: 93) 
observatory computer, it compensates without 
doubt. For ourselves, and we imagine for most 
people (adults and children), contracted methods 
in their strict form do not compensate. For us 
the best way is to calculate stolidly through, and 
at the end throw away the unnecessary figures, or 
if the numbers get very heavy, to contract to a 
modified extent, keeping, perhaps, two or three 
more figures than a strict contractionist would 
allow. 
(2) Why do the universities so carefully exclude 
Mongian geometry from their “pure” mathe- 
matical courses? To many a pure mathematician 
the happening upon such a book as Harrison and 
Baxandall’s is like the acquisition of a new sense. 
His ideas of solid geometry are those of Euclid’s 
Eleventh Book, in which he is instructed to “draw 
a plane through three given points,” or to carry 
out in three dimensions some construction that has 
been discussed in the plane, and which he had 
imagined was meant to be carried out on a sheet 
of paper with ruler and compasses. On arriving 
at the Eleventh Book he discovers that when 
Euclid says “draw a plane through three given 
points,” he only means that three points determine 
a plane; on the Book I. construction carried out 
in three dimensions he has to put such interpreta- 
tion as he can. 
In course of time he happens upon a book on 
Practical Solid Geometry and regions unknown 
to Euclid and the universities. With what joy he 
finds that it is possible to represent points in space 
upon a sheet of paper, and actually possible to 
draw a plane through them. He wishes he was 
young again so that he might follow up all the 
wonderful consequences, actually carrying out all 
the constructions and not merely talking about 
“how it is done,” as was his custom at the univer- 
sity. And Harrison and Baxandall would be a 
first-rate book to use if he could be young again. 
It contains the most alluring problems, wonderful 
in variety. There are no watertight compart- 
ments, but every branch of mathematics that can 
help is allowed to do so. The language, more- 
over, is excellent, a statement that cannot always 
be made of English mathematical writing. 
Part II]. of Messrs. Harrison and Baxandall’s 
book is Graphics, an admirable subject, neglected 
in pure mathematics in the same unaccountable 
way as Mongian geometry. The story runs that 
Prof. P. G. Tait held that the analytical method 
was always superior to the graphical, and applied 
it to ascertain the stresses in the Forth Bridge. 
Although a man of unrivalled intellectual power 
he failed that time. Those days are gone, the 
schools are learning the value of Graphic Statics, 
and perhaps in time the universities may follow. 
