124 Reviews and Notices of Books. 
row’s display of erudition will appear other than an empty tossing 
about of words. But to those who have learnt to desire to see a 
matter thoroughly sifted and examined from every conceivable 
point of view, these lectures will afford ample gratification. 
The “ Mathematical Lectures” are twenty-three in number, and 
may be divided into three parts—1. Ten lectures on First Prin- 
ciples. 2. Three lectures on the Foundation of Geometry, or ra- 
ther on that feature of it which sets forth the true meaning of 
Geometric Equality; and, 3. Ten lectures on Ratio and Pro- 
portion. 
Of the first part it is only necessary to say, that it bears evi- 
dence of the indefatigable perseverance of the author in the search 
after testimonies from ancient and modern writers. In the second 
part, the question whether the foundation of geometry can be 
better laid than on what is known as the 8th Axiom of Euclid, 
is discussed in a masterly way, and decided in the negative. We 
know of no exercise better fitted to develop the powers of a student 
than these three lectures. The lynx-eyed author has searched 
into every cranny for reasons, for opinions, for objections. He 
has handled the subject with the grasp of a giant. Only one 
point, so far as we know, escaped him. In the second part of 
Prop. 2, B. 6, Euclid proves that two triangles are equal. But 
in what way can their equality be referred to the 8th Axiom, 
‘* Magnitudes which coincide are equal?” They are not proved 
to be congruous whole by whole, or part by part, or by any pro- 
cess of transmutation or succession. Their equality rests on the 
fact, that if any multiple of the one exceed any magnitude, the 
same multiple of the other does the same ; and if any multiple of 
the one falls short of any magnitude, the same multiple of the 
other does so likewise. Now, this fact establishes congruity 
only subject to the same objections which are raised against the 
application of the axiom, that equals taken from equals leave 
equals. And the objections are not easily removed ; Barrow has 
not even alluded to them. Perhaps he had intended to discuss 
this species of equality when he came to Proportionality, but we 
can find no indication of such intention. We should like to have 
had the opinion of the learned editor of the present edition on 
the whole of this question, There are few men besides himself 
whose opinion on such a subject would be entitled to much weight. 
Mathematical studies have taken a turn in the present day, which 
leads them, after the first stage, too much into the byways of 
mental training. We hope to see them brought back. Dr 
Whewell has done much for the improvement of the studies of 
the University of Cambridge, and the edition before us is proof 
that he is still engaged on the right side. 
The third part of the “ Mathematical Lectures” is occupied 
with the subject of Ratio and Proportion. As an instance of the 
