ARTHUR LEFEVRE — PEDAGOGICAL NOTE ON MENSURATION. 43 
of two sects to the square on any third sect* is the product of the ratios 
of the two given sects to the third sect. That is to say, taking the “third 
sect” for the unit-line: — the area of the rectangle, if the unit-surface he 
the square on the unit-line, is the product of the lengths of two adjacent 
sides. 
Similar remarks apply to volumes and cubes. 
~Now, it seems plain that 'confusions about the essential meaning of 
area and volume , and about the warrants for chosen methods of cal- 
culation, would be obviated by a little systematic 'comparison of surfaces 
with some other unit besides the square on the unit-line, and of solids 
with some other unit besides the cube on the same line. 
I would suggest the regular triangle whose side is a unit-line, and 
the regular tetrahedron whose edge is the same line. One reason for 
choosing these rather than some others is the fact that they are respect- 
ively the minimum regular surface and the minimum regular solid de- 
terminable by a unit-line. 
The suggested comparisons, besides clearing up the confusions already 
mentioned, would aid in elucidating the whole doctrine of geometric 
similarity. Every thoughtful student of geometry knows better; but I 
find the notion prevalent that the essence of the matter is expressed in 
the statement that, similar surfaces have the ratio of the squares on 
corresponding lines determined by the surfaces; and that similar solids 
have the ratio of cubes on such lines. Of course the truth is, similar 
surfaces have the ratio of the regular triangles, or any other similar sur- 
faces constructed on the said lines; and similar solids, the ratio of the 
regular tetrahedra, or any other similar solids constructed on the said 
lines. The plan suggested would compel a more fundamental and gen- 
eralized concept and definition of similarity j than that in vogue, and 
would clearly display the truth that the essence of the relations just 
considered is numerical. The fundamental statement of the theorems 
would plainly appear to be, that similar surfaces have a ratio which is 
the second power of their ratio of similitude; and that similar solids 
have a ratio which is the third power of their ratio of similitude. All 
along the line of elementary mathematical instruction I find much re- 
tardation due to confusions between the numerical and geometric mean- 
* Definite piece of straight line. 
f Similarity should be discerned as a special case of projection. The 
essential definition, from which all properties of similarity flow in a true or- 
ganic sequence, is: ‘Two figures (two-dimentional or three-dimentional) are 
similar , if they can so be made perspective that the ratio of corresponding 
perspective sects is constant?’ This constant ratio of corresponding perspec- 
tive sects is called the ratio of similitude of the figures. 
