GEOMETRICAL INSTRUMENTS AND MODELS. 39 



quoted, is intelligible to any one who knows what a straight line 

 and a plane are, and may be verified by any one who has a sheet 

 of paper, a pencil, and a straight edge. 



Comte proposed to define geometry as " la science qui a pour 

 but la mesure des grandeurs." As a scientific statement this 

 definition is probably insufficient, because a great part of geometry 

 consists, as we have seen, in propositions which have no imme- 

 diate connection with measurement. It must, indeed, be admitted 

 that by far the most important applications of geometry to natural 

 science, and to the business of life, turn on the metrical pro- 

 perties of figures. But, in a purely theoretical point of view, there 

 is reason to believe that the graphical properties of space are the 

 more universal, and deeply-seated in the nature of things, notiora 

 natures, as Lord Bacon would have said ; and that the metrical 

 properties are, in a certain sense, secondary and derivative. As 

 an example of the character of universality, which we thus attribute 

 to graphical properties, we may take the general principle of the 

 duality (as it has been termed) of geometrical figures. This prin- 

 ciple asserts that all purely graphical theorems are twofold ; i.e. 

 that any graphical proposition relating to points and planes in 

 space gives rise to another, which is correlative to it, but in 

 which the points have been replaced by planes, and vice versd : 

 the line joining two points being replaced by the line of intersec- 

 tion of two planes. 



We proceed to indicate the principal classes of material 

 appliances which are of use in geometrical investigations, or in the 

 applications of geometry to the arts, or lastly, in its employ- 

 ment as a means of education. 



We shall mention successively . 



A. Instruments used in geometrical drawing or mapping, and 

 in copying geometrical drawings or maps. 



B. Instruments used in tracing special curves. 

 C. Models of figures in space. 



