38 SCIENTIFIC APPARATUS. 



examples of such properties ; and, what is of more importance 

 may be expected to suggest entirely . new points of view in a 

 branch of inquiry, which, more than almost any other within the 

 range of pure mathematics, is dependent on direct observation. 



2. The Graphical Properties of space are those which involve 

 the conceptions of the straight line and .plane, but do not involve 

 any conception of magnitude, or of measurement. The Elements 

 of Euclid will be searched in vain for an example of a purely 

 descriptive theorem, though it would seem that one of the lost 

 treatises of that great geometer the " Porisms " was devoted to 

 this part of geometry. In modern times researches into the de- 

 scriptive properties of figures were revived by Blaise Pascal, and 

 his elder contemporary, Desargties. By a strange fatality, the 

 purely geometrical works of these two eminent men were lost, or 

 wholly neglected, for more than a century, and it is only in com- 

 paratively recent times that they have received the attention 

 which they merited. We may take as a simple instance of a 

 graphical theorem the proposition of Desargues : " If two triangles 

 lying in the same plane are such that the lines joining their ver- 

 tices taken in pairs meet in a point, the three intersections of 

 the pairs of sides opposite to these vertices lie in a straight line; 

 and conversely." 



3. Lastly, the Metrical Properties of space are those which 

 involve, implicitly or explicitly, the consideration of magnitude. 

 Thus the old proposition of Pythagoras, "The square of the 

 hypothenuse in a right angle is equal to the squares of the sides 

 containing the right angle :" and the theorems of Archimedes, 

 " The surface of a sphere is equal to the curved surface of its 

 circumscribing cylinder ; the volume of the sphere is two-thirds of 

 the volume of that cylinder," are metrical propositions. They 

 could not be made intelligible to a person who had not the con- 

 ception of the equality of geometrical magnitudes ; nor verified by 

 any one who had not the means of making exact quantitative 

 measurements ; whereas the proposition of Desargue/?, above 



