On Definitions. 269 



ported Plane and Solid Geometry, Conic Sections, Trigonometry, 

 and, in some measure, all the sciences illustrated by the proportion- 

 ality of lines and figures. 



In order to show, with great brevity, how such a superstructure 

 can be raised out of definitions alone, let us take one very plain ex- 

 ample from the square. The square may be defined, a parallelogram 

 having two adjacent sides equal, and one angle a right angle. By 

 its being a parallelogram, the opposite sides are equal, and two of 

 the adjacent ones being equal, they must thus be all equal. Also, 

 one of the angles being right, and the two opposite angles of a par- 

 allelogram being equal to two, the opposite one must be right ; and 

 the two remaining angles being equal, and amounting to two right 

 angles, must be severally right angles. Its diagonals are equally the 

 property of triangles having two sides, and the included angle in the 

 one equal to the corresponding parts in the other. 



In the sciences denominated Mixed Mathematics, a much greater 

 number of principles must be admitted, in particular all those which 

 are learned from experiment. In Mechanics, we must take for 

 granted, the inertia of matter, its motion by impulse in a straight 

 line, the equality of the motion produced to the impulse, the com- 

 position of forces, and their resolution, with some others. Upon 

 these and a few other assumptions, and by means of accurate defini- 

 tions, the various branches of the science of forces is founded. 



In Astronomy, in addition to the other principles of dynamics, or 

 forces, it is admitted that the power of gravitation diminishes as the 

 square of the distance, and that it acts equally on every particle of 

 matter. On these admissions, and by the aid of a powerful analysis, 

 founded on accurate definitions, and the co-operation of innumerable 

 observations, is founded the most splendid of all the sciences. 



In Optics must be admitted, the motion of light in straight lines, 

 the reflection of the rays from the plane surfaces of non-transparent 

 surfaces, and refraction to or from the perpendicular, according to the 

 change of medium. 



It is remarkable of all these principles of the mixed sciences, that 

 they admit of strict definitions. The truth which they announce, 

 and which is ascertained by experiment, is coupled with the defi- 

 nition. 



In those sciences which treat of Fluids, whether elastic or non- 

 elastic, there is considerable difficulty in fixing upon proper defini- 

 tions, and those that have been fixed upon, or adopted by all inquir- 



