436 INDUCTION. 



tions which resolve themselves into the addition of the equals to them- 

 selves or to other equals; we cease to wonder that in proportion as a sci- 

 ence is conversant about equality, it should afford a more copious supply 

 of marks of marks ; and that the sciences of number and extension, which 

 are conversant with little else than equality, should be the most deductive 

 of all the sciences. 



There are also two or three of the principal laws of space or extension 

 which are unusually fitted for rendering one position or magnitude a mark 

 of another, and thereby contributing to render the science largely deduc- 

 tive. First, the magnitudes of inclosed spaces, whether superficial or solid, 

 are completely determined by the magnitudes of the lines and angles which 

 bound them. Secondly, the length of any line, whether straight or curve, 

 is measured (certain other things being given) by the angle which it sub- 

 tends, and vic^ versa. Lastly, the angle which any two straight lines make 

 with each other at an inaccessible point, is measured by the angles they 

 severally make with any third line Ave choose to select. By means of these 

 general laws, the measurement of all lines, angles, and spaces whatsoever 

 might be accomplished by measuring a single straight line and a sufficient 

 number of angles ; which is the plan actually pursued in the trigonometrical 

 survey of a country ; and fortunate it is that this is practicable, the exact 

 measurement of long straight lines being always difficult, and often impos- 

 sible, but that of angles very easy. Three such generalizations as the fore- 

 going afford such facilities for the indirect measurement of magnitudes 

 (by supplying us with known lines or angles which are marks of the mag- 

 nitude of unknown ones, and thereby of the spaces which they inclose), 

 that it is easily intelligible how from a few data we can go on to ascertain 

 the magnitude of an indefinite multitude of lines, angles, and spaces, which 

 we could not easily, or could not at all, measure by any more direct process. 



§ 9. Such are the remarks which it seems necessary to make in this 

 place, respecting the laws of nature which are the peculiar subject of the 

 sciences of number and extension. The immense part which those laws 

 take in giving a deductive character to the other departments of physical 

 science, is well known ; and is not surprising, when we consider that all 

 causes operate according to mathematical laws. The effect is always de- 

 pendent on, or is a function of, the quantity of the agent ; and generally of 

 its position also. We can not, therefore, reason respecting causation, with- 

 out introducing considerations of quantity and extension at every step ; 

 and if the nature of the phenomena admits of our obtaining numerical data 

 of sufficient accuracy, the laws of quantity become the grand instrument for 

 calculating forwai'd to an effect, or backward to a cause. That in all other 

 sciences, as well as in geometry, questions of quality are scarcely ever inde- 

 pendent of questions of quantity, may be seen from the most familiar phe- 

 nomena. Even when several colors are mixed on a painter's palette, the 

 comparative quantity of each entirely determines the color of the mixture. 



With this mere suggestion of the general causes which render mathe- 

 matical principles and processes so predominant in those deductive sciences 

 which afford precise numerical data, I must, on the present occasion, con- 

 tent myself; referring the reader who desires a more thorough acquaint- 

 ance with the subject, to the first two volumes of M. Conite's systematic 

 work. 



In the same work, and more particularly in the third volume, are also 

 fully discussed the limits of the applicability of mathematical principles to 



