Page:Observations on Man 1834.djvu/210

 cruelty, treachery, &c. separately taken, affect the mind; and yet, since all reasoning upon them is to be founded on their definitions, as will be seen hereafter, it seems best to refer them to this third class.

Lastly, the particles the, of, to, for, but, &c. have neither definitions nor ideas.

. II. This matter may be illustrated by comparing language to geometry and algebra, the two general methods of expounding quantity, and investigating all its varieties from previous data.

Words of the first class answer to propositions purely geometrical, i.e. to such as are too simple to admit of algebra; of which kind we may reckon that concerning the equality of the angles at the basis of an Isosceles triangle.

Words of the second class answer to that part of geometry which may be demonstrated either synthetically or analytically; either so that the learner’s imagination shall go along with every step of the process painting out each line, angle, &c. according to the method of demonstration used by the ancient mathematicians; or so that he shall operate entirely by algebraic quantities and methods, and only represent the conclusion to his imagination, when he is arrived at it, by examining then what geometrical quantities the ultimately resulting algebraical ones denote. The first method is in both cases the most satisfactory and affecting, the last the most expeditious, and not less certain, where due care is taken. A blind mathematician must use words in the last of these methods, when he reasons upon colours.

Words of the third class answer to such problems concerning quadratures, and rectifications of curves, chances, equations of the higher orders, &c. as are too perplexed to be treated geometrically.

Lastly, words of the fourth class answer to the algebraic signs for addition, substraction, &c. to indexes, coefficients, &c. These are not algebraic quantities themselves; but they alter the import of the letters that are; just as particles vary the sense of the principal words of a sentence, and yet signify nothing of themselves.

Geometrical figures may be considered as representing all the modes of extension in the same manner as visible ideas do visible objects; and consequently the names of geometrical figures answer to the names of these ideas. Now, as all kinds of problems relating to quantity might be expounded by modes of extension, and solved thereby, were our faculties sufficiently exalted, so it appears possible to represent most kinds of ideas by visible ones, and to pursue them in this way through all their varieties and combinations. But as it seems best in the first case to confine geometry to problems, where extension, and motion, which implies extension, are concerned, using algebraic methods for investigating all other kinds of quantity, so it seems