PROPORTION. There must be in the mind of every person, 'antecedently to all mathematical instruction, a perfect conception of proportion, though not perhaps of the manner of measuring magnitnder with a view to express their proportions by means of numbers. All who can trace the resemblance of a drawing to the original, or have the least notion of the use of a map, are in possession of the funds mental notions on which a theory of proportion can by founded. The term Itarto is that under which the first part of the subject should be treated, and the article cited will contain matter preliminary to the present one. It will be sufficient for our present purpose to state that the ratio or relative magnitude of two magnitudes is to be measured by the number of times or parts of times which one is contained in the other, whenever the two are commensurable; and we shall now confine ourselves to the purely mathematical treatment of the theory of proportion, and shall avoid, as much as the nature of the subject will admit, all discussion of the notion of ratio considered as s magnitude.
We cannot well explain the nature of the difficulty which occurs in the theory of proportion, without a prior reference, first, to the purely arithmetical treatment of the subject, and secondly, to the practical sufficiency of this method which is the necessary consequence of out physical constitution.
If all magnitudes of the same kind were necessarily commensurable that is, if any one among theta being taken as a unit, the rest were all expressible by multiples, aliquot parts or aubmultiples, and multiply of aliquot parts, of the unit chosen, no difficulty would arise in making the subject of proportion purely arithmetical. For lot a and b repro sent the nnita, parts of units, or both, in two magnitudes of the gam( kind (as two lines); any sufficient demonstration of the rule for the division of one fraction by another will show that a contains 6 precisely a number of times and parts of times. If then we any that a is to / in the same proportion as c to ri, we mean that a contains 6 precisely such times and parts of times as c contains d : that is, we assert the equation a f ; the mathematical treatment of which Is ao may, that no one who can solve a simple equation can be stopped for a moment by the difficulty of any consequence of it. The following proposition, which may be proved generally, contains all the consequences which are most useful. Let a, 6, c, and d be (in the arithmetical manse) proportional : take any two functions of a and 6, which are homogeneous and of the same dimension (such as air+ and (0 Take corresponding functions of c and d (which are cd+ di and then the four numbers so obtained aro also proportional (that is, a6 + bl contains as many times and parts of times as al + contains In measuring magnitudes of which the numerical representatives are afterwards to be submitted to calculation, it necessarily follows, from the imperfections of our senses, that some imperceptible amount of magnitude must always be neglected or added ; so that, for example, that which wa call a line of 3 inches long mean. something between
and 3.1, or 219 and or and 3.001, according to the degree of accuracy of the measurement. All magnitudes therefore are practically commensurable; for suppose, In the ease of weights for example, that a grain is taken as the unit, and that the ten-thousandth part of a unit is considered as of no consequence; then by taking every weight only to the nearest ten-thousandth of a grain, they may every one be expressed arithmetically with a conventional degree of precision, which for every purpose of application will do as well as though it were perfectly accurate.
The discovery of mcommessoname magnitudes, one of the most striking triumphs of reason over the imperfection of the senses, was made at a very early period ; since the demonstration of their exist ence, tho classification (to a certain extent) of their species [Iunattotrar. QusanTries], and the means of overcoming the difficulties which they present, appear in the writings of Euclid. A moment's consideration will show that a property of numbers, a relation of figura in geometry, a general law of nature, may be inferred from induction with a degree of probability which will amount to moral certainty, both as to the exactness and universality of the property, relation, or law. But the existence of incommensurable magnitudes can never be made certain, except by absolute deduction : no attempt at measurement, a risibele existing, could show the non-existence of any common measure, however amaIL Suppose for instance that, having provided means of measurement which can always be depended on to show the thousandth of an inch, but nothing less, a person should accurately (as the word is commonly used) lay down squares of one, two, &c., inches in the side, with a view to render the existence of a common measure to the side and diagonal exceedingly improbable by experiment. if not before, ho would be baffled by the square) whose side is 2378 inches, the diagonal of which could not by his measurers be distinguished from 3363 inches, from which it differs only by about the fire-thousandth, part of an inch. And let any greater degree of exactness be attained In the means of measurement, short of positive accuracy, a reasoner on the subject could still predict a square which should defeat the attain ment of the object sought.