We do not deny that a mind well versed in the doctrine of limits would have a process of its own, by which it would rigorise the method of Lacroix, arguing as follows :—The same means which show the pro positions of geometry to have no error perceptible to our senses, would also show them to have no error perceptible to any imperfect senses, however near to perfection they might be; and that error which is less than any assignable error, must he no error at. all. All this, when properly extended, we might admit ; but how are we to suppose, in the student who has just left the fourth book of Euclid, a perception of the truths of the doctrine of limits, the notorious want of which creates the difficulties of the differential calculus ? Souud teaching makes a true theory of proportion one, out of many previous helps, to the attainment of the differential calculus : but the inversion of the process not only adds difficulty to the latter, by intercepting proper illustration, but introduces falsehood into the former; and such teaching is the most vicious of all vicious circles, because it propagates its kind.
The mathematical writers of this country have, taken altogether, shown a superiority in exactness of demonstration over those of any other of modern times, and the deep and early root which the sound principles of Euclid have taken has been mainly the cause of this. If those principles be abandoned, that superiority will cease to exist ; but this of itself would be of little consequence ; not so the loss of a large amount of formation of accurate habits, which would certainly follow the substitution of a gross and (so called) practical instrument, of cal culation for an exact and liberal science.
Those to whom the mathematical sciences are taught as aids to the power of distinguishing truth from falsehood, logic from fallacy, the exact consequence from all incorrect inference, very many times exceed in number those who only wish for an instrument to be used in the study of physics and the arts of life. If then practical mathematics mean those mathematics which best answer the purpose of the great majority of learners, the more they are rigorous the more they are practical. But the enticing word practical has been otherwise appro priated ; and "exact enough for practical purposes " is the phrase applied to many a result of which the practical use belongs to the astronomer, mechanician, surveyor, engineer, or computer. To such a meaning of the word practical there is no objection as opposed to liberal or disciplinatory, when it is knoiele.dgc or science which is spoken of ; hut as applied to art, practical is opposed to unpractical, which cannot be carried into practice, or useless, for that which cannot be practised is useless in art. But those who would consider the use of knowledge in steadying the mind and making it a judge between the true and the false, and a safe guide to the methods of finding truth, must beware, in mathematics, how they allow the notions which art attaches to the word practical any influence over their method of studying science. For want of such a cautiou many have missed all comprehension of the higher branches even of the art to which they aspired, to say nothing of the loss of that science to which it should seem they did not mean to aspire.