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their oversights and mistakes alone, to imprint on the mind of the young mathematical student the necessity of close reasoning, and for the purpose of showing the consequences of coming to hurried and undigested conclusions; also, that the student might be made aware that a difficulty does exist: the nature of such difficulty he should likewise know; that he might, by the consideration of a sufficient number of examples, acquire confidence in the results of his demonstrations.
Legendre, the great French geometer, does not give proportion a place in his Elements of Geometry, for he was of opinion that the subject belonged to arithmetic, and algebra, not to geometry at all. Now this opinion must be totally wrong, for it is almost universally allowed, that the most refined specimen of human reasoning is to be found in Euclid's doctrine of geometrical proportion; while no such concession can be made in favour of the subject when treated of either algebraically or arithmetically. It has been justly observed, that “any system of geometry made less by geometrical proportion must be miserably defective."
Sir David Brewster, in his Translation of Legendre’s Geometry, falls into a notable error, insomuch that he makes assertions which are not at all true. In speaking of Legendre, he says, “ the author has provided for the application of proportion to incommensurable quantities, and demonstrated every case of this kind as it occurred, by means of the reductio ad absurdum.” This assertion Professor Young very justly
questioned, and has given examples from Brewster's translation, where the inference does not hold good.
Professor Leslie's “ Elements of Geometry” is remarkable for false demonstrations; and in his fifth
; book he demonstrates the propositions on proportion to be true, when the magnitudes are commensurable. The fact that those demonstrations do not hold good when the magnitudes are incommensurable, seems well known to Mr. Leslie; but that those magnitudes should also be homogeneous is altogether neglected by the learned professor.
Mr. Keith, desirous of applying a new demonstration to Proposition XVIII, in his edition of Euclid, falls into an egregious error, as he employs alternation to quantities whose antecedents might be heterogeneous: this mistake appears to be very common among Euclid's modern improvers, and to account why it is so is rather difficult; but a deviation from truth is easily committed, as there is but one truth, and many seeming truths.
Bonnycastle, in two of the principal propositions of his fifth book, gives demonstrations undoubtedly intended for general ones, which only apply to cases where all the magnitudes are of the same kind. That those demonstrations were intended for general ones there can be no doubt; for in his notes, page 257, Bonnycastle finds fault with Euclid's method of composition and division of ratios, as not being sufficiently general; and quotes Thomas Simpson as very properly making this remark: however, this assertion of Bonnycastle, both with respect to T. Simpson and Euclid, has been very justly contradicted by Professor Young, after the lapse of thirty years.
Dr. Austin, in his “ Examination of Euclid,” commits the same error as Mr. Keith, that of allowing demonstrations which only apply to particular cases to be substituted for general ones: most probably Keith adopted his demonstration from Austin, who recommends it in a very high degree; yet it is surprising how such errors can exist in the writings of such men, or how one can copy them from another without detecting them; since to bear in mind that quantities of a different kind can have no ratio to each other would have prevented such oversights.
That able mathematician and ingenious elementary writer, Professor Young, who so ably criticised our modern writers on geometry, especially on this subject, in cultivating the ideas of M. da Cunha, falls himself, if not into an error, into a very great inconsistencythat of discarding Euclid's doctrine of ratios from his fifth book. He undoubtedly treats of geometrical proportion without using the term ratio ; but he gives other terms of a more lengthened nature, which precisely convey the same meaning: now, to do away with ratio here, is to do away with it in every subject that follows, or through a whole course of mathematics; and any such attempt should not be entertained, for it is not so very difficult to define what is intended to be expressed by the term ratio.
It is a different thing to have a clear conception of
what the technical term “ratio" is meant to convey, from knowing that what is intended by the term cannot be exactly expressed in many cases by numbers. Professor Young will not deny (for they are his own words) that the term in reality denotes the quotient arising from the division of one magnitude or quantity by another of the same kind (or the multiple or submultiple which an antecedent is of its consequent); it is accurately assignable (in numbers) when the magnitudes are commensurable, but unassignable (in numbers) when they are incommensurable." When this simple fact is known, what is to be understood by the term cannot be misconstrued, although we do allow that in many cases the exact ratio of one magnitude to another of the same kind cannot be expressed by numbers; this
may be a fault in our present system of notation, or in the plan adopted for finding a common measure, and not in our geometrical notion of that which is to be conveyed by the term. And the impossibility of a person rightly understanding what is meant by saying, as A is to B, so is C to D, without embodying the idea of what is here expressed by the term ratio cannot be denied; it matters not by what phrase, word, form, or mode of language the idea is conveyed to the mind The student will readily perceive that the term ratio is not intended to convey a real and substantial essence, but merely a simple conception of the mind, which can be well defined, and not, as some writers would have it, an “ill-defined, or unknown term.” No; it is a term so interwoven through almost every part of mathematics,
that to expunge it would be almost impossible; nor can any real good proceed from substituting another term in its stead.
These strictures could have been carried much further: indeed, they might include many mathematical writers of much more repute than those alluded to, and several of a minor consideration, whose names should not be ranked among geometers; in the former class Dr. Simpson, the great restorer of ancient geometry, would not be exempt; nor even Newton, for in the 17th lemma of his “Principia,” edit. 1713, and in other places, he uses given ratios, and ratios that are always the same, for one and the same thing ; but such mis
, takes should not be admitted, as they may lead to other errors. Among the latter class, above referred to, may be mentioned the author of a tract entitled, “The Connection of Number and Magnitude,” which is a curious mixture of “good and evil.” To a young student it would be difficult to determine what doctrine the Author advocates, as he seems to be one of those who, to guard against objections, take shelter in obscurity, and leave the meaning doubtful.
Why so many writers on geometry wish to depart from Euclid's method of treating proportion is hard to be accounted for, yet how few of them have a correct notion of geometrical proportion! it may partly arise from their unwillingness to acknowledge that Euclid, nearly 2,130 years ago, had arrived at an ultimate stage of perfection;, and they would feign set up some contrivance