Oliver Byrne's Doctrine of Proportion
Prefaces are seldom read, because they are oftener written to magnify the merits of the books they introduce, or the doctrines which they advocate; to point out the defects, or run down the merits of others; to gratify the pride of an author, in showing the public how much he knows of the subject upon which he writes; to forward the views of publishers, in creating a sale for their publications, rather than to prepare the mind for the reception of the subject about to be introduced, by giving a general view of the matter treated of, and the manner in which it is treated, or to set the reader on his guard, that he may steer clear of the difficulties and obstructions which may retard him in his progress, as all proemial matter ought to do. For these latter purposes alone this preface is written. Therefore all who intend to study these pages would do well to read attentively the following directions and observations; for the subject upon which they are written is considered one really difficult.
Why it should be considered so, will readily be conceived when such men as Legendre, Leslie, Keith, Bonnycastle, Austin, Brewster, Young, and in fact every one who has attempted to treat the doctrine of Geometrical Proportion on any plan differing from Euclid's, have committed errors, overlooked mistakes, retrenched the generality of Euclid's reasonings, fallen into logical absurdities, or confined the general application of a subject which pervades a whole course of mathematics; while there is not one mistake, oversight, or logical objection in the whole of Euclid's Fifth Book. "In fact Euclid's Fifth Book is a master piece of human reasoning."
Censure on the works of others should be avoided as much as possible, because it shows the want of knowledge; those who know least, censure most: to correct a copy is easier than to produce an original; for men acquire criticism before ability, and it is mostly from those who possess no judgment that the most sweeping judgment comes.
But we wish to impress on the reader not to consider for a moment that, while we thus point out the defects of others, any wish is entertained to detract from the well-earned reputation of such men as are here mentioned; for we should rather praise them for their worth, and admire and adopt their beauties, than condemn them for a few faults: any attempt on our part to detract from the merits of such men would be presumptuous, arrogant, and unjust. We point out 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 inconsistency - that 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 mistakes 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 to show that their powers of improvement were not exhausted.
While we uphold the opinion that it is indispensably necessary to treat the doctrine of proportion geometrically, and so highly esteem Euclid's method, let it not be supposed that we condemn the arithmetical or algebraical system of treatment, for we are convinced that the doctrine of proportion should be fully and fairly developed in every complete elements of arithmetic, of algebra, and of geometry; because it is a subject which belongs to one as much as to either of the other two; and on these three elementary sciences a whole course of mathematics is founded, through the entire of which the doctrine of proportion is mingled, and it matters not whether the structure be raised upon numbers, symbols, or lines, &c, the well being of the combination must wholly depend on the capacity and firmness of the foundations.
For if we question the propriety of letting x, y, z, etc, stand for magnitudes, regardless whether they be incommensurable or not, we must question the whole of our beautiful system of analysis, which is just as certain in its results as plane geometry, and much more extensive in its application.
Nor do we defend the system of showing how inadequate numbers or letters are to express magnitudes, or their relations to one another; for it is admirable to see how the parallel propositions agree, although managed by means essentially different. To thoroughly understand the doctrine of proportion, it should first be acquired arithmetically, then algebraically, and both these methods made subservient to the right understanding the development of geometrical proportion.
The introduction of symbols into works on geometry is every day becoming more general; and as by their assistance the demonstrations can be more perspicuously arranged, and the train of arguments exhibited more systematic and concise, it would therefore be unnecessary to offer any remark on their adoption in the present performance; besides, symbols, while recording each stage of the proposition faithfully, relieve the mind to contemplate the absolute quantities. But the symbols used in geometry must be considered not only as appropriate emblems of the quantities themselves, but also as expressive; and not as any measures or numerical values of them.
JOC/EFR April 2016
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