Doctrine of proportion (mathematics)

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Discussions and debates over the doctrine of proportion or doctrine of proportionality, as contained in Euclid's Elements Book V, has had an interesting and varied history. Euclid's circuitous presentation of the theory of proportionality in Book V was presented independently from arithmetic proofs (discussed elsewhere in his treatise) due to commensurability issues. As methods of algebra were devised and improved upon, and the definitions and classifications of number and magnitude evolved, discussions continued. Some of that history is contained in this article.

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ordered chronologically
  • The doctrine of Proportion, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it.
  • Those persons who desire to see the doctrine of Proportion treated according to a general method which is plainer than Euclid's, and equally accurate, may consult the geometry of Playfair, [Alexander] Ingram, Leslie, Cresswell, and J. R. Young. Hutton, and other recent writers have adopted the algebraical method in their elements of geometry. Proportion is not properly a geometrical subject.
    • Francis Nichols, Elements of Geometry being Chiefly a Selection from Playfair's Geometry, with Additions and Improvements (1829)
  • On an attentive examination of the methods adopted by modern elementary writers, in laying down the first principles of ratios and proportion, and especially in commenting upon Euclid, I long since experienced a conviction of the extremely unsatisfactory nature of most of their views; and this chiefly, as appearing to me to involve inadequate ideas of Euclid's real principle in treating of proportionals in his 5th book, and of the nature of the quantities which form the subject of investigation.
    • Rev. Baden Powell, On the Theory of Ratio and Proportion (1836)
  • The doctrine of ratios and proportion is introduced by Euclid as a part of his system of geometry; and the student seldom fails to remark, that in the treatises on algebra, the same subject is presented under a considerably different form; though he is usually quite unable to determine wherein the essential difference consists; and would probably find but few teachers who could precisely point out the distinction to him.
    • Rev. Baden Powell, On the Theory of Ratio and Proportion (1836)
  • Various relations being established in geometry between lines constituted under given conditions, as parts of geometrical figures, if we choose to adopt the idea of expressing these lines by numerical measures, we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their geometrical relations however are absolutely general, and do not refer to any such distinction.
    • Rev. Baden Powell, On the Theory of Ratio and Proportion (1836)
  • Much of the confusion of ideas which has arisen on these subjects has been occasioned by not observing that when we say "two lines are incommensurable," the phrase is in fact elliptical, and we ought always to consider as understood, if not expressed, that "two lines, if referred to numbers, "are incommensurable." The deficiency of exact comparison in such cases is not in the geometrical relation of the quantities, but in the powers and capacities of our numerical system to express them. These observations may be necessary to enable us to appreciate better the opinions which have since been prevalent among mathematicians; more especially as regards the doctrine of proportionals.
  • The author's object in this tract is to defend Euclid from the charges of inconsistency which have been brought against him by Sir John Leslie and others, in consequence of the introduction of the doctrine of ratio and proportion as part of his system of geometry. Most of the best writers of geometry (as Legendre) omit this part in their elementary systems, and most teachers in this country pass over the 5th book, and adopting the doctrine of proportionals from algebra, proceed to apply it to the theorems of the 6th book. Professor Powell treats the subject in detail, stating the objections which have been urged against Euclid, and presenting answers to these objections. He begins with a general statement of the question; he then proceeds to the consideration of Euclid's method, or the doctrine of commensurables and incommensurables. He shews that Euclid, in his earlier books, does not even imply the idea of incommensurability. Neither is this introduced in the 5th and 6th books, and it is not till we arrive at the 10th that this edition in geometrical magnitudes, expressed by numerical measures, is broached. In the 11th and 12th books all reference to this distinction is dropped, recurrence being made to the principles of the 5th book. It is again, however, resumed in the 13th book, and is applied to various properties. The author observes, "that much of the confusion of ideas which has arisen on these subjects, has been occasioned by not observing that when we say two lines are incommensurable, the phrase is, in fact, elliptical, and we ought always to consider as understood, if not expressed, that two lines if referred to numbers are incommensurable. The deficiency of exact comparison in such cases is not in the geometrical relation of the quantities, but in the powers and capabilities of our numerical system to express them. Mr. Powell then proceeds to discuss the views of the earlier geometers and of later mathematicians. He points out the misapprehension under which they all labour, from the common mistake of considering that definitions describe the thing defined instead of fixing the meaning of terms. He shews that the mistake must be corrected before reasoning can be admitted on the subject. The nature of abstract quantity is next ably treated of, and the paper concluded in the same philosophic spirit which pervades it throughout.
    • Robert Dundas Thomson & Thomas Thomson Records of General Science Vol.4 (1836) p.67-68 in their review of book: On the Theory of Ratio and Proportion, as treated by Euclid, including an inquiry into the nature of quantity by Rev. Baden Powell
  • The object of the edition now offered to the public, is not so much to give to the writings of Euclid the form which they originally had, as that which may at present render them most useful. One of the alterations made with this view, respects the Doctrine of Proportion, the method of treating which, as it is laid down in the fifth of Euclid, has great advantages, accompanied with considerable defects; of which, however, it must be observed, that the advantages are essential, and the defects only accidental. To explain the nature of the former, requires a more minute examination than is suited to this place, and must, therefore, be reserved for the Notes; but, in the mean time, it may be remarked, that no definition, except that of Euclid, has ever been given, from which the properties of proportionals can be deduced by reasonings, which, at the same time that they are perfectly rigorous, are also simple and direct. As to the defects, the prolixness and obscurity, that have so often been complained of in the fifth Book, they seem to arise chiefly from the nature of the language employed, which being no other than that of ordinary discourse, cannot express, without much tediousness and circumlocution, the relations of mathematical quantities, when taken in their utmost generality, and when no assistance can be received from diagrams. As it is plain, that the concise language of Algebra is directly calculated to remedy this inconvenience, I have endeavoured to introduce it here, in a very simple form however, and without changing the nature of the reasoning, or departing in any thing from the rigour of geometrical demonstration. By this means, the steps of the reasoning which were before far separated, are brought near to one another, and the force of the whole is so clearly and directly perceived, that I am persuaded no more difficulty will be found in understanding the propositions of the fifth Book, than those of any other of the Elements.
  • 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, [Thomas] 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."
  • In matters connected with geometry, nothing is to be taken upon trust: mere opinion, unsupported by reasonings which elevate it into proof, must be regarded, in this subject, as of but little worth.
  • It would, of course, be a easier task for me to transfer Euclid's fifth book into these pages. I could find little to remark upon in it, as the ancient Geometer has displayed so much sagacity and penetration in this, the most elaborate of all his writings, that he has left to moderns little or no room for improvement: it must be studied just as it is (in Simson's restoration), or else be superseded in instruction by a treatise of equal generality, but greater simplicity. You will understand, therefore, that I do not displace fifth Euclid's book because of its imperfections, or because of its inadequacy to completely its objects; but solely because of its great difficulty to a beginner.
    • John Radford Young, "Proportion. A Treatise..." in The mathematical sciences (1860)
  • The subject of Euclid's fifth book is PROPORTION—universal proportion; that is, not numerical proportion merely, but proportion in reference to all magnitudes and quantities whatever, whether numbers, lines, surfaces, solids, or concrete quantities of any kind. With proportion in numbers you are already familiar:—this will be a help. I hope, too, by this time you are also somewhat acquainted with proportion in Algebra: this will be a greater help; for proportion in Geometry really accomplishes no more for things in general than the same doctrine in arithmetic and algebra accomplishes for what the notation of those sciences specially represents; and if this kind of proportion would do for geometry, the fifth book of Euclid would become a very easy matter indeed.
    • John Radford Young, "Proportion. A Treatise..." in The mathematical sciences (1860)
  • The obstacle... is that geometrical magnitudes when compared together are in many cases found to be incommensurable;—that is to say, two such magnitudes may be quite incapable of a common measurement—they may be of a nature not to admit of being both measured by one and the same unit of measurement, however minute the measuring unit be taken, and consequently, both cannot be represented by numbers. I have already adverted to an instance of this kind... in the side and diagonal of a square, and to another in the diameter and circumference of a circle. ...That they are so, could not have been found out by such practical or experimental tests as those here adverted to for illustration: they are proved to be so by geometrical reasoning.
    • John Radford Young, "Proportion. A Treatise..." in The mathematical sciences (1860)
  • It may be as well to caution you here that you must not speak of a line or quantity, by itself, as being incommensurable; this would be absurd. The diagonal of a square is not itself incommensurable, since it has, of course, its third part, fourth part, hundredth part, &c. and is therefore measurable by each of those parts; but as none of them will also measure the side, the two, considered together, are incommensurable: there exists no measure common to both. In the same way in reference to the circle—the circumference itself is not incommensurable any more than the diameter; for each has its fourth part, sixth part, &c. but it is incommensurable with its diameter: no length whatever can measure both.
    • John Radford Young, "Proportion. A Treatise..." in The mathematical sciences (1860)
  • Now although Euclid makes no mention of incommensurable quantities in his fifth book, he was well aware of their existence; and therefore, to render his theorems on proportion general, he had to take care that this class of quantities should be comprehended in his reasonings. But proportion limited to numbers, or to the symbols for numbers, would necessarily exclude incommensurables; he therefore had to proceed quite independently of arithmetic, and to secure to his propositions such a universality that each theorem should rigorously apply, whether the quantities or magnitudes spoken of be measurable, or beyond the powers of numerical representation. He has executed his difficult task with consummate ability; for as Dr. Barrow remarks, "there is nothing, in the whole body of the Elements, of a more subtile invention,—nothing more solidly established, and more accurately handled, than the doctrine of proportionals."
    • John Radford Young, "Proportion. A Treatise..." in The mathematical sciences (1860)
  • I have resolved to replace the fifth book by the following treatise. I cannot promise that you will find the study of it easy; but it will certainly be much less difficult than the corresponding portion of Euclid's work; and yon will enter upon it with considerable advantage, if you postpone the attempt—as I here recommend—till you have read, as far, at least, as page 220 of the Algebra. It will, indeed, facilitate your progress, and agreeably diversify your mathematical labours, if you commence the elementary algebra upon closing the fourth book of Euclid, and read the PRINCIPLES through, before you begin the following treatise.

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