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implying or imagining anything not rigidly warranted by the terms of our definition?
We will not press this distinction too closely, because we are aware the author is referring rather to the best practical mode of illustration to a learner.
However, the conception being thus formed, certain self-evident properties are seen to belong to it. Some of these are stated, which are essentially numerical, and therefore follow rigidly from the numerical definition, without regard to any other conception; but he says the fifth definition of Euclid, and the demonstrations founded upon it, are usually considered by learners as obscure and confused. He conceives “ that this impression arises, in part, at least, from the attempt, which in this case was made by Euclid himself, to reduce the subject to definition alone.”
The author afterwards goes on to observe :
“ If proportion had been separately defined, so as to bring the conception of it before the mind, I conceive that the assertion of this fifth definition would be assented to without difficulty, as an axiom. But then, what definition of proportion shall we take ?”
In order to remove the difficulty attending the choice of a definition, the author considers the introduction of a few examples will suffice, or a reference to the definitions of ratio and analogy.
“ If we take any other course, we either run into the apparent confusion and complexity which, as has been stated, arises from mixing, in the fifth definition, the character of definition and of axiom, or, on the other, taking as our definition, the one first mentioned—and running from it to cases to which it does not apply, we transform our mathematics from a praxis of logic, into an example of the most loose and inconsequent reasoning possible.”
Mr. Powell puts his general statement of the question thus:
“The doctrine of ratio and proportion, then, 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. Some of the best modern writers on geometry (as Legendre), omit it altogether in their elementary systems; and most teachers in this country, though, in general, they follow Euclid, yet pass over the 5th book; and adopting the doctrine of proportionals, from algebra, proceed to apply it to the theorems of the 6th book. Euclid, however, for some reason, thought it necessary to proceed otherwise. He establishes, by a totally different method, some of the same properties of proportionals as the algebraic writers do; but where he stops short (not even proceeding to certain of the simplest and most universally important properties), they, on the other hand, continue the subject, or make those properties fundamental. Of Euclid's design and principles of investigation, various opinions have been held among the moderns; and some of those who most profess to be his admirers and followers, have made attempts, as they conceive, to improve upon his method; and have devised various plans for treating this portion of the subject, in order to avoid what they consider the unnecessary abstruseness and prolixity of the 5th book.
“ It might, perhaps, in the first instance, be imagined that the method adopted by Euclid in his 5th book, was the first and imperfect attempt of
science, as yet in its infancy, to give a general investigation of the doctrine of proportion, which, however prolix and cumbrous, ought yet, on that ground, to claim our respect and admiration; but that, the advance of modern science having furnished us with the more comprehensive method of Algebra, we may properly discard the older and more difficult process, as doubtless Euclid would have done, bad the better method been known to him.
“ But what is the fact? We have only to look into his 7th book, and we there find the whole doctrine, as applied to the case of numbers, demonstrated quite independently of anything in the 5th book, on principles, though presented in a different form, yet in substance almost identically the same as those used in modern algebraical treatises: and it would have required no more than a very obvious extension of the method, so far to generalize it, as to make it applicable to the geometrical figures considered in the 6th book, in the same way as the modern writers have done."
Now in many relations established in geometry, as subsisting between straight lines forming parts of certain geometrical figures, as, for example, the side of a square and its diagonal, we find, that if we conceive the lengths of the lines measured by any numerical scale, and express one line by an exact whole number, we shall only be able to express the other by a square root, or some number which has a fractional remainder ; however small this remainder be, yet it makes the relation not mathematically exact, and two quantities so circumstanced, are called “incommensurables.” Of all this, however, nothing appears in Euclid's mode of treating the subject, and it arises wholly out of the adoption of the arithmetical measures.
Thus, 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 metrical relations, however, are absolutely general, and do not refer to any such distinction.
Euclid, in his earlier books, when treating of such properties as those alluded to, does not introduce, either explicitly or by implication, the term or the idea of incommensurability: neither is it introduced in the 5th or 6th books. After having in the 7th treated of the properties of arithmetical proportionals, where the terms are all supposed expressed in exact finite numbers, and having in the 8th and 9th books continued this subject to various properties of numbers, he comes in the 10th book, for the first time, to introduce the notion of a distinction between commensurables and incommensurables, in geometrical magnitudes expressed by numerical measures.
In the 11th and 12th books again, he drops al reference to this distinction, recurring to the principles of the 5th book : but resumes the consideration of it in the 13th, and applies it to various properties.
In Euclid's mode of treating proportionals, we find the same total and systematic exclusion of all reference to numerical measures. Yet many mathematicians have contended that the idea of number enters essentially into our conception of ratio and proportion, and must be supposed, in order to a right understanding of Euclid's method; which
they contend was adopted for the express purpose of including the cases both of commensurables and incommensurables. The quantities considered by Euclid in this book, it is true, may be of either kind; but he does not specifically regard them with any reference to this distinction. On the contrary, without pointing at all to any such consideration, it seems to be his object to regard quantity in a far more general point of view. “In a word,” says the Savilian professor, “I conceive it clearly evident, that Euclid's specific object in the 5th book was, not to include incommensurables, but to EXCLUDE all reference to the idea of NUMBER.”
But how is this done? The author's opinion is, that the nature of the quantities here discussed, must be understood according to the most general and abstract notion implied by the term : in his own words ;
“Now our idea of QUANTITY IN GENERAL is very easily understood if we simply consider it in the same light as other general ideas, and only bear in mind and apply to this case the same process of abstraction by which we arrive at those ideas. We form ideas of so much in length, so much in area, so much in solidity, so much in duration, so much in number, so much in velocity, &c. From all these particular ideas we abstract what belongs to the particular nature of length, of surface, of capacity, of duration, of number, &c., and thus form our abstract idea of 'so much,'or of quantity in general, a comprehensive generic term, including under it all the particular species of quantity.
“ In the first part of geometry, (i.e, in the first four books of Euclid,) we consider quantity of extension, and its properties in regard to linear and superficial space as referring to certain geometrical figures. And it is essential to observe, that the estimate we form of the magnitudes, consists in simply considering them as so much length or so much surface, without assuming any particular scale or standard of measurement: we compare such lines and surfaces together, and establish relations of equality between them as constituted under certain given conditions. And these form certain relations which are strictly and properly geometrical as not being determined with reference to any other assumed standard of measurement.
“We proceed in like manner, in the 5th book, to investigate certain more varied and comprehensive relations which subsist between quantities considered all along in the most general and abstract point of view, and which give the properties of proportionals. Nothing is introduced which refers to any of those particular characteristics by which the different species of quantity are distinguished: the conclusions, therefore, are absolutely general, and may subsequently be applied indifferently to any of the particular species. This is accordingly done in the 6th book, with regard to linear and superficial extension. And in the 11th and 12th to solid extension. The very essence of the subject treated in the 7th and other books, requires a different and more restricted mode of investigation, as applying to properties peculiar to one species of quantity; viz, number: and to certain highly curious relations which subsist when number is employed to measure extension.”
But the discussion of these properties, which arise wholly out of the numerical system of measurement adopted, is studiously kept entirely distinct from the purely geometrical investigations. Those of Euclid's books, in which the numerical principle is introduced, may be taken as an entirely distinct arithmetical treatise : and there can be little doubt, originally did actually form a separate work. There is never the least intermixture of the methods, nor is anything in the geometrical books in the smallest degree dependent upon these others. Here all such ideas are totally excluded.
In elementary algebra mere addition and subtraction is not limited to any particular species of quantity; nor does it of necessity suppose the quantities concerned to be numbers. But, in the common systems, as soon as we come to introduce the idea of multiplication, there, by the very definition, we of necessity limit our consideration to quantities, one at least of which is a number, and usually both. This refers to algebraical proportionals and to all their applications, that is, to the whole superstructure of modern mathematical science. The doctrine of numerical proportionals, as established in common algebra, comprises corresponding particular cases of all the properties delivered by Euclid of quantities in general. The investigation, of course, gains in brevity what it wants in generality. But it extends to a greater number of properties: and is principally distinguished in that it embraces the equality of the products of the extremes and means of a proportion. This, with others dependent on it, since they involve the idea of multiplication, of course have no place in Euclid's system. The analogous property of rectangles (established in the sixth book) does not supply the deficiency, because it applies only to one species of quantity, viz.-superficial area, and not to anything equivalent or analogous to a product between abstract quantities.
The whole subject has a direct bearing on the question to which we at first referred, as to the essential distinction between the nature of geometrical and algebraical investigation. And the question is well illustrated by a passage in the mathematical correspondence of Dr. R. Simson (given in his life by Dr. Traill); his friend and pupil Mr. Scott had remarked, in a letter to him,
“The fifth book is not peculiar to geometry, but is equally applicable to quantity as it is to magnitude: nor does the doctrine of proportion receive any evidence from its being expounded by lines. The symbols of algebra would, I think, be of use here, &c."-(p. 115.)
To this Dr. Simson replies,
As to the 5th book, I do not see that the demonstrations in it can receive any help from algebra : and the straight lines made use of in it, make the demonstrations clearer and easier than they would be without them. I do not understand your meaning, when you say any analogy may be called algebraical, as well as geometrical. The expressing lines by a single letter does not make analogy or anything else algebraical, any more than when they are expressed by the two letters at their ends, nor do I think any thing can be called algebraical, when no operation peculiar to algebra is made use of. I should be glad to see an algebraical demonstration of some proposition in the 5th book, eq. the 17th, &c.”—(p. 121.)
The author, in quoting this passage, observes,
“The question, What operations are peculiar to algebra in reference to proportionals 2 still remains to be answered. “Such as depend on the numerical measure of ratio' is my reply; and I am convinced would have been Dr. Simson's.”—(note p. 60.)
He remarks also,
“ Dr. Trail (in continuation of a passage before quoted) appears to me to come very near upon the truth, without actually explaining it, when be says;—an analysis or demonstration in which many combinations of ratio are employed, may generally, indeed, be much shortened by admitting multiplication, division, or other operations peculiar to arithmetic or algebra ; but the investigation or demonstration then becomes truly algebraical.”
Upon the whole we must observe, that to do justice to this subject, considered in an abstract point of view, it would be necessary to enter much more deeply into it than would be consistent with our limits. We here content ourselves with a mere attempt to lay before our readers the views of the writers we have referred to.
In a practical point of view, the question is a very important one to all who are interested, whether in preserving the exactness and logical utility of mathematics, as a study for the improvement of the reasoning powers, or in simplifying its acquisition as preparatory to its practical applications in all branches of science. For the latter purpose it is obvious to us, that by far the easiest method is for the student to acquire merely a few of the most fundamental geometrical truths from the first book of Euclid : then apply algebra (assuming the approximate process for incommensurables), and so dispense, in fact, with all the remaining geometrical demonstrations, and go at once to trigonometry and co-ordinates, which will be an ample preparation for all elementary applications in mechanics, astronomy, &c. To the leisurely academical student of course a different and more strictly logical process is highly important, if not alsolutely essential, for deriving any of the benefits of a praxis of reasoning
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