a very elegant explanation of the uses of the different parts of the retina, examined in its radial section after the manner of M. H. Müller, and of those of the choroid coat,-structures which are without meaning on the ordinary theory of vision. University, New York, Yours truly, J. W. DRAPER. XXIV. Remarks on the Theory of Parallels. THE appearance of several short articles on the doctrine of parallels in recent Numbers of the Philosophical Magazine, recalls my attention to a subject which I have long and attentively considered. While agreeing in general in the views of Drs. Whewell, Whately, Lardner, and Messrs. Mill and Hennessy, on the nature of geometrical definition, that we do not conclude from mere names, but that there lies at the basis of the reasoning a something assumed which is not susceptible of proof in its simplest form, call it axiom or conception, or what you will, it appears to me that this something, to be of any avail, must be self-evident, universally and invariably commanding assent, and incapable of being disputed, however its nature may be expressed in different verbal propositions. The question for consideration is, do the definitions of straight line, circle, or parallel line, contain distinct and independent assumptions? and if so, are these all seen to be equally necessary, or is there but one and the same assumption inevitable in the present constitution of the human mind, and embracing and implying these various modes of expression, lying at the ground of our deductions? Is not the conception of space common to all men, and does it not contain in it all that is necessary for a science of geometry? If so, there should be nothing disputable or open to cavil in the minds of competent thinkers. An axiom, to be valid, must express some ultimate truth involved in this conception, and should be incapable of further proof, otherwise it is superfluous. If, after repeated attempts to analyse a proposition already so far elementary that we are intuitively confident of its truth, we find ourselves driven to modes of explanation which are in themselves more difficult than the deductive processes to which geometry is applied, such, for instance, as the use of limits or arguments from continuity or inconceivableness, we may be pretty sure that we have passed from the region of deduction and proof into that of fundamental conception. There is an endless variety of such * Communicated by the Author. proofs, as Legendre's (by limits), Colonel Thompson's, Mr. Exley's, and lastly Mr. Stevelly's, the first being probably the best in respect of simplicity. No one denies the adequacy of the proof, but there is always some break or chasm in the argument which we must bridge over by an appeal to the ultimate conception of space, and which may therefore as well be done at once without all this din and preparation. It may, after all, be doubted whether Euclid's twelfth axiom is not incomparably the best of all that have been proposed, when the scope and requirements of elementary geometry are duly considered. I cannot agree with Mr. Hennessy, that the doctrine of parallel lines would be complete without it; for although by his definition parallel lines would never meet, yet it is necessary for many of the demonstrations of problems by construction, to know that there are no other but parallel lines which do not meet, to say nothing of the proof of the value of the three internal angles of a triangle. We want an axiom which includes the essential conception of angle as well as parallel straight line, and there is no advantage in separating them. In one sense parallelism is only a particular case of lines which meet it is the limit of angularity, so that parallel and right angle are only two particular extreme terms when the angle equals 0° and 90°. By parallel lines, therefore, we mean not only two straight lines which never meet, but lines one of which cannot be cut by a straight line without its cutting the other also. Let us imagine that Euclid was endeavouring to define an angle,-how that it was made by the intersection of two lines which continually diverged, and how this divergence necessarily implies unsymmetrical conditions on the two sides of any other line cutting them transversely. Two straight lines can be constructed so as to cut a third, both at right angles, making the angles on each side of the cutting line respectively equal to two right angles. These lines are parallel, and never meet. Any change of position of these lines by which the two ends on one side were made to approach, would be accompanied by a departure of the ends on the other side, by an enlargement of the angles on the latter, and a contraction on the former side. What more self-evident or obvious, than that as the ends have been brought nearer to one another, the lines will meet at some finite distance if continued? And why should not Euclid make choice of this as the most useful and convenient form of the assumption, seeing that if by parallel you mean not meeting, you cannot prove that the angles on one side the cutting line are exactly two right angles, but only that if they make these angles such, they are amongst the number of parallel lines? Mr. Hennessy's definition, that parallels are such that if they meet a third right line the two interior angles on the same side will be equal to two right angles, is open to the objection, that it is not proved, or certainly known, that there are any lines which will always exhibit this property, however cut, and therefore that it contains covertly what Euclid openly assumes; and I do not think that because Euclid's definition of a square is allowed to pass unquestioned, it is therefore superfluous to investigate the grounds of the parallel theory. We all know that what settles the one settles the other; it is not necessary to revive the strife every time the question recurs. Many instances could be adduced where Euclid has thought fit to be scrupulous in one instance, and to neglect such over-nicety in others equally requiring rigidity of proof; but this is only saying, that, with all his won derful discernment and care, he is not perfect. To return, however, to the main question. Some fourteen or fifteen years ago, I published two small pamphlets on the subject of the present discussion, in which I attempted to derive the doctrine of paral lels, as well as that of Proportion, from the consideration of similar triangles, which I regarded as identical, or as undistinguishable by the understanding. Geometry has not to do with empirical or assigned space magnitude. A figure once given is the same whether the space it fills is large or small. We do not by a specific triangle mean one which has one spatial dimension rather than another; but just as a limited space is a part of infinite space, a specific form may have regard to any spatial dimension, and thus there is always an infinite number of similar figures of every kind. The understanding recognizes these only as identical. Form has regard only to angularity, and the relation of the bounding sides to one another measured by an arbitrary unit. I regarded parallel lines to be the corresponding boundaries of these identical figures, and thus deduced all the leading facts of geometry by attempting to show that the difficulties owe their origin to a confusion of empirical and merely intelligible conditions. The same thing might have been accomplished differently by insisting on the absolute identity of parallel lines, as regarded by the understanding, without reference to their passing through different points in space which are likewise undistinguishable by an act of pure thought*. But further than this, the geometric conception of a circle really assumes all that is supposed in the rotation of a line round any point in itself; and as this may take place indifferently at all points in space, for anything we can conceive to the contrary, there can be no objection to the introduction of such a postulate. As every bounded figure can be conceived to be produced by the rotation of one of its sides into The full development of this line of argument is here omitted as requiring too much letter-press, all the positions of the other sides till it returns into itself, the external angles of every right-lined figure must always be equal to four right angles, and therefore the interior angles of a triangle to two. My argument then would assume this form: that if there be any result which the pure conception of space leaves unexplained, we have not seized that conception aright, or there is no basis for a complete science. If we have, no arbitrary result can be admitted which is not explicable, or we have the principle of the sufficient reason to go upon. If a change of position in space can alter the truth of a proposition respecting the relation of lines and angles, or the parts of a specific form, or I might rather say, the evidence for the truth of such a proposition, then is there no complete universal conception of space. I defend Euclid's twelfth axiom, therefore, as the best, most simple and natural mode of presenting the case in an elementary and practical treatment of it, that is, if we are to omit the deduction from the ultimate conception of straight line and angle. In other words, while I would contend for a strictly philosophical analysis of the question, which I believe to be possible, I do not think that any construction of the proof by means of lines or schemes, or any mental intuition of this sort, will make the axiom more self-evident. This is all that can be meant when we deny that a geometrical proof has been given; but the necessary truth of the axiom can be shown from our conception of space by reasoning in which lines and schemes give no assistance, call the proof metaphysical, or what you will. All we require is, to agree upon the best way of presenting the subject in a form which shall not be discreditable to the present advanced state of philosophy. Clifton, January 1857. XXV. On the Polarization of Diffracted Light. By Professor G. G. STOKES, M.A., Sec. R.S. &c.* ΟΝ of M. Holtzmann on this subject, I am induced to make the following remarks. In the more common phænomena of diffraction, in which the angle of diffraction is but small, we know that the character of the diffracting edge, and the nature of the body by which the light is obstructed, are matters of indifference. This was made the object of special experimental investigation by Fresnel; and its truth is further confirmed by the wonderful accordance which he found between the results of the most careful measurements * Communicated by the Author. |