and "breadth") cannot be understood without a previous knowledge of the meaning of the word line. Compare the definition of a line with that of a square-"a four-sided figure which has its sides equal and its angles right angles;" and the difference is at once apparent in the former the definition is performed by means of words of an equal order of difficulty, in the latter by means of an analysis of the name into others simpler than itself. Mr. Wilson is plainly not contented with Euclid's definition; but he does not mend matters by giving the definition of a line thus :-" A line has position but neither breadth nor thickness." The fact is that the difficulty cannot be got over. There are many words of which we all know the meaning, but which we cannot define: such words as red or blue are instances; and so are a point, a line, a superficies, an angle. Mr. Wilson's account of a straight line is quite a curious instance of how nearly a writer may get to a distinction and yet miss it. He tells us :-"Def. 5. A straight line is such that any one part must, however placed, lie wholly on any other part if its extremities are made to fall on that other part." This is plainly a property of a straight line, or rather of two straight lines, not a definition; it is in fact the second geometrical axiom with its wording slightly altered. If a definition is to be attempted at all, it would be hard to produce a better than the old one-" A straight line is that which lies evenly between its extreme points;" but, of course, the word evenly as much requires definition as the word straight. Mr. Wilson adds to his definition the remark, "A stretched string suggests the notion of straightness, which is in fact incapable of elucidation" (p. 6). If he had said that "a string stretched between two points suggests the notion of a straight line, a term which is in fact incapable of definition," we should have entirely agreed with him; but as to the notion of a straight line (or straightness) being incapable of elucidation, that is quite another point. The notion is elucidated in Whewell's Philosophy of the Inductive Sciences,' vol. i. pp. 96, 97, as Mr. Wilson will find on reference. (3) We have hitherto dwelt only on axioms and definitions. We will now advert to a fault which pervades the whole book, though want of space compels us to consider it with regard to only a single instance. The fault is impatience of the restrictions of Elementary Geometry; the instance is the treatment of angles. Euclid, as is well known, does not recognize any angle which might not be an angle of a triangle. There seems good reason for this: the notion of an angle that the learner brings with him to the subject is that of a knee or corner; to make him thoroughly familiar with the notion and able to reason upon it correctly is enough in the first instance, and the more as the notion is fully sufficient for all the requirements of the first four books. Mr. Wilson thinks otherwise; and so to angles right, acute, and obtuse he adds the "straight angle" [one of two right angles] and the "reflex angle" [one greater than two right angles]. Now, in the first place, he uses no term to denote merely an angle less than a "straight angle;" consequently he always terms such an angle simply an angle; in other words, throughout the greater part of the book Euclid's restriction is tacitly reimposed, and this even in cases where, if the consideration of reflex angles is to be thought advisable at all, we should have expected to see them noticed. E. g. Cor. p. 49 [Euclid I. 32, Cor. 2] is expressly limited to any convex polygon; Ex. 2, p. 39, Ex. 5, p. 40 are not always true if the quadrilateral has a reentering angle. We raise no objection to this but want of consistency. Secondly, with regard to the "reflex angle," it is merely said to be greater than two right angles. Now take any Euclidic angle and denote it by 0; are we to understand by the corresponding reflex angle 2-0 or 2n +0? If the former, here is a restriction much more arbitrary than Euclid's; if the latter, we have a conception proposed to a beginner which is well known to be one presenting great difficulty to those who are beginning Trigonometry and have therefore made some progress in Mathematics. Mr. Wilson has an "axiom" which asserts that "an angle has one and only one bisector;" but upon these terms here is an angle A O B which has one, or one of two, or two bisectors, according as AOB means some one, or any one, or all of the series 2n +0. Thirdly, with regard to the "straight angle." A boy comes to his teacher with the notion that an angle is something resembling a corner of a square or triangle; and without delay his teacher introduces him to a notion of this kind:-Draw a straight line A B and take in it any point C; at C there are two straight angles, one on each side of the line-indeed as many as we please, for there is no occasion to restrict ourselves to a single plane. He comes with a notion that an angle is something between two straight lines that meet at a point, and he is taught without delay that he has only to draw a line an inch long and put a dot upon it to produce an infinite number of angles. This may be all very well for the teacher; but what is it for the boy? Not only, however, is it of more than doubtful expediency to put the notion of a straight angle before a beginner, but Mr. Wilson does it in a way which looks as if he had hardly considered all the consequences that ought to follow from its introduction. We will try to explain what we mean. It might perhaps be contended that the theorem "all straight angles are equal" needs distinct proof; though it is not easy to see what the theorem would add to the axiom, "two straight lines, which have two points in common, lie wholly in the same straight line." Now Mr. Wilson does not deem it necessary to give a formal treatment of this theorem ; but he does deem it necessary to do what is much more surprising. He gives Theor. I. "All right angles are equal to one another.' "If a straight line stands upon another straight line it makes the adjacent angles together equal to two right angles." Theor. III. "If the adjacent angles made by one straight line with two others are together equal to two right angles, these two straight lines are in one straight line." From Euclid's point of view this would be very proper; but from Mr. Wilson's point of view a right angle is only half a straight angle. Now let us try the effect of writing in these theorems, half a straight angle for right angle, and straight Theor. II. angle two right angles. We then see that the object of Theorem I. is to prove that the halves of straight angles are equal, i. e. it is a particular case of Axiom 5. Mr. Wilson might just as well have a theorem to prove that the halves of equal straight lines must be equal. The object of Theorem II. is to prove that if a straight line divides a straight angle into two parts, the two parts are together equal to a straight angle. With regard to Theorem III., we remark that a straight angle is that "made by a straight line with its continuation;" so that the point to be proved is that a straight line and its continuation are in one straight line. In short, from Mr. Wilson's point of view, these "Theorems" are not theorems at all; and he has failed to see it. X. Proceedings of Learned Societies. ROYAL SOCIETY. [Continued from vol. xlvi. p. 410.] April 3, 1873.-William Sharpey, M.D., Vice-President, in the Chair. was THE following communicatigy W. H. Gladstone, Ph.D., F.R.S., and Alfred Tribe, F.C.S. The galvanic battery which we are about to describe is founded on a reaction that we brought under the notice of the Royal Society last spring*. We then showed that if pieces of copper and silver in contact are immersed in a solution of nitrate of copper in the presence of oxygen, a decomposition of the salt ensues, with the formation of cuprous oxide on the silver and a corresponding solution of the copper, while a galvanic current passes through the liquid from copper to silver. We stated, moreover, that this was no isolated phenomenon, but only one of a large class of similar reactions. It seemed desirable to examine more fully the history and the capabilities of the electrical power thus produced. It was previously ascertained that the combination of the oxygen takes place only in the neighbourhood of the silver; and the following formulæ may serve to render the chemical change and transference more intelligible :— Before contact, after contact, mAg+0+Cu2NO,+Cu2NO,+nCu; mAg+Cu2O+Cu2NO,+Cu2NO,+(n−2) Cu. This action is evidently a continuous one until either the oxygen or the copper fails. Now the oxygen of the atmosphere is practically unlimited in * Proc. Roy. Soc., April 1872, vol. xx. p. 290. amount; but there is a difficulty in bringing any large quantity of it into contact at once with the silver and the dissolved salt. To facilitate this, we arrange that the silver plate should have a horizontal position just under the surface of the liquid in the cell; and, in fact, we convert it into a small silver tray full of crystals of the same metal which rise in projections to the very surface. The copper plate lies horizontally under it, separated, if need be, by a piece of muslin; and connexion is made by a wire as usual. The vertical part of the copper plate, from a little above the liquid to the bend, should be varnished; otherwise solution principally takes place there, which causes the horizontal part of the plate to drop off. Holes are made in the silver tray with the view of shortening the communication between the air-surface and the copper plate and of facilitating the movements of the salt in solution. The solution itself may be contained in a shallow trough or saucer, and the whole arrangement will be somewhat as in the sectional view here given :-- That dissolved oxygen is absolutely necessary for this chemical change has been already shown; but it was interesting to measure by a galvanometer the difference of the currents obtained by means of an ordinary, that is aërated, solution of copper nitrate, and one from which the air had been separated to the greatest possible extent. A Thomson's galvanometer was employed, which had a resistance of 2631-5 units at 18°.3 C. Two cells were prepared with vertical plates and alike in all respects, except that the one contained an ordinary 6 per cent. solution of copper nitrate, and the other a similar solution which had been deoxygenized by the means described in our former paper. Another experiment was made with a different pair of cells and an 11 per cent. solution. It was necessary to use the 1-99 shunt; and the following were the amounts of deflection: The contrast is evident. That the deoxygenized solution does give a deflection at all is due partly to the difficulty of excluding air, and partly, perhaps, at first to the oxygen condensed on the surface of the silver plate. The effect due to the water itself is inappreciable. From the nature of the reaction it might be expected that the current would gradually diminish on account of the using up of the dissolved oxygen in the neighbourhood of the silver; such a diminution always does take place, at least after the first few vibrations of the needle. It might be expected, too, that when the amount of action has run down considerably, the mere moving of the liquid so as to bring fresh parts of the solution against the silver would augment the currents. It does so. The same might be predicted from stirring up the crystals of silver in the tray so as to expose new surfaces. This also was found to be the case. And, again, it might be anticipated that if the wire were disconnected for a time so as to allow the oxygen to diffuse itself from other parts of the solution, and the connexion were made, the current would be found as strong, or nearly so, as before. That also is true in fact. A cell with the plates connected by a wire was placed under a bell-jar full of air over mercury. The mercury gradually rose inside, as might be expected from the absorption of the oxygen in the air. The necessity of oxygen and the avidity with which it is taken up are both illustrated by the following experiment:-Two cells with horizontal plates were prepared alike in every respect, except that the first was filled with a solution simply deprived of oxygen, the second with a solution through which a stream of carbonic-acid gas had been passed for some time. The first was placed in the air, the second in a vessel from which the air had been expelled by allowing carbonic-acid gas to flow into it for an hour or two. The deflections obtained were as follows, the 1-999 shunt being used and the temperature being 13°-7 C: |