is mounted on the supports S, S, passing through the helices M, M', and abuts against the system of levers, which is shown enlarged in fig. 2. A micrometer-screw N drives the bar into close contact with the stud P (in fig. 2). The brass work supporting N slides along the woodwork, and can be firmly clamped to suit any length of the bar under experiment; the supports for the bar and for the helices also slide, so that the current can be sent round the bar at various positions. It was found necessary to employ wedges within M, M', to keep the bar from the lateral motion which occurred when the helices were animated. The connexions with the commutator C and the battery are evident. Fig. 2 shows the lever-frame with the front plate off. D, D are the screws of the pillars which hold the frame together. The first lever A is connected to the second B by the cross piece C; to the second lever is attached the counterpoise weight w, replacing the spring originally used. To the long arm of B is fastened a silk thread, which passes round a grooved wheel to the spiral brass spring ss; the tension of the spring can be regulated by turning the milled screw g; by this means the mirror m can be adjusted to any convenient position. The vertical scale on which is cast the reflected ray r (fig. 1) is not shown. But the difficulties in the way of making determinations with this instrument were far more than I had even anticipated. The experience of the faults of one instrument led to the construction of another, which, though yielding better results, would at times perplex me by the most anomalous behaviour. On examination, these anomalies were traced to the hair spring which held the mirror in position; being of steel, it became magnetized by the current and was displaced accordingly. This was removed and a fine spiral copper spring substituted. The first results I obtained with this improved apparatus showed that the cobalt and iron behaved in a similar manner; each displaced the light in the direction of elongation 6.5 divisions of the scale on magnetization; but with the nickel no action whatever was observed. Increasing the battery power to twenty cells of Grove, a barely discernible movement took place in the direction of elongation. The very concordant results of two or three days' work seemed satisfactory; but unfortunately little reliance is to be placed in them, owing to the fact that I discovered at their close a serious source of error had been introduced through the inductive action on an iron pin fastening the levers, by which a disturbance was caused every time the current circulated round the helices. I have had lately the apparatus entirely remade, and hope to work at the subject again; meanwhile I am endeavouring to procure a purer specimen of cobalt*. * I thought it possible a lecture demonstration of the elongation of III. I have now to allude to the deportment of iron when raised to a high temperature. Mr. Faraday has shown, in the last page of his Experimental Researches,' that a moderate degree of heat does not alter the magnetic capacity of iron, but diminishes the magnetic force of nickel and increases that of cobalt. At a greater elevation of temperature it is well known that nickel first loses its ordinary magnetic character, then iron, and finally cobalt. But Mr. Faraday has also stated (§§ 2343-2347) that though the magnetism of iron, nickel, and cobalt, as ordinarily exhibited, disappears at a high temperature, yet a feeble magnetic state remains, however exalted the temperature may be. Some time ago several specimens of very tough fibrous iron were shown to me that had been obtained directly from cast iron by bringing high magnetic power to bear upon the latter metal when in a molten state. The process, which was patented, was thus popularly described by the 'Athenæum' for April 20, 1867"The experiment has been tried at one of the leading iron-works in Sheffield, and with complete success. The mode of operation, as roughly described, is to place a fixed electromagnet opposite an opening in the side of the furnace, to excite the magnet by means of a Smee's battery, so that the magnetism thus evoked may act upon the molten metal. The effect is surprising; the metal appears to bubble and boil, the melting is expedited, which economizes fuel; and the quality of iron is so much improved that for toughness and hardness it can hardly be equalled. It appears that some, if not all, of the impurities which remain after the ordinary process are eliminated by the use of magnetism." The scheme is so opposed to the ordinary views regarding the inertness of molten iron to magnetism, that any physicist must be naturally incredulous at this report, and would expect the patent to meet with the oblivion it has received. Nevertheless is it not possible, from Mr. Faraday's experiments, that some magnetic effect, not of translation but of direction, may be impressed on the molecules of molten iron? This is an inquiry to which I hope shortly to give more attention. iron wire by magnetization might be made in the following manner :Two soft iron wires were stretched on a monochord; round one wire, and nearly its whole length, a helix was coiled, leaving free space for the stretched wire to vibrate within. The wires were tuned to perfect unison; I then hoped that when the current animated the helix and so magnetized one wire, beats would be heard, owing to the diminished tension of the magnetized wire and, hence, unequal rate of vibration of the two. But I cannot say positively that such was the case. The experiment, however, was made under unfavourable circumstances, and needs repetition. With an optical representation of the combined vibration of the two wires a better effect might be anticipated. The resumption of magnetic power by iron after being raised to bright incandescence is thus described by Mr. Faraday :"The intensity of the force did not appear to increase until the temperature arrived near a certain point; and then, as the heat continued to diminish, the iron rapidly, but not instantaneously, acquired its high magnetic power, at which time it could not be kept from the magnet, but flew to it, bending the suspending wire and trembling as it were with magnetic energy as it adhered by one end to the core" (Exp. Res. § 2345). Approximately at that temperature wherein a cooling iron wire resumes its magnetic state, a profound change occurs in the physical condition of the metal*. A momentary dilatation of the iron takes place; its thermoelectric position is reversed; a sound is emitted; and a sudden reheating, or "after-glow," is seen to diffuse itself throughout the metal just before it ceases to be incandescent; and its electric and thermal resistance at this point appears to undergo a change, though this has yet to be strictly determined. IX. Notices respecting New Books. Elementary Geometry. Books I., II., III., containing the subjects of Euclid's first four Books: following the Syllabus of Geometry prepared by the Geometrical Association. By J. M. WILSON, M.A., late Fellow of St. John's College, Cambridge, and Mathematical Master of Rugby School. Third Edition. London: Macmillan and Co. 1873. 12mo. pp. 188. THE HE first two books only of the present work have been published before; they have been revised and in several respects improved. In the composition of the third book Mr. Wilson has been assisted by Mr. Moulton. Both these gentlemen, we believe, took part in drawing up the Syllabus of the Association for the Improvement of Geometrical Teaching, in conformity with which the present work has been written. It is needless to add that great care has been bestowed on the book. Its contents, so far as the text is concerned, are substantially those of Euclid's first four books. Many exercises and deductions are given in illustration of the text. In many cases the diagrams are so drawn as to show the data of the proposition. Playfair's treatment of parallel straight lines is given in place of the method used in former editions. Geometrical Loci are admitted into the text; and their use is illustrated by several examples. On the whole, the book, though by no means free from faults, is well executed; and a student who makes himself master of the text and is able to solve the problems and exercises will have a very good knowledge of the subject. *The series of phenomena associated with this molecular change are detailed in my preliminary note on the subject in last month's Philosophical Magazine. An author who takes in hand to write a book on the Elements of Geometry labours, however, under peculiar disadvantages. His work will be compared with a high standard; and mistakes which might have passed without notice in other subjects will be challenged in this. Besides, as there are strong-we do not say conclusive reasons against the substitution of a new work for the recognized text-book, it will be expected that the author shall produce something substantially different and distinctly better. A writer who attempts to produce a synthetical Compendium of the Elements of Geometry substantially different from and distinctly better than the first four books of Euclid, undertakes a task of no ordinary difficulty; and, if we may judge from the preface to the third edition, Mr. Wilson is more aware of this than he was a few years ago. To intersperse the text with remarks and to add deductions are nothing to the point-notes and deductions to Euclid are already in existence. Nor is it much more to the point to make minor changes. Very many of Mr. Wilson's changes fall under the latter head. For instance :-the treatment of Parallels by Playfair's method; the proofs of Euclid, I. 8, I. 24, III. 26, 27, 28, 29 (Wilson, pp. 28, 29; 27; 112–120), and others; the grouping of the propositions; the separation of Problems from Theorems &c. It may be questioned whether in this there is on the whole improvement; at all events it is improvement of no great importance. To set against this there is a great deal which seems to us quite the reverse of improvement; and though we cannot exhaust the subject, we will go into details on three points. (1) Mr. Wilson makes two classes of axioms-general and geometrical. Now, of course, it is true that such propositions as "if equals be added to equals the wholes are equal" are applicable to other than geometrical magnitudes; but then the application presupposes notions which the student of geometry may or may not possess. It may be true that when equal quantities of heat are added to equal quantities of heat the wholes are equal; but the student of geometry has nothing to do with this; he may not be able to understand what is meant by equal quantities of heat; and accordingly to him the axiom means that if equal angles or lines or areas are added to equal angles or lines or areas, the wholes are equal. The axiom, in fact, is to be construed with regard to geometrical matter. The like is true of the other so-called general axioms at least as they stand in Euclid; and consequently there is nothing gained by the division. When we go further, however, and see what Mr. Wilson includes in the list of General Axioms, we cannot help suspecting the existence of some confusion of thought. Here are two of the general axioms :-" (3) If equals be added to equals the sums are equal." "(7) If it is known that ‘If A is B then C is D,' it follows that 'If C is not D then A is not B.'" Now the former of these, as above remarked, relates to the matter of geometry; and where necessary Euclid quotes it amongst the premises of his reasoning, just as he would any other antecedent theorem. The latter is merely a logical rule, and relates to the form of the reasoning. The following remark will put in a clear light the impropriety of classing (3) and (7) together under the same head. If it is right to rank as seventh amongst the axioms of geometry the rule that "a conditional proposition may be converted by negation," then the rule that "a universal affirmative proposition can only be converted by limitation" ought to be admitted into the list; and in fact it would be hard to say what rule of deductive logic could well be omitted*. It is one thing to make here and there a remark on the form of the reasoning employed in proving this or that proposition, and another to erect a rule of logic into an axiom of geometry: the latter confuses things that differ; the propriety of making the former is a matter of opinion. Considering, however, that elementary geometry should be taught to rather young boys, we are strongly inclined to think it best to keep the rules of logic out of sight, or at least that they should be noticed, if at all, in the oral instruction of the teacher, and should not form a substantial part of the text-book. It must be remembered that when we are concerned with given matter we reason correctly, or can be shown to have reasoned incorrectly without explicit reference to the rules of logic. book of geometry, therefore, logical rules are extraneous matter, and as such are nearly certain to be treated inadequately, if admitted at all. Mr. Wilson illustrates this remark by devoting a page and a half to the conversion of propositions (pp. 3, 4); but he notices only conditional propositions, and his reason for doing so is that "if A is B then Ĉ will be D is the type of a theorem," although his own enunciations often take a different form (e. g. Theorems I., VI., X., &c.). In a Mr. Wilson has four enunciations under the head of Geometrical Axioms. Two of them run thus:- 66 :- (2) Two straight lines which have two points in common lie wholly in the same straight line. (3) A finite straight line has one and only one point of bisection" (p. 9). The former of these is a true axiom; the latter admits of proof, and is therefore a theorem assumed to be true merely as a matter of convenience. Suppose Euclid had assumed the first eight and had begun his reasoning with the ninth proposition; his subsequent reasoning would have been unaffected; but surely it would have been an abuse of language if he had classed together as axioms "Two straight lines cannot inclose a space," and "The angles at the base of an isosceles triangle are equal." Yet this is just the sort of thing which Mr. Wilson has done. (2) On coming to the definitions, it may be well to notice that in definition we have to do with nothing but the meaning of words. It follows that there must be some words which are incapable of definition, and whose meanings must be arrived at by a direct appeal to the intuitions of sense-to experience, in fact. There may be good reasons for saying that a line is "length without breadth; but it is hardly a definition, as the words contained in it ("length" * Mr. Wilson's sixth axiom also is purely logical. |