dam alicujus exercitati in Græca lingua. Quid si occurrerit quod placet? Σεισάχθεια ἀριθμητικη. Nosti enim, χρεων aπоκотas sic dici. Inest vocabulo et emphasis et proprietas et similitudinis gratia, quia me Hercule novas tabulas introducis, et uno ictu liberas computatores debitis multiplicandi et dividendi inextricabilibus. Sed facile est exercitato, copiam afferre similium compositorum, ut ex iis aptius aliquod eligatur. Mihi jam plura non occurrunt." He then refers to the Tabula Tetragonica [Venice, 1592] of Maginus, and adds in a postscript "Titulus igitur talis: Zeoaxbeta sive Novæ Tabulæ, quibus Arithmetici debitis inextricabilibus multiplicandi et dividendi liberantur, ingenio, tempori viribusque ratiocinantis consulitur." 66 De § 12. It thus appears that the table was printed from a manuscript that Herwart used himself, and which very likely he had had made. As for the word prosthapheresis which occurs on the title, it is shown by the correspondence that Herwart used the table for multiplications in general, and that it was Kepler who pointed out that by means of it spherical triangles could be solved more easily than by the prosthapharesis of Wittich, and suggested the addition of the precepts for the solution of triangles, which actually occur in the preface to the work. The title suggested by Kepler was not adopted, nor was his advice about shortening it; but it must be acknowledged that Herwart succeeded in obtaining_a splendid title," which also contained a Greek word. Morgan explained the use of the word "prosthaphæresis" upon the title-page of Herwart's table, thus: "Prosthaphæresis is a word compounded of prosthesis and apheresis, and means addition and subtraction. Astronomical corrections, sometimes additive and sometimes subtractive, were called prosthaphæreses. The constant necessity for multiplication in forming proportional parts for the corrections, gave rise to this table, which therefore had the name of its application on the title-page.' The correspondence, however, shows that the table derived its title not from its general use in the calculation of astronomical prosthaphæreses, but from the special prosthaphæresis of Wittich for the solution of triangles. In a paper on Herwart's Table read before the Cambridge Philosophical Society* on October 25, 1875, which contained the greater part of the contents of §§ 9 and 11 of the present paper, I remarked that the prosthapheresis referred to seemed to be most likely a method of solving spherical triangles, in which the product of "On Herwart ab Hohenburg's Tabule Arithmetica poobapaipeσeWS universales, Munich, 1610." Proceedings of the Cambridge Philosophical Society, vol. ii. pp. 386–392 (part xvi.). two sines, or a sine and cosine, was avoided by the use of a formula such as sin a sin b={cos (a-b)-cos (a+b)}, adding that Laplace referred to such a method. This inference is shown to have been correct by the contents of § 7. Wittich does not appear to have published the method himself, though from the writings mentioned in § 7, and from Kepler's letters, it is clear that it was generally attributed to him: he ought, I suppose, to be considered the discoverer of the formulæ sin a sin b={cos (a−b)—cos (a+b)}, &c., which are really the prosthaphæretical formula. Kepler's remarks upon the difficulty of using the prosthapheresis for spherical angles, on account of the confusion between sides and angles and their complements, is interesting; and it is for this reason that I have quoted so much of the letter of October 18. The word prosthaphæresis often means the difference between the true and mean places of a body in longitude or latitude; but it seems to have been vaguely used, very much as correction" is now, to denote small quantities to be added or subtracted to quantities obtained by theory, or by a first approximation, &c.; so that without a context its signification is not precise; but I have not examined this point. In Klügel the word is derived from póσ0ev and apaipeois; but, at all events, as far as the mathematical and astronomical use of the word is concerned, De Morgan's derivation from póoleois and apaipeois seems to be certainly the true one. 66 As it happened, Herwart's employment of the word "poobapa peσews upon his title-page was not fortunate; for only σθαφαιρέσεως four years after the publication of his table logarithms were invented, all the processes of calculation were changed, and Wittich's prosthapheresis passed out of notice. Kepler, as is well known, greatly admired Napier's invention, and in 1624 published himself a table of Napierian loga rithms. It will have been noticed that Herwart describes his table as enabling multiplications to be performed "per solam additionem," and division "per solam subtractionem." These words would immediately suggest to a writer of the last or present century the method of logarithms; and it is for this reason, no doubt, that not unfrequently the methods of prosthapheresis and quarter squares have been confounded with applications of logarithms. Voisin, as mentioned in § 2, actually called his quarter squares logarithms; and this has added to the confusion. Trinity College, Cambridge. XLVI. Magnetic Figures illustrating Electrodynamic Relations. By SILVANUS P. THOMPSON, D.Sc., B.A., F.R.A.S., Professor of Experimental Physics in University College, Bristol*. [Plates VI. and VII.] N a preliminary communication to the Physical Society in February of the present year, the author announced a method of studying and illustrating the known laws of the mutual attractions or repulsions of conductors traversed by electric currents. The present paper is a complete statement of the facts obtained in the experimental research which formed the basis of that communication. While preparing a set of magnetic currents to illustrate the mutual actions of magnet-poles, it occurred to the writer that the mutual attractions and repulsion of currents might be illustrated in a similar manner by the figures formed with iron filings. He was aware† at that time that the lines of force of a straight conductor carrying a current were a series of concentric circles lying in a plane to which the conductor was normal. The series of figures now published originates, therefore, with the discovery of Faraday that the seat of the mutual actions of currents and of magnets must be sought in the surrounding medium. Since the communication of the preliminary notice, the writer has learned that one or two of the figures had been previously and independently observed by Professor F. Guthrie, but not published. Two others, Nos. 4 and 5 of the present series, are imperfectly given by Faraday in figures 18 and 19 of plate iii. in the third volume of his Experimental Researches (Series Twenty-ninth)‡, and without reference to the conclusions to be derived from their forms, which Faraday apparently overlooked §. The method employed for preserving the figures has been uniform throughout the series. Plates of glass, 3 inches long by 3 inches broad, were coated with a solution of gum-arabic and gelatine, and were then carefully dried. When the ar * Communicated by the Physical Society. +See Faraday, Experimental Researches in Electricity,' vol. iii. p. 400, § 3239, and plate iii. fig. 17; Guthrie, Magnetism and Electricity,' p. 254, fig. 225; Clerk-Maxwell, Electricity and Magnetism,' vol. ii. art. 477. And 'Phil. Trans. 1852, p. 137. $ The attention of the writer has also been drawn to a statement in the American Journal of Science for 1872, p. 263, by Professor A. M. Mayer, that he has obtained magnetic "spectra " from electric currents in a manner somewhat similar to that now described. The figures have, however, remained unpublished and undescribed, so that the writer has no means of learning how far the substance of the present communication may have been anticipated. rangement of magnets or of conducting-wires had been made for the particular case of the experiment, and the plate been laid in a horizontal position, fine filings of wrought iron previously sifted were dusted over the plate through muslin, and the plate was tapped lightly with vertical blows from a piece of thin glass rod. When the filings had arranged themselves, and the plate was still in situ, a gentle current of steam was allowed to play upon the plate, condensing upon the surface of the gum and softening it, and thus allowing the filings to embed themselves where they lay. After the gum had again become hard, the prepared face was covered by a protecting plate of glass, on which in certain cases were drawn the positions of the wires or magnets employed. The figures fixed in this manner are suitable for projection with the lantern upon the screen. They can be readily photographed for transparencies, or for paper prints; specimens of each of these methods of photographic reproduction are exhibited to the Society. Figure 1 represents the condition of the magnetic field surrounding the current in a straight conducting-wire, which was carried vertically through a hole drilled in the plate. The wire employed throughout the series was a silver one of about 8 millim. in diameter. The battery power employed for this experiment was that of 20 Grove's cells arranged in two series of ten each. In some of the succeeding experiments a less current was found sufficient. But for the imperfections of the method of experiment, these curves would be perfect circles, and the distances between two successive lines of force would be proportional to the square of the distance from the central point. The equipotential magnetic surfaces, being always normal to the magnetic lines of force, would be represented by a system of radial lines forming equal angles with one another. There appears to be no recognized name for the closed curves traced out by the lines of force around conductors carrying currents. With great diffidence I therefore beg to speak of them as isodynamic lines. They are theoretically disposed about a single straight conductor in a perfectly concentric manner, and at such distances apart as would be defined by the requirement that a parallel conductor, carrying a like current of unit strength, would do unit work in passing from one isodynamic line to the next. The absolute value of an isodynamic line would of course be determined (like magnetic and electrostatic potential) by the work done by a like element of current in passing to any point in that line from an infinite distance. No work is done in moving an element of a parallel current along an isodyna mic line, just as no work is done in moving a magnet-pole along in an equipotential surface. The isodynamic lines occupy, therefore, exactly the same relation to the element of the circuit, as do the equipotential surfaces to a magnet-pole or to an electrified point. Figure 2 represents the field above a horizontal wire carrying a current, and separated from the filings by the thickness of the glass (about 1.7 millim.). The lines cross the wire at right angles, and are really the projections of a series of such circles as exist in figure 1. Figure 3 exhibits the form assumed by the filings when the wire beneath the plate was coiled into a simple loop, a small piece of mica being inserted to prevent contact where it recrossed its path. The lines of the field within the loop run longitudinally; and their projections on the surface are mere points, as the filings show. In figures 4 and 5, two wires pass vertically through the plane of the figures, carrying parallel currrents, which in figure 4 are in the same direction, in figure 5 in opposite directions. Ampère's well-known law of the attraction in the former case, and of the repulsion in the latter, is well illustrated by the forms of the magnetic curves. In the former, where the parallel currents attract, the outer isodynamic lines are closed curves embracing both centres, the inner are distorted ovals about each centre-the whole forming a system of lemniscates, as would necessarily be the case, since the attraction at any point in the plate varies inversely as the square of the distance from each current*. In figure 5, where the parallel currents repel each other, the lines of force due to either current in no case enter or coalesce with those of the other current. They form two series of ovals of a peculiar form, flattened on the sides presented towards the opposing series. The conception of Faraday, " that the lines of magnetic force tend to shorten themselves, and that they repel each other when placed side by side," has been shown by Clerk-Maxwell, who thus concisely states it, to be perfectly consistent with the theory that explains electromagnetic force as the result of a state of stress in the medium filling the surrounding spacet. Faraday also observes that "unlike magnetic lines, when end on, repel each other, as when similar poles are face to face," and that "like magnetic lines of force," when end on to each other, coalesce. The terms "like" and "unlike," as applied 382. See Thomson and Tait, Natural Philosophy,' art. 508, vol. i. P. + Clerk-Maxwell, Electricity and Magnetism,' vol. ii. art. 645;* Faraday, Experimental Researches,' 3266, 3267, 3268. |