Percentage of gas. experiment.* The experiments showed A violent reaction takes place, due to the that the gas consisted, not of carbonic carbon of the speigeleisen combining with oxide, but of hydrogen mixed with a lit- the oxygen of the metal to form CO, and tle nitrogen. This was verified not only this probably carries off mechanically for cast steel, but for many different de- the hydrogen dissolved in the metal, scriptions of iron. Confining the inquiry which would otherwise form cavities in to steel, two cylinders of Bochum steel, the cold ingot. This was proved by made from the same perfectly sound analyzing the gases which escaped, at ingot, but one forged and the other not Bochum, in this final reaction, and which gave the following results: in two cases had the following composition: CO. CO., N. H. O. Total. Composition. Analysis No. 1.82.6 14.3 2.8 99.70 No. 2.78.55 0.86 16.38 2.52 1.32 99.63 H. Vi. C. Tot. It will be seen that hydrogen was pres ent, and in about the proportion deUnforged steel 15.5 92.4 5.9 1 1.4 99.7 manded by the theory; so that in fact Forged steelt.. 5.5 73.4 25.3 1.3 100.0 CO is not only innocent of the produc tion of cavities in steel, but may actually This proves that a certain quantity of be used to prevent them. gas is occluded among the molecules of Another mode of attaining this end the steel, even where there are no visible would be by thoroughly drying the air cavities, and that this gas consists of before blowing it through the metals, so hydrogen, with about one-fifth of nitro- as to stop the introduction of hydrogen. gen. The latter is no doubt derived from This could be done by using burnt lime, the air, and the hydrogen from the moist- with little expense. The nitrogen would, ure contained in the large quantity of of course, remain, but it is uncertain air which mixes with the metal, especially whether this, by itself, would be harmful. in the Bessemer process. The absence These discoveries of the author were of CO is remarkable, because it has been subsequently criticised by M. Pourcel, shown by Troost and Hautefeuille that who suggested that the hydrogen was it is readily absorbed by red hot iron, and due to the ingot having been quenched because it is developed in large quantities from a red heat in water; that the water in the converter. But with fluid iron it either is not absorbed at all, or in small composed, the oxygen going to oxidize had penetrated into the cavities and deproportions compared with hydrogen and the walls, while the hydrogen remained. nitrogen. It may, however, act in another This criticism, however, falls to the way, namely, by freeing the metal from dissolved gases, in the same way as when ground, because the ingots were not In fact this water is saturated with an easily soluble quenched as suggested. * could not have been done without makgas, and then exposed to a current of ing the steel too hard to bore. Moregas which will not dissolve in the water, i the latter carries away with it a large over, although red hot steel will decom pose steam, there is no reason to suppose portion of the former gas. This may that it will decompose cold water ; e.g., explain why samples taken in the middle there is no evidence that hydrogen exists of the Bessemer process are often quite in quenched iron. Nor is it easy to see sound, while those taken towards the end, how hydrogen so formed could have when the development of CO has ceased, found its way into the interior cavities are porous. It also explains the fact that at the Bochum and Hosch (Dort of the ingot, and certainly not under a mund) works, absolutely sound Besse pressure of seven atmospheres. It would mer ingots are obtained by blowing the seem as if the French metallurgists were thinking only of the central cavity, which charge completely dead, and then pour- is open to the atmosphere, and which, of ing in 7 per cent. of fluid spiegeleisen. course, was not under consideration at all. * For the process and results, see Minutes of Proceedings Inst. C.E., vol. lvi., p. 360; and vol. lx., p. * In his preliminary report to the German Chemical 494. Society, Dr. Muller mentions that the ingots were #Mr. E. Windsor Richards has obtained a very simi- cooled in water; it is to be understood, therefore, lar analysis to this for the gases in steel ingots (Proc., that this cooling was not begun till their temperature Inst. Mech. Eng. 1880, p. 402). had fallen too low to admit of hardening. THE GEOMETRICAL INTERPRETATION OF IMAGINARY QUANTITIES. Translated from the French of M. Argand by Prof. A. S. HARDY. Contributed to VAN NOSTRAND'S ENGINEERING MAGAZINE. IV. we m The treatise, by Argand, appeared angle. So for the continued proportion in the year 1806.* In Vol. IV., 1813 da : b8 : Cy ....ila : mv, 14, of Gergonne's Annales de Mathe should have matiques, appeared an article entitled 6 “New Principles of the Geometry of Position and Geometrical Interpretation b T' of Imaginary Symbols,” by J. F. Fran- and B-a=y-B=u-1. cais, Professor in the Imperial School of Artillery at Metz, of which the following By the former a =a, and 1=1 ; there He then proposed a second notation. is an abstract. The author began by calling attention fore 1:1,::a: da, or da =a.la; so that to the distinction between the magni- a directed line might also be represented tude and position of a line, and to the by the symbol a.la, a denoting its still incomplete state of the geometry length and 1, its position. of position. He proposed the notation drawn from left (right) to right (left) Lines parallel to the axis of reference da, b, to represent right lines whose absolute lengths were a, b, were distinguished as positive (negathe subscript Greek letters denoting the the axis from right to left were regarded ; () angles made by these lines with any positive (negative). This convention, in . arbitrary axis of reference. used the expression “lines given in mag. connection with the above notation, gave nitude and position,” to designate what +1=1,, -1=11n, and therefore Argand called “ directed right lines.” In +a=ax(+1)=a.1o, the term ratio he included the relative and - a=ax(-1)=a.11. position as well as the relative magnitude, And from the known relations four directed lines being in proportion +1=eOn V-1, and -1=e+T1-1, results +a=ax(+1)=A.COM V-1, when and also ß-a=0-y. In and -a=ax(-1)=a.e+#1-1. such a proportion, the absolute lengths He then proceeded to establish four are in geometrical, while the angles made theorems : with the axis are in arithmetical progres I. In the geometry of position, imaginary quansion; and the homologous sides of any tities of the form Ea V - 1 represent perpendicutwo similar complanar figures are in lars to the axis of reference, and, conrersely, perproportion. In conformity with the pendiculars to the axis are imaginaries of this above definition, the proportion form. Demonstration.—The quantity aV-1 is a an : bs ::58 : Cy mean proportional between ta and-a, that is, involves the equations between do and art ; bence by the definition of 6 mean proportional is expressed by a i or, and ß-a=y-B, b it is perpendicular to the axis and drawn either whence B={(a+y), above or below it; and we have or a mean proportional between the ta V-1=a , and -a N-1=a directed lines bisects their included Reciprocally, every perpendicular to this axis * See January, February and March Magazine. is represented, in conformity with the above VOL. XXIV.–No. 4–22. as с с + 2 π -0. + TT 2 B, v notation, by a and is, therefore, by defini V-1=log (V-1). 1 2 tion, a mean proportional between do and a 17, or between taand It is therefore an im 2. That since ag=a.la, we have also aginary of the form EaV-1. ag=a.1-1. Cor. 1. -As signs of position, I 1-1 is 3. That since ca v-1 =cosa + sinav-1, identical with 1 it follows that aq=a cosa + a siná. V-1, Cor. 2. — Moreover, since-1=1+7=eta V -1, or that to express a directed right line, we must take the sum of its projections we have also + V-1=1 in two rectangular co-ordinate axes; Cor. 3.-So-called imaginary quantities are each projection being taken with its quite as real as positive or negative quantities, proper sign of position. and differ from them only in position, being in 4. That for any such lines we may fact perpendicular to them. substitute any number, provided that M. Francais argued that this theory the sum of the projections of the latter of signs was more consistent than the is equal to the sum of the lines themordinary one of Cartesian geometry, selves: that is, we may write da, bb , . . where, as abscissas and ordinates, two mu for provided we have kinds of positive and two kinds of neg. ative quantities were admitted. He con- (A) 2e&v=1= æ.ev=1=a.eav=1 tended that having once defined positive t.... tm.eu 1-1. and negative quantities, as laid off paral X cos X==a cos a + bcos ß lel to the axis of abscissas, it was illogi t. +m cos M, cal to admit others not comprised in the or (B) xsin &=a sin a + b sin ß definition, and that the common theory t. +m sin i, was thus faulty in admitting two incom- and conversely. patible principles where one was sufficient. If the lines lla, ba, 20 etc. form a THEOREM II. — The sign of position la=eV-1. closed polygon, (B) will be satisfied, and Demonstration.- Let the semi-circumference hence for any given line may be substiof a unit circle be divided in the direction of tuted a series of others, forming with it positive arcs into m equal parts, and radii be a closed polygon; conversely for a series drawn to the points of division; these radii of lines forming an unclosed polygon will form a progression both as 10 magnit de and position, by definition. The two exiremes may be substituted the closing line. The application of these remarks to being lo=+1, and 1,=-1=eV-1 the means the theory of the composition and reso11 , 127, 1(m-1), will be lution of forces is evident. On this point M. Francais briefly says, “This N-1 "v=1; theory which has always involved some difficulties is thus reduced to a problem or, in general, 1na=em and as THEOREM III.—The sign of position le may any angle whatever, we have finally la= also be written 1 2, that is to say la=1 2* From this theorem M. Francais drew Demonstratiotu.-If the unit circle be divided the following corollaries : into m equal parts and the radii be drawn, they 1. That by taking the logarithms of will form a progression whose extremes are each member of the last equation, av-1 unity. Hence 121=1m, 144 =1m, =log (1a); showing that, in the geometry of position, arcs of circles are the 2пп logarithms of the corresponding radii, 1 2n1 =1". Let then =a; we shall have being affected with the sign V-1 since they are perpendicular to the axis of and consequently la=12. reference; explaining also the expres 2π' sion, “imaginary arcs of a circle are log- that the above radii denote the m mth roots of Cor. 1.-It follows from this theorem: 1°, arithms,” and giving a rational interpre. upity; 20, these roots are all equal, differing tation of the symbolic equation only in their positions; 3o, they are all equally ጎቢ m V-1 (m-UT e m em em . n TIL COS + sin N - 1. m 2 real, being represented by lines given both in make himself known and to publish his own magnitude and position. researches on this subject. Cor. 2.-Comparing the last two theorems we obtain at once the well-known values of In the same volume (IV. p. 71-73) of these roots, which may be expressed, in gen- the annales which contained this paper eral, by from M. Francais, a note was inserted by the editor, Gergonne, to the effect 2nt 2ηπ 1 that two years before (1811) in a letter he had written to M. de Maiziere on a He then proposes the substitution, for communication which the latter had con+, and EV -1, of 10, 117, 11, in tributed to the first volume, he had sug gested that “numerical quantities were connection with the general sign 1ta; perhaps improperly classified in a single an additional advantage over that al- series, and that, from their very nature, ready suggested being that + and it seemed as if they should be arranged will indicate addition and subtraction in a table of double argument, as folonly, and so have but one meaning. lows: THEOREM IV.-All the roots of an equation of any degree are real and may be represented by lines given in magnitude and position. ,-2+2V - 1,-1+2 V-1, Demonstration. It has been shown that every equation of any degree whatever is al +2V-1, +1+2V-1, +2+2V-1,... ways decomposable into real factors of the first or second degree, and hence it is sufficient - 2+V-1,-1+V-1,+ V-1, to show that the root of an equation of the +1+N-1, +2+/-1,... second degree can be represented by lines .,-2, -1, +0, +1, +2, given in magnitude and position. Now the roots of an equation of the second degree, be .,-2-1-1,-1-1-1,-1-1, ing of the form x=p+ Nq, can at once be con +1-V-1, +2-1-1, ... structed by the foregoing rules ; for, 1°, if I is positive, x will be the sum or difference of two positive or negative quantities, laid off on ,-2–2V-1,-1,2V - 1,-2V-1, the axis ; 2°, if q is negative, x will be a right +1-2V-1, +2—2V-1, ... line drawn from the origin, the co-ordinates of whose extremity are p and 19. M. Francais concluded his communi-So that, like Francais, he proposed that cation as follows: quantities of the form nv -1 should be Such is a very brief sketch of the new prin- laid off in a direction perpendicular to ciples on which it seems to me desirable and that in which the quantity n was. measnecessary to found the geometry of position, ured, and that quantities having other and which I submit to the judgment of geom- directions should be represented by the eters. Being in direct conflict with the commonly received ideas concerning so-called im- sum of their projections on these two. aginary quantities, I expect they will encoun. He cites also from a letter of M. de Maiter many objections ; but I dare to think that ziere the following: “ What I have adtheir thorough examination will show them to vanced on imaginary quantities is quite be well founded, and that the consequences I have drawn from them, etrange as at first sight novel, and I am sure you have they may seem, will nevertheless be found in already recognized its exactness," and harmony with the niost rigorous logic. I again : “This will cease to be a paradox ought, moreover, to acknowledge that the when I have proved that imaginaries of germ of these ideas is not my own. I found the second degree, and therefore of all it in a letter to my late brother from M. Legen. dre, in which this great geometer gave (as degrees, are no more imaginary than something he himself had derived from negative quantities or imaginaries of the another, and purely as a matter of curiosity) first degree, and that as regards the the substance of my definitions of proportion former we are exactly in the same posiII.; but the latter was a mere suggestion, and tion as were the Algebraists of the sevonly justified by a few applications. To myself, enteenth century with respect to the therefore, belongs only the credit due to the latter." M. Gergonne disclaims any manner in which these principles have been intention of depriving either Argand or set forth, and their proof, the notation, and the Francais of the credit due them, but proposal of the symbol 1a. I hope that the publicity thus given to my own results may simply called attention to the fact that, induce the real author of these conceptions to after all, these conceptions were not so 66 strange as would seem, since several had angle, a directed radius making an angle a entertained them, and in closing he re- with KĀ would be expressed, from the princimarks that M. Francais paper may be ples already laid down, by 1o; but this expressummarized in the following proposi-sion would be troublesome when a is a fraction, tion: because it would then have more than one value. This objection would be met by the When, we seek a determinate but adoption of M. Francais' notation, la; we unknown length which is supposed to should thus have KA=10, KB=1: KC=1, lie in a certain direction along a given KD=11. We have considered angles reckoned line from a given point, while it really from A above and below as positive and negalies in the opposite direction, we obtain tive. Now, if we apply to the angles the rule a negative expression ; and if this length regard imaginary angles as laid off in a direc: is not on the line at all, the expression tion perpendicular to that which corresponds will appear under an imaginary form." to real angles. Suppose the semi-circumferM. Francais' paper called forth a ence ABC to revolve about AC, the point B second article from M. Argand, which describing the circle BPDQ; since we already appeared in Vol. IV., p. 133-147 of the have Annales, wherein he called attention to angle AKB=+1=1.(+1), his previous publication, and claims to angle AKD=-=(-1), have been the person to whom Legendre we may write angle AKP = 1 V-1= 4.11; referred in his letter, he having submit- whence we conclude that ted his first treatise to Legendre's exam- KP=1; 1 =1}x=2=1#v=T= ination. This second paper is, in the main, a restatement of the views ad. This would seem to be the analytical expres (11)-T =(-1)"-1. vanced in the first; but in it he abandoned sion required. the use of the signs~and +, and returned If on the circle BPD we take the point M, so to that of IV - 1. He also added that BKM=H, we shall have, in like manner, some further remarks, which are inter- angle AKM=(cos 4+ 1-1 sin je), and writing esting as showing how he attempted to for brevity cosu+1-1 sinu=p, KM =1?p= extend his theory to tri-dimensional 1.:P =(1+ y =(N-1)cosm+ V-1 sin u will be the space, and of which the following is a general expression for all radii perpendicular translation: to the primitive radius KA. Let us now seek an expression for BKP. Let (Fig. 23) KA=+1, KC=-1, KB=+ On the circumference ABC, the angles estiN-1, KD=-V--1; any other radius KN, mated from B in either direction are positive in the same plane, will be of the form p+qN-1: and negative, and real, and the plane BKP is and, conversely, every expression of this form perpendicular to their direction; it would thus will denote a directed line of this plane. seem that the angle BKP, like AKP,={N-1, and that this should in like manner be true for Fig. 23 any angle NKP, N being on the circumference ABCD; but that this conclusion is erroneous is evident from the fact that when N and C coin cide, we should have CKP=IV-1, whereas this angle is evidently - AKP=-IN-1. To avoid this difficulty, observe that having adopted a direction for +1, there are an infin. ity of lines perpendicular to it, among which one is arbitrarily chosen as that of 1-1. The general expression for every unity taken in one of these directions is, as we have just seen, 1.p=1&p=(1-1)=(1-1]cos + *-1 sin m. Conceive at the point A, an infinite number of directions perpendicular to the circumference at that point; one of these will be that of KP; namely that one we bave taken to conDraw now from the center K a perpendicu. lar KP=KA to the plane. How shall this struct the positive imaginary angles ta N-1; directed line be designated? Is it wholly in that is, for this case we have taken p=1=KA. dependent of KA and KB, or can it be referred So, at C, the direction parallel to KP gave analytically to the prime unit KA, are KB, negative imaginary apgles -a V-1; that is, KC? Guided by analogy it would seem that, we have made p=-1=KB. Hence, with retaking the entire circumference as the unit spect to the direction from B parallel to KP, M B N K А |