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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:

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Tot.

Analysis No. 1.82.6

CO. CO. N. H. O. Total. 14.3 2.8 99.70 No. 2.78.55 0.86 16.38 2.52 1.32 99.63 It will be seen that hydrogen was present, and in about the proportion de99.7 manded by the theory; so that in fact 100.0 CO is not only innocent of the production of cavities in steel, but may actually be used to prevent them.

Another mode of attaining this end before blowing it through the metals, so would be by thoroughly drying the air as to stop the introduction of hydrogen. This could be done by using burnt lime, with little expense. The nitrogen would, of course, remain, but it is uncertain whether this, by itself, would be harmful.

These discoveries of the author were

This proves that a certain quantity of gas is occluded among the molecules of the steel, even where there are no visible cavities, and that this gas consists of hydrogen, with about one-fifth of nitrogen. The latter is no doubt derived from the air, and the hydrogen from the moist ure contained in the large quantity of air which mixes with the metal, especially in the Bessemer process. The absence of CO is remarkable, because it has been shown by Troost and Hautefeuille that 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 had penetrated into the cavities and decomposed, the oxygen going to oxidize This criticism, however, falls to the the walls, while the hydrogen remained. ground, because the ingots were not could not have been done without makquenched as suggested.*

either is not absorbed at all, or in small

In fact this

proportions compared with hydrogen and nitrogen. It may, however, act in another way, namely, by freeing the metal from dissolved gases, in the same way as when water is saturated with an easily soluble gas, and then exposed to a current of gas which will not dissolve in the water, ing the steel too hard to bore. Morethe latter carries away with it a large over, although red hot steel will decompose steam, there is no reason to suppose portion of the former gas. This may explain why samples taken in the middle that it will decompose cold water; e.g., of the Bessemer process are often quite in quenched iron. Nor is it easy to see there is no evidence that hydrogen exists 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 of the ingot, and certainly not under a that at the Bochum and Hosch (Dort

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 charge completely dead, and then pouring in 7 per cent. of fluid spiegeleisen.

*For the process and results, see Minutes of Proceedings Inst. C.E., vol. lvi., p. 360; and vol. lx., p. 494.

+ Mr. E. Windsor Richards has obtained a very similar analysis to this for the gases in steel ingots (Proc. Inst. Mech. Eng. 1880, p. 402).

is open to the atmosphere, and which, of thinking only of the central cavity, which

course, was not under consideration at all.

In his preliminary report to the German Chemical Society, Dr. Muller mentions that the ingots were cooled in water; it is to be understood, therefore, that this cooling was not begun till their temperature 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.

aa bB Cy

we should have

b с

....

=

m

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The treatise, by Argand, appeared | angle. So for the continued proportion. in the year 1806.* In Vol. IV., 1813– 14, of Gergonne's Annales de Mathematiques, appeared an article entitled "New Principles of the Geometry of Position and Geometrical Interpretation of Imaginary Symbols," by J. F. Fran- and cais, Professor in the Imperial School of Artillery at Metz, of which the following

is an abstract.

=

a b

=

ẞ—a=y-ßμ—λ.

By the former a=a, and 1,=1; there-
He then proposed a second notation.

fore 1: 1a:a: aa, or aa =a.1a; so that The author began by calling attention a directed line might also be represented to the distinction between the magnitude 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.

were distinguished as positive (nega-
the axis from right to left were regarded.
tive); angles estimated above (below)
positive (negative). This convention, in
connection with the above notation, gave
+1=1,, -1=1, and therefore
+a=ax(+1)=a.1,

of position. He proposed the notation drawn from left (right) to right (left) Lines parallel to the axis of reference aa, bs, to represent right lines whose absolute lengths were a, b, the subscript Greek letters denoting the angles made by these lines with any arbitrary axis of reference. Francais used the expression "lines given in mag: nitude and position," to designate what Argand called "directed right lines." In the term ratio he included the relative position as well as the relative magnitude, four directed lines being in proportion

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and a=ax(−1)=α.1±′′. And from the known relations +1=e0-1, and -1e-1,

results

+a=ax(+1)=α.coπ √=1,

anda ax(-1)=a.e±1. He then proceeded to establish four theorems :

I. In the geometry of position, imaginary quantities of the form a V-1 represent perpendiculars to the axis of reference, and, conversely, perpendiculars to the axis are imaginaries of this form.

Demonstration.-The quantity a V-1 is a mean proportional between +a and-a, that is, between do and an; hence by the definition of mean proportional is expressed by a #; or,

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aginary of the form a
Cor. 1.-As signs of position, ± 1 is

identical with 1

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2

Cor. 2.-Moreover, since-1=1+7=e±”

we have also + V-1=1

2

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-1,

Cor. 3.-So-called imaginary quantities are quite as real as positive or negative quantities, and differ from them only in position, being in fact perpendicular to them.

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3. That since a V-1 =cosa + sina√ −1, it follows that a,a cosa + a sina. √ — 1, or that to express a directed right line, we must take the sum of its projections in two rectangular co-ordinate axes; each projection being taken with its proper sign of position.

4. That for any such lines we may substitute any number, provided that the sum of the projections of the latter is equal to the sum of the lines themselves: that is, we may write ɑɑ, be, . . . . mu for c x & provided we have

M. Francais argued that this theory of signs was more consistent than the ordinary one of Cartesian geometry, where, as abscissas and ordinates, two kinds of positive and two kinds of negative quantities were admitted. He con- (A) x.eva.ev = 1 + b.e® v = 1 tended that having once defined positive and negative quantities, as laid off parallel to the axis of abscissas, it was illogi

cal to admit others not comprised in the or (B) definition, and that the common theory

+b.eBv=1 + +m.EM " xcosa cosa + bcos ß +....+m cos μ,

a sina sin a+b sin ẞ

was thus faulty in admitting two incom- and conversely.
patible principles where one was suffi-
cient.

THEOREM II.—The sign of position laea V-1 Demonstration.-Let the semi-circumference of a unit circle be divided in the direction of positive arcs into m equal parts, and radii be drawn to the points of division; these radii will form a progression both as to magnitude and position, by definition. The two extremes being 1。 =+1, and 17=-1=-1, the means 1(m-1), will be

1π, 12′′,

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.

1

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1. That by taking the logarithms of each member of the last equation, a-1 =log (1a); showing that, in the geometry of position, arcs of circles are the logarithms of the corresponding radii, 12n =1". being affected with the sign √-1 since they are perpendicular to the axis of reference; explaining also the expres sion, "imaginary arcs of a circle are log arithms," and giving a rational interpretation of the symbolic equation

n

a

a

= and consequently la=12′′. m 2π

that the above radii denote the m mth roots of unity; 20, these roots are all equal, differing only in their positions; 3°, they are all equally

Cor. 1.-It follows from this theorem: 1°,

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THEOREM IV.-All the roots of an equation of any degree are real and may be represented by lines given in magnitude and position.

Demonstration.-It has been shown that every equation of any degree whatever is always decomposable into real factors of the first or second degree, and hence it is sufficient to show that the root of an equation of the second degree can be represented by lines given in magnitude and position. Now the roots of an equation of the second degree, being of the form x=p+ Vq, can at once be constructed by the foregoing rules; for, 1°, if q is positive, a will be the sum or difference of two positive or negative quantities, laid off on the axis; 2°, if q is negative, a will be a right line drawn from the origin, the co-ordinates of whose extremity are p and Vq.

make himself known and to publish his own researches on this subject.

In the same volume (IV. p. 71–73) of the annales which contained this paper from M. Francais, a note was inserted by the editor, Gergonne, to the effect that two years before (1811) in a letter he had written to M. de Maiziére on a communication which the latter had contributed to the first volume, he had sugperhaps improperly classified in a single gested that "numerical quantities were series, and that, from their very nature, in a table of double argument, as folit seemed as if they should be arranged. lows:

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M. Francais concluded his communi- So that, like Francais, he proposed that cation as follows:

quantities of the form n-1 should be
laid off in a direction perpendicular to
that in which the quantity n was meas-
ured, and that quantities having other
directions should be represented by the
sum of their projections on these two.
He cites also from a letter of M. de Mai-
ziere the following: "What I have ad-
vanced on imaginary quantities is quite
novel,
and I am sure you have

Such is a very brief sketch of the new principles on which it seems to me desirable and necessary to found the geometry of position, and which I submit to the judgment of geometers. Being in direct conflict with the commonly received ideas concerning so-called imaginary quantities, I expect they will encounter many objections; but I dare to think that their thorough examination will show them to be well founded, and that the consequences I have drawn from them, strange as at first sight they may seem, will nevertheless be found in already recognized its exactness," and harmony with the most 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

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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 posiand ratio theorem I. and cor. 3 of theorem tion as were the Algebraists of the sevII.; but the latter was a mere suggestion, and only 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 1±a. 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

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angle, a directed radius making an angle a with KA would be expressed, from the principles already laid down, by 1a; but this expression would be troublesome when a is a fraction, because it would then have more than one adoption of M. Francais' notation, la; we value. This objection would be met by the should thus have KA=1,, KB=14, KC=11, KD=14. We have considered angles reckoned from A above and below as positive and nega

"When, we seek a determinate but unknown length which is supposed to lie in a certain direction along a given line from a given point, while it really lies in the opposite direction, we obtain tive. Now, if we apply to the angles the rule a negative expression; and if this length is not on the line at all, the expression will appear under an imaginary form."

we have adopted for lines, we should be led to regard imaginary angles as laid off in a direction perpendicular to that which corresponds to real angles. Suppose the semi-circumfer ence ABC to revolve about AC, the point B describing the circle BPDQ; since we already

have

angle AKB=+1=3.(+1),

11

angle AKD=-=(-1),

M. Francais paper called forth a second article from M. Argand, which appeared in Vol. IV., p. 133-147 of the Annales, wherein he called attention to his previous publication, and claims to have been the person to whom Legendre we may write angle AKP = { √ − 1 = 4.1; ; referred in his letter, he having submit- whence we conclude that ted his first treatise to Legendre's exam- KP=1, 1 =1+ √=z=1&v=1= ination. This second paper is, in the main, a restatement of the views advanced in the first; but in it he abandoned the use of the signs~and+, and returned to that of √− 1. He also added some further remarks, which are interesting as showing how he attempted to extend his theory to tri-dimensional space, and of which the following is a translation:

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This would seem to be the analytical expres(1)=(-1)-1. sion required.

If on the circle BPD we take the point M, so that BKM=", we shall have, in like manner, angle AKM=(cos + √ −1 sin μ), and writing for brevity cosμ+-1 sinμ=p, KM=1&p = 1.(1)=(√ −1)cos μ+1 sin " will be the general expression for all radii perpendicular to the primitive radius KA.

Let us now seek an expression for BKP. On the circumference ABC, the angles estimated from B in either direction are positive and negative, and real, and the plane BKP is perpendicular to their direction; it would thus seem that the angle BKP, like AKP,={√ −1, and that this should in like manner be true for 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=V-1, whereas this angle is evidently -AKP=-1-1. To avoid this difficulty, observe that having adopted a direction for +1, there are an infinity of lines perpendicular to it, among which one is arbitrarily chosen as that of -1. The general expression for every unity taken in one of these directions is, as we have just seen,

14p=1aP=( √—1)2 =( √ −1)cos μ+√−1 sin “ ̧ 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 have taken to construct the positive imaginary angles +a √−1; that is, for this case we have taken p=1=KA. So, at C, the direction parallel to KP gave negative imaginary angles a V-1; that is, we have made p=-1=KB. Hence, with respect to the direction from B parallel to KP,

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