304 
space. Francais tries to solve the problem 
by the use of imaginary angles, but frankly 
acknowledges his failure. Servois sees with 
remarkable clearness what is needed, but 
is unable to reach it. He says: 
The table of double argument which you (Ger- 
gonne) propose, as applied to a plane supposed to be 
so divided into points or infinitesimal squares that each 
square corresponds to a number which would be its 
index, would very properly indicate the length and 
position of the radii vectores which revolve about the 
point or central square corresponding to +0; and it 
is quite remarkable that if we designated the length 
of a radius vector by a, and the angle it makes with 
the real line...... 9 al S20) sEllacosac by a, the 
rectangular coordinates of its extremity remote from the 
origin by x, y, the real line being the axis of «x, the 
point would be determined by x+y )/—1...... It is 
clear that your ingenious tabular arrangement of 
numerical magnitudes may be regarded as a central 
slice (tranche centrale) of a table of triple argument 
representing points and lines in tri-dimensional space. 
You will doubtless give to each term a trinomial 
form ; but what would be the coefficient of the third 
term? For my part I cannot tell. Analogy would 
seem to indicate that the trinomial should be of the 
form p cosa+q cosf+r cosy, a, 8, and y being the 
angles made by a right line with three rectangular 
axes and that we should have 
(p cosa -+-q cos +r cosy) (p! cosa 
q' cos 3 +r’! cosy) 
= cos? a-+ cos? 3+ cos 2y = 1. 
The values of p, g, 7, p’, q’, r/ satisfying this con- 
dition would be absurd, but would they be imagi- 
naries, reducible to the general form A + B V—1? 
As we all know now, these non-reals which 
Servois could not determine may be identi- 
fied with the +7, +j, +k, —i, —j, —k, of 
Hamilton’s Quaternions. 
In 1799, in his first published paper, 
Demonstratio NOvVa theorematis omnem functionem 
algebraicam rationalem integram wnius varia- 
bilis in factores reales primi vel secundi gradus 
resolvi posse, the celebrated Gauss, then only 
twenty-two years of age, says: 
By an imaginary quantity I always understand 
here a quantity contained in the form a+d Sa 
so long as bis not zero. * * * If imaginary quantities 
are to be retained in analysis (which for many reasons 
seems better than to abolish them, provided they are 
SCIENCE. 
[N.S. Vou. VI. No. 139, 
established on a sufficiently solid foundation) it is 
necessary that they be considered as equally possible 
with real quantities, on which account I should pre- 
fer to include both real and imaginary quantities 
under the common designation possible quantities. 
* * * A vindication of these (7. e., imaginary quan- 
tities), as well asa more fruitful exposition of the 
whole matter, I reserve for another occasion. 
This occasion, however, does not seem to 
have come till more than thirty years later. 
In the Gottingische gelehrte Anzeigen of April 
23, 1831, in an account by Gauss of his own 
paper Theoria residuorum biquadraticorum, 
Commentatio secunda, we read : 
Our general arithmetic, so far surpassing in extent 
the geometry of the ancients, is entirely the creation 
of modern times. Starting originally from the notion 
of absolute integers, it has gradually enlarged its do- 
main. ‘To integers have been added fractions, to ra- 
tional quantities the irrational, to positive the nega- 
tive and to the real the imaginary. This advance, 
however, has always been made at first with timor- 
ous and hesitating step. The early algebraists called 
the negative roots of equations false roots, and these 
are indeed so when the problem to which they relate 
has been stated in such a form that the character of 
the quantity sought allows of no opposite. But just 
as in general arithmetic no one would hesitate to admit 
fractions, although there are so many countable things 
where a fraction has no meaning, so we ought not to 
deny to negative numbers the rights accorded to 
positive simply because innumerable things allow 
no opposite. The reality of negative numbers is 
sufficiently justified since in innumerable other cases 
they find an adequate substratum. This has long 
been admitted, but the imaginary quantities—for- 
merly and occasionally now, though improperly, 
called impossible—as opposed to real quantities are 
still rather tolerated than fully naturalized, and ap- 
pear more like an empty play upon symbols to which 
a thinkable substratum is denied unhesitatingly by 
those who would not depreciate the rich contribution 
which this play upon symbols has made to the treas- 
ure of the relations of real quantities. 
The author has for many years considered this 
highly important part of mathematics from a different 
point of view, where just as objective an existence 
may be assigned to imaginary as to negative quan- 
tities, but hitherto he has lacked opportunity to pub- 
lish these views, though careful readers may find 
traces of them in the memoir upon equations which 
appeared in 1799 and again in the prize memoir upon 
the transformation of surfaces. In the present paper 
