Auaust 27, 1897. ] 
Having thus justified the existence of 
negative planes, he goes on: 
But now (supposing this Negative Plain, —1600 
Perches, to be in the form of a Square ;) must not 
this Supposed Square be supposed to have a Side? 
And if so, what shall this Side be? 
We cannot say it is 40, nor that it is —40** 
But thus rather that it is V —1600, or * * 10 /—16, 
or 20 V —4, or 40 V —1. 
Where // implies a Mean Proportional between a 
Positive anda Negative Quantity. For like as Vic 
signifies a Mean Proportional between --) and +c; 
or between —b and —c ; ** So doth V—te signify a 
Mean Proportional between +b and —c, or between 
—b and +e. 
In chapter LX VII Wallis gives a geo- 
metric exemplification of a mean propor- 
tional, interpreting “bc asasine in a circle 
whose diameter =b-+c, and “—bc as a tan- 
gent in a circle whose diameter = —b+ce. 
He then finds the base of a triangle when 
the two sides and the angle opposite, and 
hence the altitude, are given. Assuming 
AP=20, PB=15, and the altitude PC=12, 
by the use of the triangle BCP, right-angled 
at (, he obtains two values for the base AB. 
Then taking AP=20, PB=12, and the alti- 
tude PC=15, he finds imaginary values for 
the base. 
These he interprets by saying : 
This Impossibility in Algebra argues an Impossi- 
bility of the case proposed in Geometry ; and that 
the Point B cannot be had, (as supposed, ) in the Line 
AC, however produced (forward or backward, ) from 
A, 
Yet there are Two Points designed(out of that Line, 
but) in the same Plain ; to either of which, if we 
draw the Lines 4B, BP, we have a Triangle ; whose 
Sides, 4P, PB, are such as were required: And the 
Angle PAC, and Altitude PC, (above AC, though not 
above AB,) such as was proposed : 
In this case he takes the triangle BCP to 
be right angled at B. Further: 
And (in the Figure,) though not the Two Lines 
themselves, AB, AB, (asin the First case, where they 
jay in the Line AC:) yet the Ground-Lines on which 
they stand, A’, Af, are equal to the Double of AC: 
That is, if to either of those 4B, we join Ba, equal to 
the other of them, and with the same Declivity ; ACa 
SCIENCE. 
299 
(the distance of Aa) will be a Straight Line equal to 
the double of AC; asis ACa in the First case. 
The greatest difference is this ; that in the first Case, 
the Points B, B, lying in the Line AC, the Lines AB, 
AB, are the same with their Ground-Lines, but not so 
in this last case where B, B are so raised above [3 3 
(the respective Points in their Ground-Lines, over 
which they stand), as to make the case feasible ; (that 
is, so much as is the versed sine of CB to the Diameter 
PC:) But in both ACa (the Ground-Line of ABc) is 
equal to the Double of AC. 
So that, whereas in case of Negative Roots, we are 
to say, The Point B cannot be found, so as is supposed 
in AC Forward, but Backward from Ait may in the 
same Line: We must here say, in case of a Negative 
Square, the Point B cannot be found so as was sup- 
posed, inthe Line AC; but Above that Line it may in 
the same Plain. This I have the more largely in- 
sisted upon, because the Notion (I think) is new; and 
this, the plainest Declaration that at present I can 
think of, to explicate what we commonly call the 
Imaginary Roots of Quadratic Equations. For such 
are these. 
From these extracts it is evident that 
Wallis possessed, at least in germ, some 
elements of the modern methods of addition 
and subtraction of directed lines. 
For the next hundred years no advance 
of importance was made. LEuler, for ex- 
ample, makes large use of the imaginary, 
but in his Algebra, 1770, he observes : 
All such expressions as Y Sh, V —2, etc., are 
consequeutly impossible or imaginary numbers, since 
they represent roots of negative quantities ; and of 
such numbers we may truly assert that they are 
neither nothing, nor greater than nothing, nor less 
than nothing, which necessarily constitutes them im- 
aginary or impossible. 
On the 10th of March, 1797, a surveyor 
named Wessel presented to the Royal 
Academy of Sciences and Letters of Den- 
mark a memoir ‘On the Analytic Repre- 
sentation of Direction,’ which was printed 
in 1798 and appeared in Vol. V, of the 
Memoirs of the Academy in 1799. 
Caspar Wessel was born June 8, 1745, at 
Jonsrud, in Norway, where his father was a 
pastor. Though one of thirteen children, 
he had a good education, for in 1757 he 
entered the high school at Christiania and 
