298 
the lower base is 10, of the upper 2, and 
the edge 9, the author endeavors to solve 
the problem when the side of the lower base 
is 28, of the upper 4, and the edge 15. In- 
stead of the square root of 81-144 required 
by the formula, he takes the square root of 
144-81 and calls it equal to 8 less =4, 7. e., he 
replaces “—1 by 1, and fails to observe 
that the problem as stated is impossible. 
Whether this mistake was due to Heron or 
to the ignorance of some copyist cannot be 
determined. 
In the solution of the problem to find a 
right angled triangle whose perimeter is 12 
and area 7, Diophantus, in his Arithmetica, 
300 A. D., reaches the equation 336a’+ 24 
=172x and says that the equation cannot 
be solved unless the square of the half co- 
efficient of « diminished by the product of 
24 and the coefficient of «’ is a square. No 
notice is taken of the fact that the value of 
« in this equation actually involves the 
square root of a negative quantity. 
Bhaskara, born 1114 A. D., in his chapter 
Viyja Ganita, was able to go a step further. 
He gave the rule: 
The square of a positive number as also of a nega- 
tive number is positive and the square root of a posi- 
tive number is twofold, positive and negative. There 
is no square root of a negative number, for this is not 
a square. 
The first mathematician who had the 
courage actually to use the square root of 
a negative number in computation was 
Cardano. At an earlier period ha had de- 
clared such a quantity to be wholly impos- 
sible, but in the Ars Magna, 1545, he 
discusses the problem of dividing 10 into 
two parts whose product shall be 40 and 
obtains the values 5+.,/—15, 5—./—15. 
These he verifies by multiplication. Such 
quantities he calls sophistic, since it is not 
permissible to operate with them as with 
pure negative numbers or others, nor to 
assign them a meaning. 
Bombelli, in his Algebra, 1572, gives a 
SCLENCE. 
[N. 8. Von. VI. No. 139. 
number of rules for the use of such quan- 
tities as a+b“ —1, but makes no endeavor 
to explain their character. 
Girard knew that every equation has as 
many roots as its degree indicates and con- 
sequently recognized the existence of im- 
aginary roots. In his Invention nouvelle en 
V algebre, 1629, while discussing the roots of 
the equation «'—47+ 3=0 he asks what pur- 
pose is subserved by such roots as —1 + /—2 
and —1—V—2 and says that they show 
the generality of the law of formation of 
the coefficients and are useful of themselves. 
Descartes, in his Geometria, 1637, gives 
us no new ideas upon the subject, but is the 
first to apply the terms real and imaginary 
by way of contrast to the roots of an equa- 
tion. 
Wallis, in his Zreatise of Algebra, 1685, 
leads the van in his endeavor to give a geo- 
metric interpretation to the square root of 
a negative number. In chapter LX VI we 
read : 
These Imaginary Quantities (as they are commonly 
called) arising from the Supposed Root of a Negative 
Square (when they happen, ) are reputed to imply 
that the Case proposed is Impossible. 
And so indeed it is, as to the first and strict notion 
of what is proposed. For itis not possible that any 
Number (Negative or Affirmative) Multiplied into it- 
self can produce (for instance) —4. Since that Like 
Signs (whether ++ or —) will produce + ; and there- 
fore not —4. 
But it is also Impossible that any Quantity (though 
not a Supposed Square) can be Negative. Since that it 
is not possible that any Magnitude can be Less than 
Nothing or any Number Fewer than None. 
Yetis not that Supposition(of Negative Quantities, ) 
either Unuseful or Absurd ; when rightly understood- 
And though, as to the bare Algebraick Notation, it 
import a Quantity less than nothing. Yet, when it 
comes to a Physical Application, it denotes as Real a 
Quantity as if the Sign were +; but to be interpreted 
in a contrary sense. 
He illustrates this by distances measured 
forward and backward upon a straight line 
in the usual way, and continues: 
Now what is admitted in Lines must, on the same 
Reason, be allowed in Plains also. 
