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WHAT IS MATHEMATICS?

ELLERY W. DAVIS.

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The definition, “Mathematics is the science of quantity,” will not stand in the light of modern developments. For example:

Let t teacher, p pupil.
Thent:p= the relation of teacher to pupil.

teacher of.
t:t= colleague of.
put= pupil of.

p:p=playmate of. We have the following multiplication table, where the relations at the left are

t:t

put

P:P
tut tit

0 0
t:p 0 Q tit t:P
put
p:p 0

0
p :P
0 0

p :P

t:P

put

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supposed multiplied into those at the top. We read

t:p Xp:t=t:t, teacher of pupil of is colleague of; while

p:t Xp:t=0, is pupil of pupil of does not exist. The rule of combination is that two relations give a new relation, that of antecedent of first to consequent of second, if consequent of first is antecedent of second; otherwise they give zero.

Using the same rule of multiplication consider the expressions, -never mind their meaning,

1 = a:a + b:b + c:c + d: d
i = a:6 - b:a tc:d
j
a:c + b :d

d:6
k a:d d: a + b : :c

d:c

= Cia

c:6

it will be found that the multiplication table is

1 i j k
1 1 i j k
i i

k
j j k 1 i
k k i

-1

precisely that of the quaternion units.

Is all this mathematics? Has the idea of quantity for a moment entered in? The example is from Charles Pierce's Logic of Relatives. He has among other algebras expressed all of the two hundred odd of his father's "Linear Associative Algebra" in this notation.

Take another example, this time from the theory of groups.

Let (ih) denote the operation that changes love to hate and hate to love, while (wp) similarly interchanges wealth and penury.

Then (1h)? = 1, i. e., leaves all as it was.
Likewise (op) = 1.
While (ih) (wp) gives both transformations at once.
Call (1h), (wp), (ih) (wp),

al, b, and c respectively. The multiplication table is

1

b 1 1

b 1

b b b

1 6

1

a

с

a

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a

с

с

a

The similarity to the quaternion table is manifest. In fact, the quaternion units are identity and three quarter-rotations, while here we could take for units identity and three half-rotations.

Any meanings whatsoever may be given to our symbols that are consistent with the purely formal laws of combination. It is not the subject-matter, but the character of the reasoning and the method of carrying it on, that makes the science rather ab..

stract. The reasoning is deductive, rather intricate, and generally carried on by an elaborate symbolism. Wherever this is so, whether in physics, chemistry, or biology, economics, logic, or philosophy, we recognize it as mathematics and we know that only the mathematical mind can successfully grapple with it.

I plead, then, that all who have, in any degree, mathematical power should, no matter what their chosen line of work, develop that power. At any time an occasion demanding the use of that power is liable to arise. I would that a large proportion of scientific men, especially, could have what Darwin has called their “sixth sense” developed. I would, too, that all mathematicians could take at least a master's course in some non-mathematical science. It seems to me that no one science can so well serve to co-ordinate and, as it were, bind together all of the sciences as that queen of them all, mathematics.

A FAMILY OF QUARTIC SURFACES.

ROBERT E. MORITZ.

The principal surface in this family was discovered during an attempt to construct the locus of a point so moving that the sum or difference of its distances from two intersecting straight lines is constant.

Setting up the equation of condition, using rectangular Cartesian co-ordinates, taking the line bisecting the angle between the directrices for the x-axis, a line perpendicular to their plane at their point of intersection for the z-axis, calling 2k the sum or difference of the distances of the running point to the directrices, and 20 the angle between the directrices, we obtain, after proper reductions,

k*r*sin 4 x*yosinobcos'ø + k*y*cos P + koz=k". If now we put sin p=e",

sin'0 cos'
**
cos:0

and
=b?,

ka the equation assumes the symmetrical form

==* [co – a*] [*°]. This quartie surface possesses the following remarkable features:

(1.) Two of the parallel systems of sections of this surface are coaxal systems of conics.

(2.) The sections parallel to the third co-ordinate plane are curves of the fourth degree, having in general four infinite branches, and, near the principal section, an oval besides. The principal section consists of two pairs of parallel lines.

(3.) The locus of the asymptotes to either system of coaxal conics forms a companion surface which is also of the fourt order. These two companion surfaces intersect in two plane

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=c",

curves.

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