MATHEMA
THE SCIENCE
JAMES BYRNE SHAW, D. Sc.
Presented to Faculty of Purdue University
as part requirement for the
Doctor's Degree.
*
- HENDERSON & DEPEw,
- JACKSON VILLE, IL [...
1895. - .










MATHEMATICS
THE SCIENCE


ALGORITHMS
JAMES BYRNIE SHAW, D. Sc.
Presented to Faculty of Purdue University
as part requirement for the
Doctor’s Degree.
HENDERSON & DEPEW,
JACKSON WILLE, ILL.
1895.
**
j < */ PREFACE.
The fundamental ideas of this paper are given in one of the Sen-
tences quoted from Professor Benjamin Peirce: “In every form of
material manifestation, there is a corresponding form of human thought;”
and in the statement of Professor Felix Klein (Inaugural Address, Bulletin
N. Y. Math. Soc., III, p. 2); “Proceeding from the idea of groups, we
learn more and more to coordinate and connect different mathematical
sciences.” That these are both true is evidenced by the bistory of mathe-
matics, pure and applied, for the last twenty years. From the thought-
side then, the thing to be studied is the group.
To do this successfully it seemed necessary to investigate the very
principles and processes themselves of Mathematics, and to trace the
subtlest threads of connectiou between its members. In the course of
this analysis of the structure of Mathematics it transpired that the prim-
ary thing of importance was anterior even to “groups.” This paper is an
attempt to bringto light these anterior elements of structure.
It appears thus that the “associative” law is one of a great many
such laws. Thus the whole field of “associative” groups and algebras is
multiplied into many fields. Again, “commutative” is found to be one of
four cases. It may be added, no example being given in the paper, that from
the statement of any one of these laws to which a set of symbols may be
subject, may be deduced all the general theorems of such sets of symbols.
It is expected to develop in much greater detail the matter here set
forth, and to push much farther the general development. It is hoped
that all branches of Mathématics in their structure, function, and use,
may be properly placed by means of this work, and the missing members
pointed out. In this way a rational philosophy of the existence of this
“Queen of all the Sciences” may be at last developed.
<!)-
§)
Z / C.
// - - CONT ENTS.
c. – Cº-
CHAPTER I.
FUNIDAMENTAL NOTIONS.
*
§
1-
2
Definition of Mathematics
* * * e º e s = º e e is a s e e º 'º e s a s tº e º 'º e º 'º is sº e º º º
º Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\ Phases of Mathematics. . . . . . . . . . .
C’) Qualitative relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§ Algorithm defined . . . . . . . . . . . . . . . .
tº
3
4
5
7. Mathematics defined as the science of Algorithms. . . . . . . . . . . .
8. Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9. Theory of Vids and Multenions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10. Theory of Facters
11. Theory of Multifacters
12. Theory of Vid-Variations
13. Theory of Multifacter-Variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14. Theory of Vid-Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15. Theory of Multifacter-Transformations
16. Theory of Variants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17. Theory of Vid-Substitutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18. Theory of Substitution-Variations... . . . . . . . . . . . . . . . . . . . . . . . . .
19. Theory of Substitution-Transformations
20. Theory of Šubstitution-Variants. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21. Theory of Substitution. Substitutions
23. Examples in the graphic phase
e e a e s a e º & 2 e s - e º 'º a s = e, e = * * * * * * * * * * * * * * * * * * * * *
• * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
• a s e e e s - e. a a e - - - a • * - e a e s a ... • * * * * * * * * * *
• e º ºr 4 a a e e º e º e º e s tº s tº º 'º -
* * * * * a e s is 2 e < * * * * * * * * * *
* * e e º e s a s e a s e e º e º e º e º 'º e º 'º
* * * * * * * * a w e º a tº e º a s e º & e º º º is a º ºs º wº
GENERAL PRINCIPLES.
I. Ideality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. Existence.,. . . . . . . . . . . . . . . . . . . . . . . . . .
I, Homology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IV. Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Heterogeneity
• * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
CEIA, PTER II.
A. GENERAL IDENTTTIES, MODULI, ZEROS,
§1-2. Definition of vid
4. Definition of facter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Definition of multenion. . . . . . . . . . . . . . . . . . . . . . . . . : - - - e º 'º e º 'º - - -
6. The cofacters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-8. Relations among the cofacters
9. Definition of field
• * * * * e º a e º e º sº e º e º e º ºr e º e s tº e º 'º e a e
* e º a c e e s a e is e a e º 'º e º e s = e e s - a s e e º ºs e º e º e º us tº e e a s
©
§
#
§: - THEORY OF VII)s, MULTENIONS, AND FIELDS.
&-
O
j
.
i
Section. Page.
10. Definition of modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
11. Definition of zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
12. Definition of complete modulus or Zero... . . . . . . . . . . . . . . . . . . . . 11
13-14. Definition of potent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
15. The infinities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
16. Deductions from the definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
17. The infinities. . . . . . . 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 12.
18. Facters as applied to numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.
B. IDENTITIES AND LIMITATIONS.
§1. Fundamental Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Limitation types of degorder (1, 2). . . . . . . . . . . . . . . . . . . . . . . . . 13
3. Deductions in these cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4. Limitation types of degOrder (1,3), . . . . . . . . . . . . . . . . . . . . . . . . . 15
5. Limitation types of degorder (2, 4). . . . . . . . . . . . . . . . . . . . . . . . . . . 16
CHAPTER III.
*=== Q
A CYCLIC TRIVID ALGORITHM.
$1. The general case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Cases with a modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19:
4. The two trivid cases with no modulus. . . . . . . . . . . . . . . . . . . . . . . . 20.
5-18. Examples of the use of this case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
19. Closing remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.
–1–
CHAPTER I.
Fundan:ental NotionS.
l, “Mathematics is the science which draws necessary conclusions.
“This definition of mathematics is wider than that which is
ordinarily given, and by which its range is limited to quantitative research.
The ordinary definition, like those of the Sciences, is objective; whereas
this is subjective. Recent investigations of which quaternions is the most
noteworthy instance, make it manifest that the old definition is too re-
stricted. The sphere of mathematics is here extended, in accordance with
the derivation of the name, to all demonstrative research, so as to include
all knowledge strictly capable of dogmatic teaching. Mathematics is not
the discoverer of laws, for it is not induction; neither is it the framer of
theories, for it is not hypothesis; but it is the judge over both, and it is the
arbiter to which each must refer its claims; and neither law can rule nor
theory can explain without the sanction of mathematics. It deduces from
a law all its consequences, and develops them into the suitable form for
comparison with observation; and thereby measures the strength of the
argument from observation in favor of a proposed law or of a proposed
form of application of a law.
“Mathematics, under this definition, belongs to every enquiry, moral
as well as physical. Even the rules of logic by which it is rigidly bound.
could not be deduced without its aid. The laws of argument admit of a
simple statement, but they must be curiously transposed before they can
be applied to the living speech and verified by observation. In its pure and
simple form the syllogism cannot be directly compared with all experience.
or it would not have required an Aristotle to discover it. It must be trans-
muted into all the possible shapes in which reasoning loves to clothe itself-
The transmutation is the mathematical process in the establishment of the
law. Of some sciences, it is so large a portion that they have been quite
abandoned to the mathematician,--which may not have been altogether to
the advantage of philosophy. Such is the case with geometry and analytic
mechanics. But in many other Sciences, as in all those of mental philoso-
phy and most of the branches of natural history, the deductions are so
immediate and of such simple construction, that it is of no practical use to
separate the mathematical portion and subject it to isolated discussion.
2. “The branches of mathematics are as various as the sciences to
which they belong, and each subject of physical enquiry has its appropri-
ate mathematics. In every form of material manifestation, there is a cor-
responding form of human thought, so that the human mind is as wide in
its range of thought as the physical universe in which it thinks. The two
are wonderfully matched. But where there is a great diversity of physical
appearance, there is often a close resemblance in the processes of deduction.
It is important, therefore, to separate the intellectual work from the exter-
n al form. Symbols must be adopted which may serve for the embodiment
—2–
of the forms of argument, without being trammeled by the conditions of
external representation or special interpretation. The words of common
language are usually unfit for this purpose, so that other symbols must be
adopted.””
3. Symbols are concrete existences having in general no inherent
connection to the idea symbolised. They are arbitrarily assigned to their
respective ideas. They may consist of letters or other designs; of lines,
points or other configurations in space; of assigned motions; of chemical
compounds, or a variety of other arbitrary entities. The symbolic side of
mathematics is called pure mathematics. That side in contact with the
thing symbolised is called applied mathematics.
4. There are as many phases of mathematics as there are varieties
of symbols. The most general phase, probably, is algebraics, in which the
symbols consist of letters, signs, designs, etc. Indeed, so general is this
phase that it has been styled formal mathematics, to the exclusion of the
other phases. The next most general phase is graphics, in which the symbols
are geometrical figures. Graphics must not be confused with geometry,
which is as much applied mathematics as is the science of force.j:
Graphics merely makes use of geometrical things to visualise its results
and processes, and depends in no way on the properties of space. The work
of modern mathematics in this phase is too well known to need more than
mention. The trend of mathematics seems nowadays to be toward the
large development of graphics; and the reactionary benefit on the subjects
so treated in the suggestion of new relations and new theories is proof of
the great value of the phase. We might with equal propriety devise
phases of mathematics in which colors are the symbols, or sounds, or
motions, or heat, etc. We can divide the phases of mathematics into the
divisions: algebraics, graphics, mechanics, chromatics, acoustics, thermics,
magnetics, electrics, chemics. It is worth noticing that historically the
first two divisions are traceable in essence to the Hindoos and to the
Greeks. Professor Cayley has made some slight use of chromatics in the
theory of groups.
5. Relations are either qualitative, or quantitative, or both. Quali-
tative relations may be considered by themselves without regard to
quantity. We have thus a system of enquiries which has been called logi-
cal algebra, of which Boole gave the first example. It is manifest that
such a system of enquiries may be considered graphically, also, and such a
system would, with equal propriety, be called logical graphics.
Quantitative relations may be considered by themselves without
regard to quality. They belong to the branch called arithmetic, or the
theory of numbers. They may also be considered graphically, and we have
point, line and circle representations of numbers.
*Benjamin Peirce, Linear Associative Algebra, p. 1.
+Ibid, p. 2. Felix Klein, Evanston Lectures, p. 46.
–3—
In almost all enquiries both kinds of relation are intermingled, and
the system of mathematics corresponding must conform to this character.
6. Any system of deduction of necessary conclusions, or transmuta-
tion of laws or ideas into other forms, is an algorithm. Thus quaternions
is an algorithm. Graphical mechanics is an algorithm.
7. We may define mathematics as the science of algorithms. It
must begin with a set of general principles, out of which issues a general
theory of algorithms. The general theory must be followed by the special
theories of, and the classification of, particular algorithms. Such an
encyclopedic work would cover the whole realm of pure mathematics.
8. It appears that the general theory of algorithms may be sub-
divided, the same plan of division applying to each phase of the subject.
"Thus we have:
Theory of Vids and Multenions.
Theory of Facters.
Theory of Multifacters.
Theory of Vid-Variations.
Theory of Multifacter-Variations.
Theory of Vid-Transformations.
Theory of Multifacter-Transformations.
Theory of Variants.
Theory of Vid-Substitutions and Functions.
Theory of Substitution-Variations.
Theory of Substitution-Transformations.
Theory of Substitution-Variants.
Theory of Substitution-Substitutions, and Multifunctions.
9. The theory of vids and multenſions treats of the abstractions for
which the mathematical symbols stand. The vid is the fundamental thing
in a mathematical treatment. Thus, in studying quantity relations, num-
bers are the vids considered. These are represented by such symbols as 1,
2, 3, #, e, V3 T, and the like (in which we consider # as mot representing an
operation but a single idea for which no other symbol has been devised, so,
also, the V2 T); so the symbols of quaternions, i, j, k, q, represent vids, or at
least a compound of vids, i. e., a multenion; the Grassman symbols for
points, lines, etc., represent vids or multenions. A multenion is a notion
compounded of several similar vids, each vid having with it a number-vid,
but dissimilar vids or vids having with them other vids than number vids
do not constitute multenions. Thus the quaternion—
Q = w -Hari + yj +2k.
is a multenion. The + does not mean addition here, but is a mere separa-
tion mark. I shall prefer throughout to use the comma for such a mark,
reserving the + to indicate only arithmetical addition; thus the quaternion
q would be written—
Q = w, aci, yj, 2k.
Here the unexpressed scalar unit, the i, j, k, are the fundamental vids, the
---4–
w, ar, y, 2 the number vids. But
'VIT, j, k,
is not a multenion in the sense in which the word is here used. It is very
properly called a biquaternion, since it involves a vid, VIT, which is dis-
similar to the quaternion vids. Sir William Rowan Hamilton was certainly
right in the distinction he drew in this matter between quaternions and
biquaternions. For the same reason, algebra treats of such multenions as
a—b, but it algebra
- a –b VIT
is not a legitimate expression any more than is the quaternion aſ-- b i.
Such expressions might be called poly multenions, but no special
name is necessary.
10. The theory of facters treats of the primary modes of combining
vids or multenions. By primary mode I mean a simple, unique, definite
law by which given two vids or multenions, a third is thereby determined.
Thus, for numbers, addition is such a facter, and the only one. For vect-
ors, “geometric” addition is such a facter. But neither the internal nor
the external multiplication of Grassman is a facter, since there is an inde-
finiteness in the product on the one hand, when two of the units become
identical, and on the other hand, when they are not identical, i. e., in the
one case there is no product of are by ye, in the other of ace by ya. Quater-
nion multiplication is a facter. So also quaternion addition is a facter.
"These two are the only facters in quaternions, for any other law may be
reduced to an expression in terms of these, while neither is, so far as qua-
termions is concerned, deducible from the other.
11. The theory of multifacters treats of those laws or processes of
combination which are derivable from a single facter. Thus, defining
arithmetical addition as usual, we derive from it multiplication, which is a
multifacter. Likewise involution—raising to a power—is a multifacter.
So in quaternions, involution is a multifacter. But multiplication is not a
multifacter, since it cannot be derived from addition in this case. In any
algorithm there may of course be several facters, and any number of mul-
tifacters. As one more example, in a substitution-group, there is one
facter and one multifacter.
12. The theory of vid-variations treats of the changes produced in
the result of a multifaction when one of the elements is changed. Thus, if
the two quaternions q, r are such that
q r = 8,
while if for r we substitute another quaternion tr, so that
q. tr. = 'w,
then the change from s to w corresponding to the change from r to tris a
subject of investigation for this theory. The change may be infinitesimal.
13. The theory of multifacter-variations treats of the result of
substituting for a multifacter, another.
—5—
14. The theory of vid-transformations treats of the result of sub-
stituting for a vid a multifaction of other vids, all the multifacters con-
cerned being based on one facter. Thus in arithmetic if in
Q & = y
we substitute for ac, a + da:, giving
a (a;+da;) = y + dy,
the change in y, dy, is a subject for investigation in this theory. This
theory includes, for example, the theory of ordinary arithmetical substi-
tutions and functions, but not that of algebraic quantities.
15. The theory of multifacter-transformations treats of the result
of substituting for a facter or a multifacter, a multifaction of other multi-
facters, -
16. The theory of variants treats of those elements which remain
unchanged when the above substitutions are made.
17. The theory of vid-swbstitutions treats of the results of putting
in place of a vid or multenion a multifaction of other vids or multenions,
this multifacter not being based on the same facter as the one designating
the mode of combination in the original expression. This includes the
theory of real and complex functions.
18. The theory of substitution-variations treats of the change in
the result of a function, when for a vid or multenion another is sub-
stituted. This includes the theory of finite differences, differential cal-
culus and calculus of variations. -
19. The theory of substitution-transformations treats of the result
of substituting for a wid or function a new expression involving other
functions and vids. It includes the theory of substitutions and the theory
of transformation-groups.
20. The theory of substitution-variants treats of those expressions
which remain unaltered by the preceding processes. It includes the
modern theory of invariants.
21. The theory of substitution-swbstitutions treats of any expression
built up as in the preceding numbers but involving three or more facters.
22. Some of the divisions given above are only partially developed,
if at all. Their complete development is a work for future investigators.
It is evident that an algorithm may consist of some or all of these theories
applied to a special field, as quaternion differentiation,
23. The graphic phase admits the same divisions as algebraics.
These have been studied less even than the corresponding divisions of
algebraics, from the theoretical standpoint, at least. But homogeneous
strain, for example, uses the graphic theory of substitution-transforma-
tions. Configurations is but a graphic study of vids, Rotations is a
—6—
graphic study of facters. Other examples might be brought forward of
the other theories also but these suffice.
24. Manifestly the freest and most convenient phase is algebraics,
The very nature of the symbolism permits of a more general treatment.
The fact must not be overlooked, however, that whether a vid is repre-
sented by a letter or a line, it is equally a vid and in thought is the same
thing. The clothing does not change the thing clothed.

General Principles.
e tº
— — . . —
I. IDENTITY. Any particular feature of a conception of the mind
may be symbolised and may enter into a mathematical process through
its symbol independently of the other features of that conception.
II. ExISTENCE. An algorithm fulfilling any assigned compatible
conditions may be devised independently of any application for it, in which
case it has an ideal existence, at least; and for every case of concrete or
abstract reasoning an algorithm exists.
III. HomoLOGY. “If we have two sets of concepts, A and B, such
that to every concept of the one set shall correspond a concept of the other
and to every relation between any two of one set a corresponding relation
between the corresponding two of the other, then all language, reasoning
and conclusions as to the one set may be applied to the other set.”
IV. SUBSTITUTION. For any idea or symbol may be substituted
another whose meaning is identically the same.
V. IDENTITY. If in any algorithm, we may for a certain idea or
symbol wherever it occurs substitute another, or vice versa, then the two
are identical so far as that algorithm is concerned.
VI. HETERoGENEITY. A symbol may represent two or more hetero-
geneous eierments, in which case the individual elements may undergo
independent change, and the algorithm will consist of a compound of two
or more independent heterogeneous algorithms.
*Simon Newcomb, Bullet in W. Y. Math. Soc., III: 98.
e gº--&Nº.3/~----.S. 3
- *\ *
—8—
CHAPTER II.
Theory of Vids, Multenions, and Fields.
A. GENERAL IDENTITIES, MODULI, ZEROS.
1. Probably the first idea of qualitative relations as a part of mathe-
matics came from geometry, and the attempt to represent by one symbol a
line. However that may be, it is true that geometry furnishes very appar-
ent cases of qualitative relations. A line is seem to possess not only length
but direction, and position in space. The essential element that makes the
line this line and not some other, is a vid. It cannot be said that the vid
is the sum total of all the non-numerical ideas that go to make up the
concept lime, for it is not so inclusive. But all the distinguishing elements
are the vid of that line. And so in other cases than geometrical notions.
It is in this differentiating process, without analysis of the content of
either the full notion or even of so much of it as constitutes the vid that
the power of mathematics lies. It matters little for the mathematical
argument what the entire meaning of the vid is—a question of philoso-
phy—but only what so much of its meaning may be as is to be considered
in the argument at hand.
2. Wids, in themselves, are rather symbolic or abstract notions, than
otherwise. Thus, consider the quaternion vid i. The printed character is
of course only a check or tag which enables the mind to keep watch of the
vid which it represents. The vid itself is one of several whose relations
constitute the formulas and transformations of the quaternion algori-
thm. The system of formulas and processes is independent of what i and
the others represent; whether, for example, i is a rotation, or a unit line in
Space of four dimensions. .
3. it frequently occurs that, given two lines in a plane, say, a third
is thereby determined. Thence, and from similar determinations, arises
the notion of considering that in general two homogeneous vids can deter-
mine a third. That this determination can take place in a variety of ways,
is evident, and that the vid so determined may thus change is also evident.
For example, it might be assigned as the method of determination that
the line determined shall be the bisector of the angle between the two
lines determining, and it can further be assigned that one line may be con-
sidered as acting on the other, in which case the acting line must be con-
sidered, say, as lying so that a positive rotation less than a straight angle
would rotate it into the position of the line acted upon; the determined
line is then uniquely determined. If the acting line and the one acted
upon change places the determined line manifestly is a perpendicular to:
the other. But the process of determination might have been differently
chosen. Thus, the determined line might be that which is produced from
the line acted upon, by the same rotation that is necessary to turn the
acting line into the position of the line acted upon.
From these examples the following is evident: First: That the
––9– *.
process of determination needs a distinctive name and symbol. This I
shall call a facter, denoting facters by capital Roman letters. Second:
That it makes a difference which vid is considered as acting on the other,
and thus each vid needs a distinctive name explaining its role in the pro-
cess. The vid acting I shall call the facient, writing it before the other,
which will be called the faciend. The result will be called the factu m.
Vids will be denoted by small Italic letters. The mark = will be used to
indicate identity. Thus we may write
F. (t ) = C,
which means, a determines from b a vid c, by a given process, or facter, F.
4. The determination must be unique and definite. If the facter F
were so indefinite that in some cases it gave no determinate result, it
would cease to be a facter, as the term is here used. Again, if F were not
unique, but were to give two or more possible results, it would cease to be
a facter as the term will be used in this paper. And finally, F must be
such that given any two of the three vids the other is in general uniquely
determinable. Any law of determination not fulfilling all of these condi-
tions can be either resolved into simple laws, or can be made definite by
further determinations being added to it. -
5. It is clear, further, that we might have taken definite measurable.
lengths along the line, in §3, thus introducing the quantitative idea.
Such segments could be represented by a vid symbol and a number Sym-
bol writtten together, thus .*
tl{{..
I shall use small Roman letters exclusively for numbers. Such a
combination of vid and number is a multemion. We might consider com-
pound notions made up of two or more multenions. Such would be bi-
nomials, multimomials, etc. Thus a quaternion would be written
w1, xi, yj. Złc.
6. It is clear now that if
F. (t b = c,
from the definition of facter, there must be some determinate proccss by
which, given any two of these, we can find the third. There are thus five
co-facters of F, which will be temporarily designated thus,
iF, rR', sE, tR', ub',
defined thus,
iF. ac = 5.
r.F. bc = n.
sR. bat = c.
th", cat = b.
uP. eb = (t.
For example, taking the second method given in $3 as an example, the
-—10–
facter F is: rotate the faciend into position of the facturm by the same rotation
that turns the facient into the faciend; the facter iP is: rotate the faciend
into the position of the factum by one-half the reversed rotation that turns
the facient into the faciend; the facter rR is: rotate the faciend into the
position of the factum by double the reversed rotation that turns facient
into faciend; the facter sR is: rotate the faciend into the factum by double
the reversed rotation that turns facient into faciend; the facter tº is: rotate
faciend into factum by one-half the reversed rotation that turns facient
into faciend; the facter u R is: rotate faciend into factum by same rotation
as turns facient into faciend.
The six facters are a set of co-facters and are all unique and definite.
r-y
7. Suppose we let in F represent that facter which has the same
relation to JF that JF has to F. We find
ii F. Cab = c,
But this is the same form as F. ab = c and hence, (gen. prin. V.) F and
iiſ' must be the same facter. Hence we may write
F = i if',
In the same way we can easily obtain
riF = up".
r).E = tº".
ir F = sh".
We may then drop all the indices but and r, and write the forms, with
their names, thus:
F, the direct facter.
iF, the inverse facter.
rF, the reverse facter.
ir F, the inn'everse facter.
riF, the reinverse facter. a
rrF, the rereverse facter.
8. Forming the whole set of cofacters of each cofacter we easily
find the results thus tabulated:

In this table it stands for the direct facter. By this table we can reduce
—11—
any combination to its simplest form. This table is very remarkable, and
in fact, regarding the wºmbrae as vids, (since they come within the defini-
tion), we have an algorithm of six vids. These are “associative,” but not
“commutative.” The table is read as a multiplication table, e. g. The
reinverse of the in reverse is the rereverse.
9. The ensemble of vids and multenions which can be constructed
in an algorithm is the field of the algorithm.
10. The }%; modulus of a field with reference to a facter
F, is that { *::::: | which used with any } #: ! of the field repro-
| faciend
| facient ſ facient \
duces that ! as factum. Thus, if m is a R faciend ) modulus,
}#} = % } a, d, any vid of the field.
11. The }%: zero of a field with reference to any facter F,
is that }}.}} which used with any } º ! reproduces itself as
| facient
| faciend * jf
facturm. Thus n is a
} #. #. = !. | a, d, any vid of the field.
12. A modulus or a zero is complete if it is both facient and faciend
modulus or zero, respectively.
13. Any combination built up with F and a given vid only, is a
potent of that vid.
14. A vid is idempotent with respect to F, if one of its potents is the
vid itself; modulopotent, if one of its potents is a modulus; 2eropotent, if
one of its potents is a zero; allopotent, if no one of its potents is ever iden-
tical with any other one of them.
15. The so-called zeros and infinities will be found to come within
the definition of a zero above.
A field in general does not necessarily possess any of these peculiar
vids; but facters having certain limitations may imply some of them.
16. Putting these definitions into symbols, and writing the con-
nected and immediately deducible identities, there results, a being any vid
of the field;
m, the facient-modulus,
F. mac =ac; iP. mac =ac; in F. am = az; prºb". a.m. = ac; rR', a.a. = m; riF. was m;
or, in words, the inverse has the same facient-modulus as the direct, and
...this is the faciend-modulus of the in reverse and the rereverse: it is found
w —12—
by taking the reverse or the reinverse with any vid as both facient and
faciend;
m, the faciend-modulus,
F. it'm—a', riH', a.m.–a'; 'F. mari-a”; irº.naca-tr; iP. a'a'-m; rºbº', ºr-m;
or, the reinverse has the same faciend-modulus as the direct, and this is
the facient-modulus of the reverse and the inreverse; it is found by taking
the inverse or the rereverse with any vid as both facient and faciend;
m, the facient-Zero,
F. na'-m; rif', ma;-m; irF. &m-m; rB'. acn-m; rr P, n m-,e; iP. m. m-ac;
or, the direct and the reinverse have the same facient zero, which is the
faciend-zero of the in reverse and the reverse; it gives, when used as both
facient and faciend of the rereverse or the inverse, an absolutely inde-
terminate result;
n, the faciend-zero,
F. ºr m=n; if', a.m.–n; irº, mir-m; r. F. maſ-m; rR', m m-irº; rib". In n-a:;
or, the inverse and the direct have the same faciend zero, which is the
facient zero of the inreverse and the rereverse; used as both facient and
faciend of the reverse or reinverse, it gives an absolutely indeterminate
result.
17. In many cases it is possible to write for the inverse with a given
facient, the direct with a different facient, thus,
iP. a = F. b.
In any such case, if n is a facient-Zero, let
iF. n= F. p ;
then F. pn=3; since F. p n = i F. nºv.
Now F. pa: cannot be a, else F. ma;-a!,
F, pa: cannot be y, another vid, else F. my=a: ,
F. .pæ cannot be m, else F. nºn-a".
The only conclusion left is
F. pa: = p, and F. n p=a.
Hence p is also a facient-Zero, and with n as facient-Zero gives an inde-
determinate result. -
Likewise, if we can write
J.F. Ja! = F. a. |),
m being a faciend-zero,
F. ap cannot be a, else F. a.m. = ºr,
F. ap cannot be y, else F. yºu = ºr,
F. a')) cannot be it, else F. Tvm = ºr,
Therefore, F. a p = p, and F. pm = r.
These p's are the so-called infinities, and from the identities given, al
their properties can be deduced in any given case.
18. In the case of multenions, F must be defined for all the vids
involved, thus, if
F. & b = C,
we can tell what F, 2d 30 is identical with, only when we know the law by
—13–
which F combines the vids 2 and 3. Thus, if F. 2, 3 is 6,
F. 2C/ 3U) = 60,
It has been pretty generally assumed so far in the treatment of
new algorithms, that every facter will multiply the numbers attached, but
it need not be so, of course.
In the case of binomial, or multinomial multenions, the facient or
the faciend or both being compound, the law of combination of the facter
must be definitely assigned also, thus, if
F. ab = c.
F. ad = e.
F. a (0, d) may = (c, e) or may equal
another vid f. Such cases have been uniformly treated so far as “distri-
butive,” but of course this method of combination is arbitrary.
B. IIDFNTITIFS AND LIMITATIONS,
1. We have in any case:
[A] F. ab=trF. ba; iF. ab= rP. ba; rR', ab=rif'. ba.
[B] iP. a. (F. ab)=b; r). F. (F. ab) d=b; r.F. b (F. ab)=a; rt F. (F. ab) U=a.
[C] F. a (iF. ab)=b; irP. (i.F.ab) as-b; r. F.b (E. ab)=a; , F.(IF, ab) b-a.
[D] irP. a (r.F. ab)=b; F. (FF. ab) a-b; rrr' b (FF. ab)=a; F. (FF. ab) b-a.
[E] rr B. a (riP. ab)=b; if'. (rif', ab) as-b; urb'. b (riF. ab)=a; F. (r. F. ab)b=a.
[F] rP. a (irP. ab)=b; riſ'. (ºr F. ab)a=b; iP. b (ºr F. ab)=a; rr F. (ºr F. ab)b-a.
[G] rif', a (rrF. ab)=b; rR. (rrF. ab) a-b; F. b (rrF. ab)=a; ºr F. (rrF. ab)b=a.
These will be referred to as the fundamental identities.
2. It might not have escaped observation that, in the example given in
A § 6, the direct and the reinverse were precisely the same law; so the
inverse and the rereverse. and the reverse and in reverse. This suggests at
once a limitation on the generality of the facter, or the vids, which is worth
investigating. The example might be stated
F. a b-ri F. ab
or, by [B] above ºr Fa (Fab)=b , or as we may write it
rR'a Fa = I, I denoting absolute identity.
Now, there are 6×6, or 36 conceivable cases of this kind, but six of them
are always true by the first column of B – G] above, and by noticing that
such a form as rR'a iPa-I is the same as FairPa-I, the remaining reduce
to four types, with six identities in each, all of the identities of a case fol.
lowing as consequences of any one of them. These are
Type I: F=uF, rF==rif' , irf'-rrF.
Type II: F=r'E=rrT , iF=rif'—ur F.
Type III: F=rif', IR-rrF, r.F.—in F.
Type IV: F =ir F, *F =rP', rulº'-rr H'.
Since each one of these has the same facient, I shall call them of degree
ome; and since there are two facters involved in each identity, I shall call
them of the second order. These are the limitation types of the first
degree and second order. Manifestly a limitation type of the first order
—14—
would reduce all the facters to the form I, and one of the second order and
Second degree would reduce to the same thing since the facients would be
utterly independent and might be any vids of the field.
3. Substituting the limitation identities of § 2 in the general iden-
tities of $ 1, we have (omitting mere restatements):
jº Type I.
[A] irP. ba = r).E. ba = F. ab = i F. ab.
riF.ba = rR'. ba = r F. ab = riP. ab.
[B] b=F. a Fab = i F, a Fab =rrF. (iF. ab) a = irf". (F. ab) a.
a=rif, bF. ab = -F, b iP. ab = rR. (F. ab)b = riſ'. (iFab) b.
[D] b-rrF. a rE. ab = irP. a riF. ab = iP. (T. ab) a- F. (riP. ab) a.
a=irF. b rR. ab = rºP. briFab = F (FF, ab) b = F. (PiF. ab) a.
[F] b=riF. a ir F. ab=rP. a rr F. ab=riF. (ºr F. ab) d=rF. (ir F. ab) a.
a=F. b irP. ab=iF. b rR', ab=ir F. (in F. ab) b-rrF (rrF ab) b.
From [A] the reverse of F is such that facient and faciend in it may
be interchanged. From [B], if we denote F. & F. a by FF, etc., we have
F. F. . . . . . . F. F., ab=U=iſ' iP. . . . . . iF iH', ab.
There can be no facient-zero in such a field; nor can there be a faciend-
modulus, except we have for every vid of the field
F. bb=m.
There may be a faciend-zero, or a facient-modulus.
Type II.
[A] irP. ba-riF. ba-iP. ba-F. ab=rF, ab=rrF. ab.
[B] b-F (F. ab) a ; al-F. b Fab–if', (Fab) b.
[C] b=iP. (iFab) a , cº-iR. b iP. ab=F (iF ab) b.
The important features of this type are in the statements in [B],
from which it appears that such fields cannot contain a Zero, and that the
modulus, if there be one, is complete. If there is a modulus,
F. aq=m,
for every vid of the field. -
Type III.
[A] irP. ba-r F. ba-F. ab=riP. ab.
ºr F. Da-iP. ba-ik", ab=rrF. ab.
[B] 0–iF. (F. ab) a ; a=F. (Fab) D.
[C] b–rF, (iF. ab) a ; q=nP. (iF. ab) 5.
[D] 0–r F. a rP. ab ; d=if'. b rh'. Cub.
[F] b–F (rR. ab) a ; a—iB (r.E. ab) 5.
Here we have
rF TH' . . . . . . +E". rF, bd=a.
There can be no faciend-zero; nor facient-modulus, unless for every vid
- F. b b = 17).
There may be a facient-Zero or a faciend-modulus.
—15–
Type IV.
[A] irP. Da-F0a–F, ab=irF. ab.
ºr F. Öa—rip'. Dai-iP. ab=r F. &I).
[B] b=rP. a. F. ab=rrF. (F. ab) as a+rF. b F ab–rrF. (F. ab) b.
[C] b=F, a rR', aſ-F. (FF. aſ a; a rºR. 5 rE. ab=r F. (FF, ab) b.
[E] D=rr F. a rºl". ab=r F (rr F. (tô) Cl; (t=F. B ºr F. &D=F (rr F. ab) b.
The principal feature is [E]
r). F a rh', ab=0; ri F rºB', ab=0.
It is to be noticed that in type I, the reverse, in tye III, the inverse,
and in type IV, the direct, are “commutative.” These three might possi-
bly be called commutative types, but the distinction appears artificial.
Type IV is of interest as being that of all ordinary addition, and of some
multiplication. It is evident that any zero or modulus is complete. This
is the only type permitting both a modulus and a zero without restricting
the vids of the field. It is to be noticed, however, that a zero or a modulus
is not necessary and systems might be constructed without them. I do
not mean to say that 0 and oo are not, for example, necessary to ordinary
addition, but that a system might be devised without them. It might
not apply to counting perhaps, as an interpretation or application of it,
but it would obey all the laws necessan'y to a commutative algorithm. Let
the purpose of the present article be emphasised again, namely; to classify
all possible algorithms; to show thus the inter-relations of those existing;
to point out gaps or unexplored regions in the whole realm of mathematics.
4. Consider next the limitations of first degree and third order.
Those of the second degree and third order manifestly reduce to a facter
which does not change any vid. The forms considered are of the form
F. C. F. & Fa-I, etc.
There are, at first glance, two hundred sixteen cases; but since the form
F. a F a F a-I, is equivalent to iE'a, iPa af'a-I, and since the three ele-
ments of any form may be changed cyclically, we need consider only those
beginning with Fa, rR'a, and rrr'a. This gives three classes:
CLASS I. FAMILY 1. Fa Fa Fa-I.
Foº Fa iH'O-I.
Fa H'a rFa-I.
Fa Fa ir Fa-I.
Foº Fa rifla-I.
Fa Fa ºr Fa-I. 6 cases.
FAMILY 2. Fo, iR'a rR'a-ſ.
Fa ib'a rrh'a-I. 2 cases.
FAMILY 3. Fa rR'a, iFa-i.
Fo, rRC, Ela-I.
Fa rR'a, irTa-I.
Fo, rFa riH'd—I.
- Fa rR'd rºR'a-I.
—16—
Fa trfa Fa-l.
Fa, irPºd, irP'a-I.
|Fa ir Fa riFal-I.
Fa, in FC, ºr Fa-I.
Fa rif'd iR'cº-I.
FC, rif'd rR'a-I.
Fo, riſ'd, irP'a-I.
Fa riH'a rif'a-I.
FC, riſ' a rr Rai-I.
For rr Fat TEq=I.
Fa rr Fa ir Fa-I.
Fa, ºr FC, rur'a-I.
Fa rºR'd, rrr'a-I. 18 cases.
CLAss II. FAMILY 1. FFa rR'a rR'a-I,
*Fa rFa ir Foſ-I.
rR'd rE'a riH'a-I.
rFa rR'a rrr'a–I. 4 cases.
FAMILY 2. rB'd, in FC, rif'cº-I.
rR'a wrP'a prFa–I. 2 cases.
FAMILY 2. YEO, rule'a, irP'a-I.
rR'd rif'd riH'a-I.
*Fº, rifa, rr R'a-I.
TEC rºR'a rif'a-I.
r.For rr Fa rFa-I. 5 Cases.
CLASS 1ſ.I. - ºr Fa rr Fa riFa-I.
rrE'a rr Fa rrºſa–T. 2 cases.
This gives a total of thirty-nine cases of this limitation type. These cases
might be investigated as those of the preceeding type were, but the com-
plete development is left for a future time. Some of these cases evidently
drop out upon farther investigation, as for example, class I, family 2, class
I, family 1, case 2, etc.
5. The next type of limitation is that of the first degree and fourth
order. But we can investigate a more general case with greater profit,
namely; the case of second degree. The two cases of fourth are, a and b
being any vids,
acFa yFb 2Fa włºb-I,
acFa yFa 2Fb wrºb–I,
since the two a's must succeed one the other or be separated by a b. But
the latter gives at once,
acFa y Fac = iwiR'b i2Fbc
tº Q a:Fa yFa = iwiR'a i2Fa, which is a form of the other.
The first form can evidently be written
facFa iyFb izſa iwiR'b=I,
and can be changed cyclically, since a and b are any vids. Hence we have,
—17—
as before, three classes, beginning respectively F, rP', rB'. The possible
twelve hundred ninety-six cases reduce then to one hundred seventy-five,
as follows:
CLASS I. FAMILY 1.
st
1
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31,
32.
FC,
FC.
FC,
FC,
FC.
Fa
FC,
FC.
FC,
FC,
FC.
FC.
FC.
FC,
FC,
FO,
FC,
FC,
Fa
FC,
FCſ
FC.
Fo,
FC.
FC.
FC,
FC,
FC.
FC.
B'a.
FG
FC.
rEa.
rR'C,
r.FO,
r.F.C.
Fb Fa tR'b=I, (t=i, ii, tr. .
Fb rb'a tRU=I, (t=i, r, ir. .
Fb irP'a tR'b=I, (t=i, r, . .
Fb riFa tR'b=I, (t= i, r . . . .
Fb rrr'a tR'a-I, (t=i, r . . .)
iFb FC, tR'b=I, (t=1, r . . ..)
iFU) r.Fa, tR'b=I, (t=r, tº . . .
*Fb irfa, tºb-I, (t=r, rr)
iFa riH'b trºb–1, (t=r)
r.FO FC. tR'b=I, (t=r, in', ri, rr)
rFb iFU trºb–I, (t=r, wr. . .)
rR'b rFa tRö==I, (t=i, r . . .)
rFb urb'a tR'b=I, (t=i, r, ir. .
rFb riFa tR'b=I, (t=i, r, . . .)
rFb rr Fa tR'b=I, (t=i, r, . . .)
trFO Fa tR'b=I, (t=ir, . . .)
ir Fö rFa tR'b=I, (t=i, r . .)
ir Fö ºr Fa tRU=I, (t=r, ir. .)
irPb riPa tR'b=I, (t=i, r . . .)
ir FU) rºR'a, trºb–I, (t=r, ir . .)
jºb FC, tR'b=I, (t=ri, Tr)
riFb iPa tR'b=I, (t=r, . . .)
rif'b rPatrºb–I, (t=r, wr . .)
rºlſ"b ºr Eq. trib-i, (t=r, 'll . . .)
.),
Fb Ra tR'b=I, (t=i, r, ir. . . .
..)
rif'b riPa tRU=I, (t=i, r, . . . .)
rif'b ºr Fa tR'b=I, (t=r, ir. . .)
rr Rºb Fo trºb–I, (t=rr)
rrFb rb'a tEb=I, (t=r, ir . .)
rºR'b irTa. tRU=I, (t=r, wr. . .)
rrFb riEa tºb-I, (t=i, r, . .)
rrE'b ºr Fa t|F b=I, (t=r, in . .)
Total:
rFb rſ'a tR'b=I, (t=r, ir ...)
rEb irP'a tRU=I, (t=ir, ri, rr)
rFO riFo, tR'b=I, (t=ir; ri, rr)
rFb rºFa, tR'b=I, (t=ur, ri, rr
rFa, in Fl) ºf 'g tEU=I, (t=ir, a'i, ºr
)
)
)
)
6 cases.
5 cases.
5 cases.
5 cases.
5 cases.
5 cases.
5 cases.
4 cases.
2 cases.
1 cases.
4 cases.
4 cases.
5 cases.
5 cases.
5 cases.
5 cases.
3 cases.
5 cases.
4 cases.
5 cases.
4 cases.
2 cases.
4 cases.
4 cases.
4 cases.
5 cases.
4 cases.
1 case.
4 cases.
4 cases.
5 cases.
4 cases.
133 cases.
5
º
rFC.
rFa
rF6.
PRO
irPb riH'a tRU=I, (t=ir, ri, rr
irPb rrr'a tR'b=I, (t=ir, ri, ºr
riFb rb'a tR'b=I, (t=ri, ºr)
riFb irPa tR'b=I, (t=ri, rr)
3.3.3.55
4 cases.
3 cases.
3 cases.
3 cases'
3 cases.
3 cases.
3 cases.
2 cases.
2 cases.
—18––
{{ 10. rR'a rif'b raſ'd tſ'U=I, (t=ir, ri, ºr) , 3 cases.
46 11, rR'a ri Fl) prR'a tRU=I, (t=ºr, ri, ry) , 3 cases.
{ % 12. Fa Prſ’b rR'a tºb-I, (t=rr) , l case.
{{ 13, rB'a rr]'b riſ'a tRU=l, (t=ir, ri, wr) , 3 cases.
& 4 14. FEa rr Rö rºRa trºb–I, (t=ri, rr) , 2 cases.
Total: 38 cases.
CLASS III. FAMILY 1. rrrº'a ºr Fö rr Fa tEb=I, (t=rr, ri) , 2 cases.
{{ 2. rºRa rr Fö rufa tºb-I, (t=ri) , , 1 case.
{ % 3. Fr. Fa rufºb rr Ed rufºb-I, 1 case.
Total: 4 cases.
This number will of course reduce to fewer, or special investigation.
These types are remarkable in that from each case we may produce
another form, which we may call an association-form, thus:
acFa yFb=Fa wrºb–I,
acFa yFbc=iwFb izBac.
By $ 1 [A], iracR. (yFbc) a-iwFb irizr'ca,
Or since a is any vid,
iracF (yFbc)=iwłP. bryºzR'c.
In case ac=ir, y=ii, w=i, 2=r, this becomes
F (Fbc)=Fb Fe,
the well-known “associative law.” The true position of this law is thus
made quite apparent. It is seen to be one among several such laws, any
one of which might furnish the fundamental basis of a whole set of algori-
thms. These algorithms would each possess peculiarities due to the law
determining them, and would have fields of application peculiarly suited
to their use.
6. The development seems now to have proceeded far enough to
furnish the elements of a science. That limitations of higher orders and
degrees might be deduced, and in the history of mathematics will be so
developed and used, is beyond question. That the vast fields of territory
yet unexplored in the realms so far opened up in this paper, will be inves-
tigated, is also beyond question. That new algorithms of exceeding power
will result therefrom, is not to be doubted. In conclusion, I shall indicate
One set of such algorithms, with some few applications.
—19–
CHAPTER III.
A Cyclic Trivid Algorithm.
1. The fundamental limitation taken for the distinguishing char-
acteristic of this group is type two of limitations of the first degree and
second order. For these F= r F=rrF,
iF=r, F=rif'.
If F. ab=c, then F. ca—b, F. bc=a,
iF. ab=e, then iF. be=a, iF. ea-b.
Thus the direct and inverse are similar in their properties. These are
cyclic, from the identities above, and are triad-cyclic from the fact that
three vids complete the cycle.
2. According to the statements of chapter II, A. §3, under type II,
there cannot be a zero. If there be a modulus, say m, then
F. aa-m,
for all vids. Thus © F. mm–m, so that m is indempotent.
Further; since if F. aq=m, F. ma–a=F. am, then every
vid is “commutative” with m. This coupled with the fact that F. mºm-m;
enables us to write for this modulus, 1, unity; thus connecting the
qualitative and the quantitative parts. [º
3. Without entering into the general formation of these algorithms
that have a modulus, the following cases of divid, tetravid, and pentavid
algorithms may be given. There is none of order three or six.
i j k l
i i j k l
7 j i l k
j k k l i j
l l k j i
i j k l m
k | }c m i j l
m m / j k i
|
The first and the second of these examples are also “associative,”

—20—
and “commutative,” So that it may easily be proved that in these cases
F=uF-FF=l rR'—rulº'-y). F.
4. Supposing there is no modulus, we find that there is no case of a
divid algorithm of this sort. Of the twelve possible trivid algorithms, there
are but two, for if the vids are a, b, c, then F. aa must equal a, or b (tak-
ing it equal to c amounts to the same thing). In the first case, F. boi=b or
c. But, filling out the possible relations of the latter, we easily find that
the case is impossible. In the former we find

In the other case, in which F. aq=b, we get at once F. ba-a, F. ab=a.
Therefore F. bb=c, and the diagram is
a D c

a b & C
b | C. C. b
G | C b C.
It is to be remembered, of course, that no row and no column can contain
the same letters twice, by the definition of facter.
5. As an example (interpretation, application) of one of these algori-
thms, I shall choose the first of those in the preceding section:
i j ſc
7 i le j
j le j i

k j i lº
I shall give this a purely geometrical interpretation.
6. Let three points, A, B, C, form a triangle. Attach to A the
“affix” i, to B, j; to C, le. Let the affix of any point in the plane be
p=xi, yj, Zic,
determined thus; divide AB, in the ratio y:X; join this point to the point
ſ —21—
C, and divide the segment so obtained in the ratio Z:y-HR; the point so
found is P, affix p. As examples. the median point is (i, j, k); the incenter
is ai, bj, cle); the excenters are (–ai, bj, cle), (ai, -bj, cle), (ai, bj, —ck); the
orthocenter is (tan A. i, tan B. j, tan C. le); the symmedian point is (a”, b%j,
c2k); the circumcenter is sin 2A. i, sin 2B.j, sin 20, k); the Brocard points
are cºa?i, a2bºj, b2C2k) and (aºb?i, b2C3), c*a*k).
7. As in the analytic method of homogeneous coordinates, many
propositions may be deduced from these fundamental assumptions, in the
same way as Hamilton exemplifies the use of vectors in his early chapters
in his Elements of Qwatermions. This development, however, must give
place to the matter more nearly at hand.
8. It will always be supposed that
X--y–H Z=1,
in the affix of any point.
9. The point
F. i (xi, yj, zk)=F. ie
is (xi, Zj, yk).
This is evidently a point on the ray from A to the isotomic conjugate of
the point (yj, Zk) with respect to B and C. It is that point at which a line
through P parallel to BC cuts this ray, since this ray and the one from A
to (yj. Zk) are divided in the same ratio.
10. The point
F. (ai. b), cle) (xi, yj, zle)
will be defined as that point determined by finding the three points
F. i (xi, yj, Zk),
F. j (xi. yj, Zk),
tº F. le (xi, yj, zk),
and dividing the lines joining them in the ratios a:b:c. The point so
determined is manifestly
(ax+bz-Hcy) i. (az-i-by--cx).j, (ay--bx +cz) k,
since this point is that represented by
a (xi+Zj-Hyk), b (zi-Hyj +xk), c (yi-Hs; +zk),
and xi+Zj+yk, etc., are the points determined above, as F. (xi, yj, zle,
etc. By the definitions given in $6, if P and R are two unit points, then
ap, bft is found by dividing the distance between them in the ratio b:a.
11. The isotomic conjugate of P is
l i _1 , | 1 k
x ‘’ y J; Z t =
For, since it is on the ray isotomically conjugate to AP, it is represented
by
wi,-- j, -—le.
'y J; Z.
—22—
But being on the other rays isotomic to BP, and CP, it is olso represented
by
. . 1
-1, V), TZT le,
1 l .
- t; — ), ule.
x y
These, all representing the same point,
S
W=—=—,
X
+=tv=+.
y y
1
—E -—=SUI.
Z
Therefore t-s=1, and w—- v=+u-4- &
* X y Z
12. In § 10 if a=b=c, the point ai, bj, cle, becomes the median point
of the triangle ABC. But in this case the popht
F. (i, j, k) (xi, yj, zk)=(i, j, k)
Whatever x, y, z, Hence, considering the method of determining it we
have the following theorem: -
If the point P be joined to the vertices of a triangle and the rays
from the vertices isotomically conjugate to these be drawn to meet rays
from P parallel to the opposite sides of the triangle, the points of intersec-
tion of each isotomic ray and the corresponding parallel determine a tri-
angle whose median paint is that of the original triangle. For any point
can thus be found a triangle co-medial with a given triangle.
13. The median point is clearly its own isotomic conjugate by § 11
as are also the points • *
(—i, j, łc),
(i, —j, k,),
(?, j, —,ſc),
which are the points determined by drawing at each vertex a line parallel
to the opposite side. The isotomic conjugate of the incenter is
(bci, caj, able);
of the orthocenter is (ctn A. i, ctn B, j, ctr, C. k), of the symmedian point is
b?cºi, c2a3), a2b^k,
of the circumcenter is
cSc 2A, i, csc 2.B. 7, csc 2C. ſc,
of the Brocard points are -
b°i, cºj, aºk and cºi, ałj, bºk.
14. I shall name the set of three points determined from P by the
process of § 9 the triad of P with respect to the fundamental triangle,
—23–
A B C. It has already been shown that every triad so determined has
the same median point as A B C, giving thus a doubly-infinite series of
co-medial triangles for A B C.
15. The triad of the median point is the median point. The triad
of each of the three points —i, j, k; i, -j, le; i, j, -k, is these three points.
The triad of the incenter is ai, CJ, ble;
ci, bj, ale;
bi, aj, ch:
The triad of the symmedian point is the set of points constituting the first-
Brocard triangle.
15. The six points
Xi, yj, Złc; (
Xi, Zj, y/c; (
yi, Xj, Z/c; (
yi, Zj, xk; (4
Zi, Xj, yk; (5
Zī, yj, xk; (6
are so related that the triad of
(1) is (2) (3)(6)),
(2) is (1) (4) (5),
(3) is (1) (4) (5)},
(4) is {(2) (3) (6)},
(5) is (2) (3) (6)},
(6) is (1) (4) (5)}.
They can then be arranged in two sets of three, each set being the
triad of any point of the other set.
It follows that every triad determines three points, of any one of
which, it is the triad.
The hexad of points above is called a self-conjugate hexad.
17. The general case of comediality would imply, if li, mi, nk'; pi,
q.j, rle,; ui, vſ, whe; are the three vertices of the triangle under considera-
tion that l-Hp-Hu-m--q+-v=n-Hr-i-w.
When this condition is fulfilled, the triangle is comedial with A B C.
Thus the two Brocard points and the point
b?c2i, cºaºj, a2bºk
form a triangle comedial with A B C. The latter point is the isotomic
conjugate of the symmedian point.
18. Propositions like the following are quite easily deduced: Find
the point of concurrency of rays from the vertices of a triangle to the
points of tangency of the opposite exscribed circle. This point is
(s—a) i, (s—b).j, (s—c) k,
i. e. S (i.j, k)–(ai, bj cle).
—24—
Hence it is collinear with the center of the inscribed circle and the median'
point.
lf L and M are two of the excenters,
L= —ai, bj, cle;
M= ai, -bj, cle,
they are collinear with C, for
L–|-M=2ck.
Therefore C divides LM in the ratio
LC:CM=a—-b-|-c:—a-i-b-i-c.
The ray from A to X on BC through K, the symmedian point, is-
divided at K in the ratio
AK; KX=b2+-c2:a?.
The orthocenter may be written in unit form
ctn B ctn C. i, ctn C ctn A. j, ctm A ctn B. le.
Since sin 2A-Hsin 2/3-i-Sin 20=sin A sin B sin C the circumcenter in unit.
form is
*śroºm.a.ſ.º.
But this is also equivalent to
—%(ctm B ctn (; i, ctn C ctn A. j, ctn A ctn B, k)
––3-2 (%i 36.j, Žák).
Hence the median point divides the line joining the circumcenter and the
orthocenter in the ratio of 1 to 2.
19. Enough has been written to show the extent and the applica-
bility of the method. The chief purpose here is not so much to develop
a new algorithm, but to show the importance of a few of those deduced in
the first two chapters. The aim has been to show on what principles rest the
very processes of mathematics themselves. It is an effort to grasp the secret
of Life of the subject. Perhaps the dissection has been a vivisection unto
“death, but it is hoped that it will not prove so, That the mathematician
that is, he who would be a master of the subject, and who would make,
polish, temper, and design the tools which the practical investigators in
the subject use, must himself first understand the general theory of tools
and tool-making, is so self-evident as to need no proof.
JAMES BYRNIE SHAw,
March 1, 1895. Jacksonville, Illinois.
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