I make use of my last notation for relatives. I write li to mean that two objects l and i are connected. These two objects generally pertain to different universes; thus, l may be a character and i a thing. But there is no reason why I should not, instead of li, write (l, i), except that the first way is more compact. A line over an expression negatives it, so that
I use the signs of addition and multiplication as in the modified Boolian calculus. Thus, limj means that l and i are connected and also m and j. While li + mj means that either l and i are connected or else m and j.
I further use the symbols Σ and Π as follows. Σi means that i is something suitably selected from the universe of i’s, while Πi means that i may be anything however taken from the universe of i’s. For example, ΣiΠjlij will mean that an object i can be found such that, taking any object j whatever, i and j are related in the way signified by l. While ΠiΣjlij will mean that any object whatever, i, being taken, an object j can then be found such that i has the relation to j signified by l.
Numbers are characters of collections; or, since every collection may be considered as a class defined as having a certain essential character, we may say that numbers are certain characters of characters. For example, if we write 2l to signify that the collection l is 2 in number, we may write
That is to say, take any collection l, then, either this is not 2 in number, or if it is, take any objects, i, j, k, then either i or j or k does not belong to the collection l, or if they all do, taking any character m whatever, either i and j both possess or both want this character, or if they differ in this respect, taking any character n whatever, either i and k both possess or both want this character, or if they also differ, then taking any character p whatever, j and k either both possess or both want this character. This is merely given as an example of the mode of writing a proposition so as to analyze its meaning completely. We should ordinarily express it by saying that if any objects i, j, k all belong to a collection of two objects, some pair of them are identical. The fact that 2 is the character of a character is shown by its subjacent index, l, appearing on the line, itself having indices, i, j, k.
The comparison of collections with regard to their number depends upon putting their members into correspondence with one another. How, then, shall we write that for every l there is an m? We must say that there is a relation, r, such that every l is in this relation to some m and no two different ones to the same m.
This does not appear to differ from the broader
As an abbreviation for this we may write Glm, that is, there are at least as many m’s as l’s.
16
Promptuarium of Analytical Geometry
| c. 1890 | Houghton Library |
Let P1 and P2 be any two points.
Now consider this expression
λP1 + (1 − λ)P2
where λ is a number. P1 and P2 are not numbers, and therefore the binomial cannot be understood exactly as in ordinary algebra; but we are to seek some meaning for it which shall be somewhat analogous to that of algebra. If λ = 0, it becomes
0P1 = 1P2
and this we may take as equal to P2, making 0P1 = 0 and 1P2 = P2. Then if λ = 1, the expression will become equal to P1. When λ has any other value, we may assume that the expression denotes some other point, and as λ varies continuously we may assume that this point moves continuously. As λ passes through the whole series of real values, the point will describe a line; and the simplest assumption to make is that this line is straight. That we will assume; but at present we make no further assumption as to the position of the point on the line when λ has values other than 0 and 1. We may write
λP1 + (1 − λ)P2 = P3.
Transposing P3 to the first side of the equation and multiplying by any number, we get an equation of the form
a1P1 + a2P2 + a3P3 = 0
where a1 + a2 + a3 = 0. Such an equation will signify that P1, P2, P3 are in a straight line; for it is equivalent to
Let P1, P2, P3 be any three points, not generally in a straight line. Then λP1 + (1 − λ)P2 may be any point in the straight line through P1 and P2 and μ(λP1 + (1 − λ)P2) + (1 − μ)P3, where μ is a second number, will be any point in a line with that point and with P3. But that plainly describes any point in the plane through P1, P2, P3 so that
P4 = a1P1 + a2P2 + a3P3
where a1 + a2 + a3 = 1 denotes any point in that plane. By transposing and multiplying by any number, we can give this the form
b1P1 + b2P2 + b3P3 + b4P4 = 0
where b1 + b2 + b3 + b4 = 0. To avoid the necessity of the second equation, we may put for b1, b2, b3, b4, four algebraical expressions which identically add up to zero; and may write
(a − b − c)P1 + (− a + b − c)P2 + (− a − b + c)P3
+ (a + b + c)P4 = 0.
This expression will signify that P1, P2, P3, P4 lie in one plane.
I will now give an example to show the utility of this notation. Let
P1, P2, P3, P4 be any four points in a plane. Assume for the equation connecting them
(a − b − c)P1