that 'x varies as y' is written y kx, where k is called a factor of proportionality, which enables Us to convert a statement of variation into an equation. If y varies inversely as *, we write A multiplication by a
constant thus introduces a relation of proportionality, hence the importance of proportionality in a world where constants are present.
It must also be noticed that the two fundamental operations of the calculus are linear without being equivalent to multiplication by a constant. These are: 'the derivative of the sum is the sum of the derivatives', that is, , and 'the integral of the sum is the sum of the integrals', that is . But as the fundamental notion of the
calculus is to substitute for a given function a linear function, in other words, to deal with curves as the limits of vanishingly small straight lines, this linearity underlies structurally all fundamental assumptions of the calculus, and one might say with Weyl that 'one here uses the exceedingly fruitful mathematical device of making a problem linear by reverting to infinitely small quantities'.1
A vector is defined roughly as a line-segment which has a definite direction and magnitude, and any quantity which can be represented by such a segment is defined as a vector quantity.
The addition of vectors is defined by the law of the parallelogram, as in the case of two forces. It should be noticed that because of this definition the sum of two vectors differs in general from the arithmetical sum of the lengths, and only collinear, or parallel vectors obey the arithmetical summation law.
The introduction by definition of mathematical entities which obey different laws from the usual arithmetical laws is an important structural and methodological innovation. It gives us the useful precedent of defining our operations to suit our needs. The vector calculus accepted as the definition of the sum of two vectors the law established experimentally in physics for the sum of two forces; and so the vector calculus from the beginning was structurally a particularly useful language in physics. Only since Einstein has the value and importance of the vector calculus for physics become generally appreciated.
If we have two vectors, a and b, starting from a common origin O and complete the parallelogram as in Fig. 1, then the diagonal of the parallelogram will be the required sum, a+b, by definition.
If we choose two co-initial vectors of unit length, one on the X axis, and the other on the Y axis, and call them i and j, we can always represent any vector x as the sum of two vectors, one of which is the projection
of x on the X axis, and the other the projection of x on the Y axis. (See Fig. 2.) Let us call these vectors x' and x" respectively. Then, by definition.
But differs from i in length only, hence it can be obtained by multiplying i by 39