Chalana-Kalana: By motion or by applying a shift

Chalana-Kalana, by motion, is about how quantities change, captured by differences: look at how each term of a sequence shifts from the last, and patterns emerge, an idea the book extends all the way to calculus.

Theory

How things change

Many of the sutras so far were about a single calculation. Chalana-Kalana, 'by motion', is about something more dynamic: how a quantity changes as you move from one value to the next. The most accessible form of this idea is the study of differences, how much each term of a sequence shifts from the one before.

The book extends this principle all the way to calculus (rates of change), but here we meet its clear, checkable arithmetic form: successive differences. As always, we take it from Bharati Krishna Tirtha's 1965 book and direct you to the prescribed text for the fuller calculus treatment.

Theory

Successive differences

Take the sequence of square numbers: 1, 4, 9, 16, 25.

Look at the first differences, each term minus the previous one: 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9. So the first differences are 3, 5, 7, 9 (the odd numbers).

Now take the second differences, the differences of those: 5 - 3 = 2, 7 - 5 = 2, 9 - 7 = 2. They are constant at 2.

That constant second difference is a signature: it reveals that the original sequence is quadratic (the squares). Studying how a sequence 'moves' through its differences exposes its underlying pattern, that is the spirit of Chalana-Kalana.

Practical

Successive differences in Python (verified)

def differences(seq):
    return [seq[i+1] - seq[i] for i in range(len(seq) - 1)]

squares = [1, 4, 9, 16, 25]
d1 = differences(squares)   # [3, 5, 7, 9]  first differences
d2 = differences(d1)        # [2, 2, 2]     second differences (constant)

print(d1)   # [3, 5, 7, 9]
print(d2)   # [2, 2, 2]  -> constant, revealing a quadratic pattern

This example runs in Gri-Learn on the web, where you can edit it and see the output.

Formula

Differences are the discrete side of change

The idea of measuring change by differences is the everyday, whole-number cousin of the calculus idea of a derivative (an instantaneous rate of change). Where calculus asks 'how fast is this changing at a point?', successive differences ask 'how much did it change from one step to the next?'.

The book's Chalana-Kalana connects these, using the pattern of change to compute and to reason. We stay with the concrete difference form here; for the calculus applications, consult the prescribed text. The takeaway: to understand a sequence, study how it moves.

Quiz

For the square numbers 1, 4, 9, 16, 25, what are the first differences (each term minus the previous)?

2, 2, 2, 2, all equal
3, 5, 7, 9, the odd numbers
1, 4, 9, 16, the squares again
5, 5, 5, 5, all equal to 5

Show the answer

3, 5, 7, 9, the odd numbers

Subtract each term from the next: 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9, giving 3, 5, 7, 9, the odd numbers. Option A describes the SECOND differences (the differences of 3, 5, 7, 9 are 2, 2, 2), not the first. Option C just repeats the original sequence, which is not what differencing does. Option D is a wrong constant. First differences measure the step-to-step change; for the squares they turn out to be the odd numbers, and differencing again gives the constant 2 that marks a quadratic.

Think first

Why do constant second differences signal a square (quadratic) pattern?

The squares gave constant second differences of 2. Why does that constancy reveal a quadratic? Then tap.

Show the answer

Because differencing a sequence is the discrete version of differentiating a function, and each difference lowers the 'degree' by one, so a quadratic needs exactly two rounds of differencing to become constant. Think about the parallel with calculus. If a quantity grows like n squared (a quadratic, degree 2), its rate of change (first derivative) grows like 2n (linear, degree 1), and the rate of change of THAT (second derivative) is a constant (2). Successive differences behave the same way on whole-number sequences: the FIRST differences of the squares (1, 4, 9, 16, 25) come out as 3, 5, 7, 9, which is a LINEAR pattern (they increase by a fixed step), and the SECOND differences of a linear pattern are CONSTANT (here 2). So reaching a constant after exactly two differencings is the fingerprint of a degree-2 (quadratic) sequence, just as a constant second derivative marks a quadratic function. More generally, a sequence whose k-th differences first become constant is a polynomial of degree k: one round for linear, two for quadratic, three for cubic. This is why analysts and mathematicians use difference tables to detect and identify the hidden formula behind a sequence of numbers. Constant second differences equals quadratic; the depth at which differences flatten reveals the degree. Change, differenced enough times, gives up the pattern.

Summary

Key takeaways

Chalana-Kalana ('by motion') concerns how quantities change; its accessible form is successive differences.
First differences of a sequence are each term minus the previous one.
For the squares 1, 4, 9, 16, 25: first differences are 3, 5, 7, 9 (the odd numbers).
Second differences (the differences of those) are constant at 2, marking a quadratic pattern.
Differencing is the discrete cousin of the calculus derivative (rate of change); the book extends the idea to calculus (see the prescribed text).
A sequence whose k-th differences first become constant is a degree-k polynomial.
Memory hook: study how a sequence moves, difference it; constant second differences mean a square pattern.