Cube Root

The
cube root of a given number is the value that, when multiplied by itself three
times, yields the given number. For example, is the cube root of since . In his first numerical
example, Macdonald shows that the cube root of 436036824287 is 7583.

Macdonald
does not define cube roots. Instead, he launches immediately into an algorithm
to find the cube root of a number. I think at this point Macdonald was blindly
copying material without care since, for example, he runs the first three
sentences together without punctuation and pluralizes a word in what should be
the second sentence when the word should have been singular. Here is his
algorithm as he has written it:

Divide the number
into periods of 3 figures each find the greatest cube contained in the left
hand periods the root of it, will be the first figure of the root; subtract the
root from the period, and annex the second period to the remainder.

Multiply the square
of the figure already found by 300; the product will be the first part of the
divisor.

Consider how often
the divisor is contained in the dividend, the number of times will be the
second figure of the root.

Multiply the first
figure by the second figure by 30; the product will be the 2^{nd} part
of the divisor.

Square the 2^{nd}
figure for the 3^{d} part of the divisor

Add the 3 parts of
the divisor together multiply the sum by the 2^{d} figure of the root;
subtract the product from the dividend and annex the third period to the
remainder ―

Proceed in this
manner till the given number of the period be exhausted always taking 300 times
the square of the root already found in the 1^{st} part of the divisor;
30 times the product of the part already found by the figure you are finding
for the 3^{d} product.

If there be a
remainder the root may be continued to decimals by annexing cyphers to every
successive remainder.

Note 1^{st} In the extraction of decimals & fractions &c will
proceed in a similar manner to that direct in square root

Note 2^{d} The labour of finding the cube root may be often abridged by
taking a factor or factors of it.

The
algorithm is the same as, but worded differently from, an algorithm given in Nicholas Pike’s
*A New and Complete System of Arithmetick*. As his first example, Pike calculates the cube
root of 436036824287 so that Macdonald’s teacher, George Baxter, probably based
his lessons on Pike’s book.

Here
are calculations for the cube root of 436036824287 that Macdonald carried out:

1. Dividing the number
into periods of the three yields .

2. The greatest number
yielding a cube less than is , Hence the cube root is of the form .

3. Subtract from 436 to get .

4. Bring down the next
three digits to get .

5. The way Macdonald
has written the next set of steps is a little confusing. When he says “Consider
how often this divisor is contained in the dividend …” he means the divisor
that consists of the three parts that are described in the next three lines of
the algorithm. One wonders if Macdonald really understood what he was doing. Suppose
is the next digit so that the cube root is now
of the form . It is not stated,
but it can be deduced from looking at modern algorithms that we need to find
the greatest number such that divides . In his working of
the problem Macdonald has with no intermediate
attempts at other numbers. At the divisor is . The cube root is now
of the form .

6. Calculate
. This yields .

7. Bring down the next
three digits to get .

8. Now find the
greatest number such that divides . Macdonald has with again no intermediate attempts at other
numbers. At the divisor is . The cube is now of
the form .

9. Calculate
. This yields .

10. Bring down the last
three digits to get .

11. Now find the
greatest number such that divides . Macdonald has which yields a divisor of . Since then is the exact value of the cube root.

The fact
that Macdonald has run through a fairly complex algorithm for a twelve year old
without providing any intermediate calculations says one of two things: (1) He did
a lot of work on a slate on the side and then made a condensed fair copy
afterward; or (2) he just copied something that was given to him, perhaps a
complete cyphering book from another student. The second possibly was not
cheating on Macdonald’s part. Teachers often gave old cyphering books to their
students as guides for the students to follow. This was near the end of
Macdonald’s mathematics course and he may have been losing interest, especially
with such a complex subject. His lack on interest
shows in his treatment of the final topic in his notebook, “Application and Use of the Cube
Root.”