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 2nd part of the divisor.

Square the 2nd figure for the 3d part of the divisor

Add the 3 parts of the divisor together multiply the sum by the 2d 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 1st part of the divisor; 30 times the product of the part already found by the figure you are finding for the 3d product.

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

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

Note 2d 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 PikeA New and Complete System of Arithmetick. As his first example, Pike calculates the cube root of 436036824287 so that Macdonald抯 teacher, George Baxter, probably based his lessons on Pike抯 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 揅onsider how often this divisor is contained in the dividend 厰 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抯 part. Teachers often gave old cyphering books to their students as guides for the students to follow. This was near the end of Macdonald抯 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.