Passages from the Life of a Philosopher. Charles Babbage

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СКАЧАТЬ papa fail to inform him, let him go to mamma, who will not fail to find means to satisfy her darling’s curiosity. In the meantime the author of this sketch will endeavour to lead his young friend to make use of his own common sense for the purpose of becoming better acquainted with the triangular figures he has formed with his marbles.

      〈SECOND DIFFERENCE CONSTANT.〉

      In the case of the Table of the price of butchers’ meat, it was obvious that it could be formed by adding the same constant difference continually to the first term. Now suppose we place the numbers of our groups of marbles in a column, as we did our prices of various weights of meat. Instead of adding a certain difference, as we did in the former case, let us subtract the figures representing each group of marbles from the figures of the succeeding group in the Table. The process will stand thus:—

Number of the Group.	Table.	1st Difference.	2nd Difference.
Number of Marbles in each Group.	Dif­fer­ence be­tween the num­ber of Mar­bles in each Group and that in the next.
1	1	1	1
2	3	2	1
3	6	3	1
4	10	4	1
5	15	5	1
6	21	6
7	28	7

      It is usual to call the third column thus formed the column of {53} first dif­fer­ences. It is evident in the present instance that that column represents the natural numbers. But we already know that the first difference of the natural numbers is constant and equal to unity. It appears, therefore, that a Table of these numbers, representing the group of marbles, might be constructed to any extent by mere addition—using the number 1 as the first number of the Table, the number 1 as the first Difference, and also the number 1 as the second Difference, which last always remains constant.

      Now as we could find the value of any given number of pounds of meat directly, without going through all the previous part of the Table, so by a somewhat different rule we can find at once the value of any group whose number is given.

      Thus, if we require the number of marbles in the fifth group, proceed thus:—

Take the number of the group	5
Add 1 to this number, it becomes	6
Multiply these numbers together	2)30
Divide the product by 2	15
This gives 15, the number of marbles in the 5th group.

      If the reader will take the trouble to calculate with his pencil the five groups given above, he will soon perceive the general truth of this rule.

      We have now arrived at the fact that this Table—like that of the price of butchers’ meat—can be calculated by two different methods. By the first, each number of the Table is calculated independently: by the second, the truth of each number depends upon the truth of all the previous numbers.

      〈TRIANGULAR NUMBERS.〉

      Perhaps my young friend may now ask me, What is the use of such Tables? Until he has advanced further in his {54} arithmetical studies, he must take for granted that they are of some use. The very Table about which he has been reasoning possesses a special name—it is called a Table of Triangular Numbers. Almost every general collection of Tables hitherto published contains portions of it of more or less extent.

Above a century ago, a volume in small quarto, containing the first 20,000 triangular numbers, was published at the Hague by E. De Joncourt, A.M., and Professor of Philosophy.14 I cannot resist quoting the author’s enthusiastic expression of the happiness he enjoyed in composing his celebrated work:

      “The Trigonals here to be found, and nowhere else, are exactly elaborate. Let the candid reader make the best of these numbers, and feel (if possible) in perusing my work the pleasure I had in composing it.

      “That sweet joy may arise from such contemplations cannot be denied. Numbers and lines have many charms, unseen by vulgar eyes, and only discovered to the unwearied and respectful sons of Art. In features the serpentine line (who starts not at the name) produces beauty and love; and in numbers, high powers, and humble roots, give soft delight.

      “Lo! the raptured arithmetician! Easily satisfied, he asks no Brussels lace, nor a coach and six. To calculate, contents his liveliest desires, and obedient numbers are within his reach.”

14 ‘On the Nature and Notable Use of the most Simple Trigonal Numbers.’ By E. De Joncourt, at the Hague. 1762.

      〈SQUARE NUMBERS.〉

      I hope my young friend is acquainted with the fact—that the product of any number multiplied by itself is called the square of that number. Thus 36 is the product of 6 multiplied by 6, and 36 is called the square of 6. I would now recommend him to examine the series of square numbers

      1, 4, 9, 16, 25, 36, 49, 64, &c.,

      {55} and to make, for his own in­struc­tion, the series of their first and second differences, and then to apply to it the same reasoning which has been already applied to the Table of Triangular Numbers.

      〈CANNON BALLS.〉

      When he feels that he has mastered that Table, I shall be happy to accompany mamma’s darling to Woolwich or to Portsmouth, where he will find some practical illustrations of the use of his newly-acquired numbers. He will find scattered about in the Arsenal various heaps of cannon balls, some of them triangular, others square or oblong pyramids.

      Looking on the simplest form—the triangular pyramid—he will observe that it exactly represents his own heaps of marbles placed each successively above one another until the top of the pyramid contains only a single ball.

      The new series thus formed by the addition of his own triangular numbers is—

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Number.	Table.	1st Dif- ference.	2nd Dif- ference.	3rd Dif- ference.
1	1	3	3	1
2	4	6	4	1