〈GROUPS OF MARBLES.〉Let us now consider another form of Table which might readily occur to a boy playing with his marbles, or to a young lady with the balls of her solitaire board.{51}The boy may place a row of his marbles on the sand, at equal distances from each other,thus—He might then, beginning with the second, place two other marbles under each,thus—He might then, beginning with the third, place three other marbles under each group, and so on; commencing always one group later, and making the addition one marble more each time. The several groups would stand thusarranged—He will not fail to observe that he has thus formed a series of triangular groups, every group having an equal number of marbles in each of its three sides. Also that the side of each successive group contains one more marble than that of its preceding group.Now an inquisitive boy would naturally count the numbers in each group and he would find themthus—136101521He might also want to know how many marbles the thirtieth or any other distant group might contain. Perhaps he might go to papa to obtain this information; but I much fear papa would snub him, and would tell him that it was nonsense—that it was useless—that nobody knew the number, and so forth. If the boy is told by papa, that he is not able to answer the question, then I recommend him to pay careful attention to whatever that father may at any time say, for he has overcome two of the greatest obstacles to the acquisition{52}of knowledge—inasmuch as he possesses the consciousness that he does not know—and he has the moral courage to avow it.1313The most remarkable instance I ever met with of the distinctness with which any individual perceived the exact boundary of his own knowledge, was that of the late Dr. Wollaston.If 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 sameconstantdifference 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 standthus:—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.111123213631410415155162167287It is usual to call the third column thus formedthe 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, proceedthus:—Take the number of the group5Add 1 to this number, it becomes6Multiply these numbers together2)30Divide the product by 215This gives 15, the number of marblesin 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.14I 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 numbers1, 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 numbersis—Number.Table.1st Dif-ference.2nd Dif-ference.3rd Dif-ference.1133124641310105142015653521656He will at once perceive that this Table of the number of cannon balls contained in a triangular pyramid can be carried to any extent by simply adding successive differences, the third of which is constant.The next step will naturally be to inquire how any number in this Table can be calculated by itself. A little consideration will lead him to a fair guess; a little industry will enable him to confirm his conjecture.〈NUMBER IN EACH PILE.〉It will be observed at p. 49 that in order to find{56}independently any number of the Table of the price of butchers’ meat, the following rule wasobserved:—Take the number whose tabular number is required.Multiply it by the first difference.This product is equal to the required tabular number.Again, at p. 53, the rule for finding any triangular numberwas:—Take the number of the group5Add 1 to this number, it becomes6Multiply these numbers together2)30Divide the product by 215This is the number of marbles in the 5th group.Now let us make a bold conjecture respecting the Table of cannon balls, and try thisrule:—Take the number whose tabularnumber is required, say5Add 1 to that number6Add 1 more to that number7Multiply all three numbers together2)210Divide by 2105The real number in the 5th pyramid is 35. But the number 105 at which we have arrived is exactly three times as great. If, therefore, instead of dividing by 2 we had divided by 2 and also by 3, we should have arrived at a true result in this instance.The amended rule istherefore—{57}Take the number whosetabular number isrequired, saynAdd 1 to itn+ 1Add 1 to thisn+ 2Multiply these threenumbers togethern× (n+ 1) × (n+ 2)Divide by 1 × 2 × 3.The result is(n(n+ 1)(n+ 2))/6This rule will, upon trial, be found to give correctly every tabular number.By similar reasoning we might arrive at the knowledge of the number of cannon balls in square and rectangular pyramids. But it is presumed that enough has been stated to enable the reader to form some general notion of the method of calculating arithmetical Tables by differences which are constant.〈ASTRONOMICAL TABLES.〉It may now be stated that mathematicians have discovered that all the Tables most important for practical purposes, such as those relating to Astronomy and Navigation, can, although they may not possess any constant differences, still be calculated in detached portions by that method.Hence the importance of having machinery to calculate by differences, which, if well made, cannot err; and which, if carelessly set, presents in the last term it calculates the power of verification of every antecedent term.
〈GROUPS OF MARBLES.〉
Let us now consider another form of Table which might readily occur to a boy playing with his marbles, or to a young lady with the balls of her solitaire board.{51}
The boy may place a row of his marbles on the sand, at equal distances from each other,thus—
He might then, beginning with the second, place two other marbles under each,thus—
He might then, beginning with the third, place three other marbles under each group, and so on; commencing always one group later, and making the addition one marble more each time. The several groups would stand thusarranged—
He will not fail to observe that he has thus formed a series of triangular groups, every group having an equal number of marbles in each of its three sides. Also that the side of each successive group contains one more marble than that of its preceding group.
Now an inquisitive boy would naturally count the numbers in each group and he would find themthus—
136101521
He might also want to know how many marbles the thirtieth or any other distant group might contain. Perhaps he might go to papa to obtain this information; but I much fear papa would snub him, and would tell him that it was nonsense—that it was useless—that nobody knew the number, and so forth. If the boy is told by papa, that he is not able to answer the question, then I recommend him to pay careful attention to whatever that father may at any time say, for he has overcome two of the greatest obstacles to the acquisition{52}of knowledge—inasmuch as he possesses the consciousness that he does not know—and he has the moral courage to avow it.13
13The most remarkable instance I ever met with of the distinctness with which any individual perceived the exact boundary of his own knowledge, was that of the late Dr. Wollaston.
13The most remarkable instance I ever met with of the distinctness with which any individual perceived the exact boundary of his own knowledge, was that of the late Dr. Wollaston.
If 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 sameconstantdifference 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 standthus:—
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.111123213631410415155162167287
It is usual to call the third column thus formedthe 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, proceedthus:—
Take the number of the group5Add 1 to this number, it becomes6Multiply these numbers together2)30Divide the product by 215This gives 15, the number of marblesin the 5th group.
5
6
2)30
15
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.14I 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.”
“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.
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 numbersis—
Number.Table.1st Dif-ference.2nd Dif-ference.3rd Dif-ference.1133124641310105142015653521656
He will at once perceive that this Table of the number of cannon balls contained in a triangular pyramid can be carried to any extent by simply adding successive differences, the third of which is constant.
The next step will naturally be to inquire how any number in this Table can be calculated by itself. A little consideration will lead him to a fair guess; a little industry will enable him to confirm his conjecture.
〈NUMBER IN EACH PILE.〉
It will be observed at p. 49 that in order to find{56}independently any number of the Table of the price of butchers’ meat, the following rule wasobserved:—
Take the number whose tabular number is required.
Multiply it by the first difference.
This product is equal to the required tabular number.
Again, at p. 53, the rule for finding any triangular numberwas:—
Take the number of the group5Add 1 to this number, it becomes6Multiply these numbers together2)30Divide the product by 215
5
6
2)30
15
This is the number of marbles in the 5th group.
Now let us make a bold conjecture respecting the Table of cannon balls, and try thisrule:—
Take the number whose tabularnumber is required, say5Add 1 to that number6Add 1 more to that number7Multiply all three numbers together2)210Divide by 2105
5
6
7
2)210
105
The real number in the 5th pyramid is 35. But the number 105 at which we have arrived is exactly three times as great. If, therefore, instead of dividing by 2 we had divided by 2 and also by 3, we should have arrived at a true result in this instance.
The amended rule istherefore—{57}
Take the number whosetabular number isrequired, saynAdd 1 to itn+ 1Add 1 to thisn+ 2Multiply these threenumbers togethern× (n+ 1) × (n+ 2)Divide by 1 × 2 × 3.The result is(n(n+ 1)(n+ 2))/6
n
n+ 1
n+ 2
n× (n+ 1) × (n+ 2)
(n(n+ 1)(n+ 2))/6
This rule will, upon trial, be found to give correctly every tabular number.
By similar reasoning we might arrive at the knowledge of the number of cannon balls in square and rectangular pyramids. But it is presumed that enough has been stated to enable the reader to form some general notion of the method of calculating arithmetical Tables by differences which are constant.
〈ASTRONOMICAL TABLES.〉
It may now be stated that mathematicians have discovered that all the Tables most important for practical purposes, such as those relating to Astronomy and Navigation, can, although they may not possess any constant differences, still be calculated in detached portions by that method.
Hence the importance of having machinery to calculate by differences, which, if well made, cannot err; and which, if carelessly set, presents in the last term it calculates the power of verification of every antecedent term.