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.〉