Popular Science Monthly/Volume 10/April 1877/The Origin and Curiosities of the Arabic Numerals


By D. V. T. QUA.

IN an article on the "Origin of the Numerals," published in The Popular Science Monthly for January, 1876, the writer remarks: "Having never met with any explanation of the origin of the numerals, or rather of the figures symbolizing them, perhaps I am right in supposing that nothing satisfactory is known of it."

The history of the Arabic or decimal notation is somewhat as follows: The characters of this notation were introduced into Europe, during the tenth century, by the Crusades. From the Arabic, these characters have been traced to the sacred books of the Brahmans of India. It was long supposed that for our modern arithmetic we were indebted to the Arabians. But this, as we have seen, is not the case. The Hindoos communicated a knowledge of it to the Arabians, and we have been unable to trace it beyond the Hindoos: hence we must concede the honor to them of its invention.

To the Arabians, however, belongs the honor of introducing arithmetic into Europe. It was the Arabians who took the torch from the Orient and passed it along toward the Occident, when "westward the star of empire took its way."

The origin of the characters came, undoubtedly, from the fact that the Orientals first learned to count on their fingers and thumbs, and from this originated the ten characters employed, and originally called digits, from the Latin word digitus, signifying finger. In keeping accounts among the Orientals, one mark represented one finger, or number, thus: PSM V10 D757 Numerals 1.png. Two horizontal marks, with a connecting line, stood for two, thus: PSM V10 D757 Numerals 2.png. Three horizontal marks, with connecting lines, would stand for three, thus: PSM V10 D757 Numerals 3.png; and four marks in the form of a square, or a triangle, would stand for four, thus: PSM V10 D757 Numerals 4.png. Five marks in this form, PSM V10 D757 Numerals 5.png, was the original figure five in this notation; six marks, thus, PSM V10 D757 Numerals 6.png, the original figure six. The figure seven was made by marks representing two squares with one of the lines wanting, thus: PSM V10 D757 Numerals 7.png. The figure eight was made by placing two squares near each other, thus: PSM V10 D757 Numerals 8.png; and nine by adding one or more marks to the two squares representing eight, thus: PSM V10 D757 Numerals 9.png. The zero, or cipher, was originally a circle, and seems to have come from counting around the fingers and thumbs. Hence, once around was denoted by one finger, or character, representing one, thus: PSM V10 D757 Numerals 1.png and PSM V10 D757 Numerals 0.png; twice around, by PSM V10 D757 Numerals 2.png and PSM V10 D757 Numerals 0.png . From this last arrangement seems to have come the fundamental law of the decimal notation in which its superior utility consists, and upon which quite recently has been based the metric system of weights and measures. By placing any of the digits in the place of the zero to make the numbers between ten and twenty, we have the law established. The science of arithmetic, like all other sciences, was very limited and imperfect at the beginning, and the successive steps by which it has reached its present extension and perfection have been taken at long intervals, and among different nations. It has been developed by the necessities of business, by the strong love for mathematical science, and by the call for its higher offices by other sciences, especially that of astronomy. In its progress, we find that the Arabians discovered the method of proof by casting out the 9's, and that the Italians early adopted the practice of separating numbers into periods of six figures, for the purpose of enumerating them. The property of the number 9 affords an ingenious method of proving each of the fundamental operations in arithmetic, and it seems to be an incidental attribute of this number. It arises from the law of increase in the decimal notation. It universally belongs to the number that is one less than the radix of the system of notation. And in this connection it may not be irrelevant to state some facts or curiosities with regard to this number 9. It cannot be multiplied away, or got rid of in any manner. Whatever we do, it is sure to turn up again, as was the body of Eugene Aram's victim. One remarkable property of this figure (said to have been discovered by W. Green, who died in 1794) is, that all through the multiplication-table the product of 9 comes to 9. Multiply any number by 9, as 9×2=18, add the digits together, 1+8=9. So it goes on until we reach 9×11=99. Very well add the digits 9+9=18, and 1+8=9. Going on to any extent it is impossible to get rid of the figure 9. Take any number of examples at random, and we have the same result. For instance, 339×9=3,051. Add the digits 3+0+5+1=9. Take one more, 5,071×9=45,639, and the sum of the digits, 4+5+6+3+9=27, and 2+7=9.

The French mathematicians found out another queer thing about this number, namely: if we take any row of figures, and, reversing their order, make a subtraction, and add the digits, the final sum is sure to be 9. For example, 5,071-1,705=3,366; add these digits 3+3+6+6=18, and 1+8=9. The same result is obtained if we raise the numbers so changed to their squares or cubes. Starting with 62, and reversing the digits, we have 26, then 62-26=36, and 3+6=9. The squares of 26 and 62 are respectively 676 and 3,844, and 3,844-676=3,168; add 3+1+6+8=18, and 1+8=9. This may be exemplified in another way. Write down any number, as, for example, 7,549,132, subtract the sum of its digits 7+5+4+9+1+3+2=31, and 7,549,132-31= 7,549,101. Add these digits, 7+5+4+9+1+0+1=27, and 2+7=9.

But we have extended already this article to a greater length than we intended, simply wishing to give the origin and history of the decimal notation as far as it can be traced, and will close by stating that this notation is every way adapted to the practical operations of business, as well as the most abstruse mathematical investigations. In whatever light it is viewed, the decimal notation must be regarded as one of the most striking monuments of human ingenuity, and its beneficial influence on the progress of science and the arts, on commerce and civilization, must win for its unknown author the everlasting admiration and gratitude of mankind.