Popular Science Monthly/Volume 16/December 1879/Early Methods in Arithmetic

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EARLY METHODS IN ARITHMETIC.
By E. O. VAILE.

IN our day arithmetic is looked upon as a science of which every boy at fourteen ought to be master. Such was not the case a century or so back. In England, as well as upon the Continent, arithmetic was long considered a higher branch of science, and a university study, like geometry. In part, this is accounted for by the strong conviction which has always possessed mankind until within the last two hundred years, that numbers have about them very potent and mystical properties. During the middle ages this science had its skilled professors. The partial title of a work gives an idea of its exalted claims even after the time of Shakespeare and Bacon. The book appeared in London in 1624. Its title-page read thus: "The Secrets of Numbers according to Theological, Arithmetical, Geometrical, and Harmonical Computation. Pleasing to read, profitable to understande, opening themselves to the capacities of both learned and unlearned; being no other than a Key to lead Man to any Doctrinal Knowledge whatsoever."

But, in addition, there was difficulty and complexity in the science as practiced then that made it no boy's play. Even making allowance for the great advantage of "being used to a thing," the middle-age processes in the fundamental rules were often much more intricate than those practiced nowadays. In his incomparable history of the science of arithmetic, in the "Encyclopædia Metropolitana," Dr. Peacocke gives many interesting illustrations, some of which will doubtless strike the reader as novel. Some of their steps are easily explained, but others are by no means so simple. It might prove of interest and advantage to test the higher grades in some modern schools in regard to their actual comprehension of the first four rules by requiring them to explain the philosophy, not the process merely, of a few of these mediæval "sums." Explanations further than a description of the process are purposely omitted.

In subtraction they usually began at the left hand instead of the right. Inconvenient as it is, the method was continued as late as the end of the sixteenth century. The difference was placed above the numbers instead of below.

Process
18769 remainder.
54612 minuend.
35843 subtrahend.
1111

Process
06779 remainder
2991
30024 minuend.
23245 subtrahend.

Example 1. Subtract 35843 from 54612. When the digits in the subtrahend are greater than those in the minuend, units are placed beneath them as in the example; 3 being increased by the unit the next place to the right, and similarly for 5, 8, and 4.

Example 2. Subtract 23245 from 30024. Of course with such an arrangement it is of no consequence whether the operation proceeds from right to left or from left to right. It will be easily seen how the substituted minuend is obtained, with the exception of the one ten. Suppose the figure 4 in the subtrahend had been 1; then to what device would the boys and girls of the time of Luther and of Queen Elizabeth have had to resort to save their credit?

There is reason for thinking that the modern method of subtraction was the invention of an English mathematician of the first part of the seventeenth century, by the name of Gath.

In multiplication there were some ten or twelve different processes in practical use; but, strange to say, our present mode is not found among them. A few of the subjoined examples are easily intelligible. A little study will make the others plain:

Example 1. Multiply 135 by 12.

Process 1. Process 2.
 
PSM V16 D220 Old Mathematical calculation 1.jpg
 
Process 3.
PSM V16 D220 Old Mathematical calculation 2.jpg The commentator considers this method as difficult, and not to be learned by dull scholars without instruction.
 

Example 2. Multiply 15 by 12.

15 = 4 + 5 + 6. 12 = 2 + 4 + 6.

Process.

PSM V16 D220 Old Mathematical calculation 3.jpg
 

 Example 3. Multiply 30 by 46.

 
Process.
PSM V16 D220 Old Mathematical calculation 4.jpg This method was called "crosswise," from the manner in which the partial products to be added were obtained. It is not improbable that our present sign of multiplication was derived from the crossing of the lines in this process, as being somewhat indicative of the operation.
 

Here is a larger example worked by the cross-method:

 

Example 4. Multiply 456 by 456.

Process.
PSM V16 D220 Old Mathematical calculation 5.jpg To indicate the successive steps the linking-lines are numbered, so as to show the groups in which the products are to be taken for addition.
 
Process.
PSM V16 D220 Old Mathematical calculation 6.jpg Let the products be found and properly grouped and 456 added mentally, and one will better appreciate how much 456 depend upon mere mechanism in our own mental operations.

The method by the little castle was much practiced at Florence. Why the name was given to it is not very clear.

Example 5. Multiply 9876 by 6789.

Process.

PSM V16 D221 Old Mathematical calculation 1.jpg
 

The method by the square was regarded as elegant, not requiring the operator to attend to the places of the figures.

Process.

PSM V16 D221 Old Mathematical calculation 2.jpg
 

Latticed Multiplication.

Process.

PSM V16 D221 Old Mathematical calculation 3.jpg
 

Some authors wished to elevate the character of the study, so as to save the labor of carrying tens. Here are two processes, or rather one process under different forms, which save that labor :

Example 6. Multiply 234 by 234.

Example 7. Multiply 5142 by 43.

Processes.
Ex. 6. Ex. 7.
PSM V16 D222 Old Mathematical calculation 1.jpg

Though the multiplication-table was in use by the Arabians and Italians at an early date, no notice was taken of it during the middle ages in the rest of Europe. It may give us more charity for the boys and girls who are now wrestling with it—although nowadays it does not seem to require the struggle that it used to—to know that grown men, and wise men probably, sought for devices by which the labor might be avoided which we go through in childhood. Outside of Italy, many writers considered it necessary to relieve the memory from retaining the products of digits above five. The principal rule—known as the "sluggard's rule"—given for this purpose during the last half of the sixteenth century, the half century after the time of Luther, Melanchthon, and Erasmus, was this: Subtract each digit from ten, and write down the differences ; multiply the differences together and add as many tens to their product as the first digit exceeds the second difference, or the second digit the first difference.

Examples. 7 X 8 = (3 X 2) + (7 — 2 = 5) tens = 56.
 6 X 9 = (4 X 1) + 5 tens = 54.

The method which we call short division was largely used in the middle ages, as was also the method of dividing by using the factors of the divisor. The process by long division was known, but was not so commonly used as others. It was called the process "by giving," since after subtraction we give or add (bring down) one or more figures to the remainder. Here is an example set down after the fashion of those times:

Example 8. Divide 97335376 by 9876.

Divisior. Proveniens.
PSM V16 D222 Old Mathematical calculation 2.jpg

De Burgo, the most noted mediæval writer on arithmetic, thinks this last process—our long division—much less pleasant than the following method. Surely tempora mutantur, et nos mutamur in illis.

Example 9. Divide 97535399 by 9876.

PSM V16 D223 Old Mathematical calculation 1.jpg
 

In this work the divisor is placed next below the dividend, and removed one place to the right since it is not contained in the first four figures of the dividend. The process with the first figure of the quotient, placed as usual at present, is as follows: The first number of the divisor, 9, is contained in 97 nine times with a remainder 16. The first figure of the divisor having been used is canceled; as are also the first two figures of the dividend. (The "scratches" or canceling-marks are omitted in the illustration.) The remainder, being of the same denomination as the first two figures of the dividend, is put directly above them. The next number to be used is 165. Multiplying the second figure of the divisor, 8, by 9, and subtracting from 165, 93 remains; 165 and 8 are now canceled, having been used. The remainder 93 is placed above in the proper orders, the 6th and 7th places. So it continues, leaving, after completing the work with the first figure of the quotient, the remainder 8651399. The divisor is now set down again, taking one place to the right as it should to correspond to the highest order now in the dividend: the last figure being raised to the line above, probably for symmetry. The process is continued as before.

All writers upon arithmetic appear to have agreed in commendation of this method as late as the end of the seventeenth century. It was, in fact, the only method thought necessary to notice. The English arithmeticians, from evident cause, called it the "scratch way" of division. Our present method was known specifically as Italian division, and was not introduced until the beginning of the last century.

One writer on arithmetic, a pious monk, furnishes a good illustration of mediæval logic. He is embarrassed by the usage and meaning of the term "multiplication" in the case of fractions in which the product is less than the multiplicand, and he proposes the question, "Whether the multiplication of fractions is an increasing process?" In order to prove that to multiply means to increase, he bases his argument on Scripture, and clinches the whole by quoting the promise to Abraham, "I will multiply thy seed like the stars of the firmament." To this devout logician there would be no joke in the common conundrum that proves Abraham to have been a mathematician because he increased and multiplied on the "face of the earth." But how is this to be reconciled with the numerical result in the cases under consideration? He supposes the units of the product to be of greater virtue and significancy than those of the factors: thus, if ½ and ½ represent the sides of a square, their product will represent the area of the square.

The first actual mention of real decimal fractions is in a Flemish work published in 1590. There the mixed number 27·847 is written

(0) {1} {2} {3}
2 7 8 4 7.

To the present advocates of the metric system it may afford encouragement to know that Stevinus, in this work, enumerates the advantages which would result from the decimal subdivision of the units of length, area, capacity, value, etc.

In 1619 the contents of the Flemish book were embodied in an English work—"The Art of Tens, or Decimal Arithmetike, exercised by Henry Lyte, Gent., and by him set forth for his Countries Good." After enlarging upon the value of his system to all classes, he adds: "If God spare me life, I will spend some time in most cities of this land for my countries good to teach this art. I hold the lively voice of a meane speculator somewhat practised, furthereth ten fold more in my judgement than the finest writer that is," Rather severe on those "meane speculators," his contemporaries, Francis Bacon and William Shakespeare.

 
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