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| ISBN-13: | 9780486158297 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/09/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 736 |
| File size: | 30 MB |
| Note: | This product may take a few minutes to download. |
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A SOURCE BOOK in MATHEMATICS
By DAVID EUGENE SMITH
Dover Publications, Inc.
Copyright © 1959 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-15829-7
CHAPTER 1
SOURCE BOOK IN MATHEMATICS
I. FIELD OF NUMBER
The First Printed Arithmetic
Treviso, Italy, 1478
(Translated from the Italian by Professor David Eugene Smith, Teachers College, Columbia University, New York City.)
Although it may justly be said that mere computation and its simple applications in the lives of most people are not a part of the science of mathematics, it seems proper that, in a source book of this kind, some little attention should be given to its status in the early days of printing. For this reason, these extracts are selected from the first book on arithmetic to appear from the newly established presses of the Renaissance period. The author of the work is unknown, and there is even some question as to the publisher, although he seems to have been one Manzolo or Manzolino. It is a source in the chronological rather than the material sense, since the matter which it contains had apparently but little influence upon the other early writers on arithmetic. The work is in the Venetian dialect and is exceedingly rare. The copy from which this translation was made is in the library of George A. Plimpton of New York City. As with many other incunabula, the book has no title. It simply begins with the words, Incommincia vna practica molto bona et vtilez a ciascbaduno cbi vuole vxare larte dela mercbadantia. cbiamata vulgarmente larte de labbacbo. It was published at Treviso, a city not far to the north of Venice, and the colophon has the words "At Treviso, on the 10th day of December, 1478."
Here beginneth a Practica, very helpful to all who have to do with that commercial art commonly known as the abacus.
I have often been asked by certain youths in whom I have much, interest, and who look forward to mercantile pursuits, to put into writing the fundamental principles of arithmetic, commonly called the abacus. Therefore, being impelled by my affection for them, and by the value of the subject, I have to the best of my small ability undertaken to satisfy them in some slight degree, to the end that their laudable desires may bear useful fruit. Therefore in the name of God I take for my subject this work in algorism, and proceed as follows:
All things which have existed since the beginning of time have owed their origin to number. Furthermore, such as now exist are subject to its laws, and therefore in all domains of knowledge this Practica is necessary. To enter into the subject, the reader must first know the basis of our science. Number is a multitude brought together or assembled from several units, and always from two at least, as in the case of 2, which is the first and the smallest number. Unity is that by virtue of which anything is said to be one. Furthermore be it known that there are three kinds of numbers, of which the first is called a simple number, the second an article, and the third a composite or mixed number. A simple number is one that contains no tens, and it is represented by a single figure, like i, 2, 3, etc. An article is a number that is exactly divisible by ten, like iO, 20, 30 and similar numbers. A mixed number is one that exceeds ten but that cannot be divided by ten without a remainder, such as ii, i2, i3, etc. Furthermore be it known that there are five fundamental operations which must be understood in the Practica, viz., numeration, addition, subtraction, multiplication, and division. Of these we shall first treat of numeration, and then of the others in order.
Numeration is the representation of numbers by figures. This is done by means of ten letters or figures, as here shown, .i., .2., .3., .4., .5., .6., .7., .8., .9., .0.. Of these the first figure, i, is not called a number but the source of number. The tenth figure, 0, is called cipher or "nulla," i. e., the figure of nothing, since by itself it has no value, although when joined with others it increases their value. Furthermore you should note that when you find a figure by itself its value cannot exceed nine, i. e., 9; and from that figure on, if you wish to express a number you must use at least two figures, thus: ten is expressed by i0, eleven by ii, and so on. And this can be understood from the following figures.
To understand the figures it is necessary to have well in mind the following table:
i times i makes i i times i0 makes i0
i times 2 makes 2 2 times i0 makes 20
i times 3 makes 3 3 times i0 makes 30
i times 4 makes 4 4 times i0 makes 40
i times 5 makes 5 5 times i0 makes 50
i times 6 makes 6 6 times i0 makes 60
i times 7 makes 7 7 times i0 makes 70
i times 8 makes 8 8 times i0 makes 80
i times 9 makes 9 9 times i0 makes 90
i times 0 makes 0 0 times i0 makes 0
And to understand the preceding table it is necessary to observe that the words written at the top give the names of the places occupied by the figures beneath. For example, below 'units' are the figures designating units, below 'tens' are the tens, below 'hundreds' are the hundreds, and so on. Hence if we take each figure by its own name, and multiply this by its place value, we shall have its true value. For instance, if we multiply i, which is beneath the word 'units,' by its place,—that is, by units,—we shall have 'i time i gives i,' meaning that we have one unit. Again, if we take the 2 which is found in the same column, and multiply by its place, we shall have 'i time 2 gives 2,' meaning that we have two units, ... and so on for the other figures found in this column ... This rule applies to the various other figures, each of which is to be multiplied by its place value.
And this suffices for a statement concerning the 'act' of numeration.
Having now considered the first operation, viz. numeration, let us proceed to the other four, which are addition, subtraction, multiplication, and division. To differentiate between these operations it is well to note that each has a characteristic word, as follows:
Addition has the word and,
Subtraction has the word from,
multiplication has the word times,
Division has the word in.
It should also be noticed that in taking two numbers, since at least two are necessary in each operation, there may be determined by these numbers any one of the above named operations. Furthermore each operation gives rise to a different number, with the exception that 2 times 2 gives the same result as 2 and 2, since each is 4. Taking, then, 3 and 9 we have:
Addition: 3 and 9 make i2
Subtraction: from 9 leaves 6
Multiplication: 3 times 9 makes 27
Division: 3 in 9 gives 3
We thus see how the different operations with their distinctive words lead to different results.
In order to understand the second operation, addition, it is necessary to know that this is the union of several numbers, at least of two, in a single one, to the end that we may know the sum arising from this increase. It is also to be understood that, in the operation of adding, two numbers at least are necessary, namely the number to which we add the other, which should be the larger, and the number which is to be added, which should be the smaller. Thus we always add the smaller number to the larger, a more convenient plan than to follow the contrary order, although the latter is possible, the result being the same in either case. For example, if we add 2 to 8 the sum is i0, and the same result is obtained by adding 8 to 2. Therefore if we wish to add one number to another we write the larger one above and the smaller one below, placing the figures in convenient order, i. e., the units under units, tens under tens, hundreds under hundreds, etc. We always begin to add with the lowest order, which is of least value. Therefore if we wish to add 38 and 59 we write the numbers thus:
59
38
Sum 97
We then say, '8 and 9 make i7,' writing 7 in the column which was added, and carrying the i (for when there are two figures in one place we always write the one of the lower order and carry the other to the next higher place). This i we now add to 3,making 4, and this to the 5, making 9, which is written in the column from which it is derived. The two together make 97.
The proof of this work consists in subtracting either addend from the sum, the remainder being the other. Since subtraction proves addition, and addition proves subtraction, I leave the method of proof until the latter topic is studied, when the proof of each operation by the other will be understood.
Besides this proof there is another. If you wish to check the sum by casting out nines, add the units, paying no attention to 9 or 0, but always considering each as nothing. And whenever the sum exceeds 9, subtract 9, and consider the remainder as the sum. Then the number arising from the sum will equal the sum of the numbers arising from the addends. For example, suppose that you wish to prove the following sum:
.59.
.38.
Sum .97. | 7
The excess of nines in 59 is 5; 5 and 3 are 8; 8 and 8 are i6; subtract 9 and 7 remains. Write this after the sum, separated by a bar. The excess of nines in 97 is 7, and the excess of nines in 7 equals 7, since neither contains 9. In this way it is possible to prove the result of any addition of abstract numbers or of those having no reference to money, measure, or weight. I shall show you another plan of proof according to the nature of the case. If you have to add 816 and 1916, arrange the numbers as follows:
1916
816
Sum 2732
Since the sum of 6 and 6 is 12, write the 2 and carry the 1. Then add this 1 to that which follows to the left, saying, '1 and 1 are 2, and the other 1 makes 3.' Write this 3 in the proper place, and add 8 and 9. The sum of this 8 and 9 is 17, the 7 being written and the 1 carried to the other 1, making 2, which is written in the. proper place, the sum being now complete. If you wish to prove by 9 arrange the work thus:
1916
816
The sum 2732 | 5
You may now effect the proof by beginning with the upper number, saying '1 and 1 are 2, and 6 are 8, and 8 are 16. Subtract 9, and 7 remains. The 7 and 1 are 8, and 6 are 14. Subtract 9, and 5 remains,' which should be written after the sum, separated by a bar. Look now for the excess of nines in the sum: 2 and 7 are 9, the excess being 0; 3 and 2 are 5, so that the result is correct.
Having now considered the second operation of the Practica of arithmetic, namely the operation of addition, the reader should give attention to the third, namely the operation of subtraction. Therefore I say that the operation of subtraction is nothing else than this: that of two numbers we are to find how much difference there is from the less to the greater, to the end that we may know this difference. For example, take 3 from 9 and there remains 6. It is necessary that there should be two numbers in subtraction, the number from which we subtract and the number which is subtracted from it.
The number from which the other is subtracted is written above, and the number which is subtracted below, in convenient order, viz., units under units and tens under tens, and so on. If we then wish to subtract one number of any order from another we shall find that the number from which we are to subtract is equal to it, or greater, or less. If it is equal, as in the case of 8 and 8, the remainder is 0, which 0 we write underneath in the proper column. If the number from which we subtract is greater, then take away the number of units in the smaller number, writing the remainder below, as in the case of 3 from 9, where the remainder is 6. If, however, the number is less, since we cannot take a greater number from a less one, take the complement of the larger number with respect to 10, and to this add the other, but with this condition: that you add one to the next left-hand figure. And be very careful that whenever you take a larger number from a smaller, using the complement, you remember the condition above mentioned. Take now an example: Subtract 348 from 452, arranging the work thus:
452
348
_____
Remainder 104
First we have to take a greater number from a less, and then an equal from an equal, and third, a less from a greater. We proceed as follows: We cannot take 8 from 2, but 2 is the complement of 8 with respect to 10, and this we add to the other 2 which is above the 8, thus: 2 and 2 make 4, which we write beneath the 8 for the remainder. There is, however, this condition, that to the figure following the 8 (viz., to 4), we add 1, making it 5. Then S from 5, which is an equal, leaves 0, which 0 we write beneath.
Then 3 from 4, which is a less from a greater, is 1, which 1 we write under the 3, so that the remainder is 104.
If we wish to prove this result, add the number subtracted to the remainder, and the result will be the number from which we subtracted. We may arrange the work as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Now add, 4 and 8 are 12; write 2 under the 4 and carry 1; then 1 and 4 are 5; write this 5 under the 0; then add 1 and 3, making 4, and write this 4 under the 1, and the work checks. Thus is found that which was promised you, as you can see ...
Having now explained the third operation, namely that of subtraction, the reader should give attention to the fourth, namely that of multiplication. To understand this it is necessary to know that to multiply one number by itself or by another is to find from two given numbers a third number which contains one of these numbers as many times as there are units in the other. For example, 2 times 4 are 8, and 8 contains 4 as many times as there are units in 2, so that 8 contains 4 in itself twice. Also the 8 contains 2 as many times as there are units in 4, and 4 has in itself four units, so that 8 contains 2 four times. It should be well understood that in multiplication two numbers are necessary, namely the multiplying number and the number multiplied, and also that the multiplying number may itself be the number multiplied, and vice versa, the result being the same in both cases. Nevertheless usage and practice demand that the smaller number shall be taken as the multiplying number, and not the larger. Thus we should say, 2 times 4 makes 8, and not 4 times 2 makes 8, although the results are the same. Now not to speak at too great length I say in brief, but sufficiently for the purposes of a Practica, that there are three methods of multiplication, viz., by the tables, cross multiplication, and the chess-board plan. These three methods I will explain to you as briefly as I am able. But before I give you a rule or any method, it is necessary that you commit to memory the following statements, without which no one can understand all of this operation of multiplication ...
I have now given you to learn by heart all the statements needed in the Practica of arithmetic, without which no one is able to master the Art. We should not complain, however, at having to learn these things by heart in order to acquire readiness; for I assure you that these things which I have set forth are necessary to any one who would be proficient in this art, and no one can get along with less. Those facts which are to be learned besides these are valuable, but they are not necessary.
Having learned by heart all of the above facts, the pupil may with zeal begin to multiply by the table. This operation arises when the multiplier is a simple number, and the number multiplied has at least two figures, but as many more as we wish. And that we may more easily understand this operation we shall call the first figure toward the right, units; the second toward the left, tens, and the third shall be called hundreds. This being understood, attend to the rule of working by the table, which is as follows: First multiply together the units of the multiplier and the number multiplied. If from this multiplication you get a simple number, write it under its proper place; if an article, write a 0 and reserve the tens to add to the product of the tens; but if a mixed number is found, write its units in the proper place, and save the tens to add to the product of the tens, proceeding in the same way with all the other orders. Then multiply together the units of the multiplier with the tens; then with the hundreds, and so on in regular order ...
In order to understand the fourth operation, viz., division, three things are to be observed, viz., what is meant by division; second, how many numbers are necessary in division; third, which of these numbers is the greater. As to the first I say that division is the operation of finding, from two given numbers, a third number, which is contained as many times in the greater number as unity is contained in the less number. You will find this number when you see how many times the less number is contained in the greater. Suppose, for example, that we have to divide 8 by 2; here 2 is contained 4 times in 8, so we say that 4 is the quotient demanded. Also, divide 8 by 4. Here the 4 is contained 2 times in 8, so that 2 is the quotient demanded.
Second, it is to be noticed that three numbers are necessary in division,—the number to be divided, the divisor, and the quotient, as you have understood from the example above given, where 2 is the divisor, 8 the number to be divided, and 4 the quotient. From this is derived the knowledge of the third thing which is to be noted, that the number which is to be divided is always greater than, or at least is equal to, the divisor. When the numbers are equal the quotient is always 1.
(Continues...)
Excerpted from A SOURCE BOOK in MATHEMATICS by DAVID EUGENE SMITH. Copyright © 1959 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
AUTHOR'S PREFACE,I. THE FIELD OF NUMBER,
II. THE FIELD OF ALGEBRA,
III. THE FIELD OF GEOMETRY,
IV. THE FIELD OF PROBABILITY,
V. FIELD OF THE CALCULUS, FUNCTIONS, QUATERNIONS,
Index,