Music Welcome to Snewscast, the podcast is on to help you fall asleep. Find us at Snewscast.com and if you enjoy our show, please share it with a friend. This episode is brought to you by Proportionality. Tonight we'll read from Elements of Arithmetic written by Augustus Day Morgan and first published in 1846. Demorgan was a pioneering British mathematician and logician, remembered not only for his clear explanations, but also for his sharp wit. He introduced the world to what we now call De Morgan's Laws in Logic, and was the first to formally define and use the term mathematical induction. Because he was a unitarian and refused to subscribe to the 39 articles of the Anglican Church, he was denied a fellowship at Oxford and Cambridge. This principle stands, however, did not hinder his influence. He went on to become the first professor of mathematics at the newly founded University College of London. His legacy is honored not only in mathematics, but on the moon itself, where a crater bears his name. Elements of Arithmetic was one of his most widely-read works, offering both beginners and more advanced students a foundation in the science of numbers. Arithmetic, the branch of mathematics that studies numbers and their operations forms the bedrock of numbers. Arithmetic, the branch of mathematics that studies numbers and their operations, forms

the bedrock of mathematics, bridging the practical art of calculation with the deeper theories that underpin algebra and number theory. Let's get cozy.

Close your eyes. Let's get cozy.

Close your eyes. Relax your body into the softness of your bed. Now, take a few deep breaths. Imagine a multitude of objects of the same kind assembled together. For example, a company of horsemen. One of the first things that must strike a spectator, although unused to counting, is that to each man there is a horse. Now, though men and horses are things perfectly unlike, yet, because there is one of the first kind to every one of the second, one man to every horse, a new notion will be formed in the mind of the observer, which we express in words by saying that there is the same number of men as of horses. Someone who had no other way of counting might remember this number by taking a pebble for each man, out of a method as rude as this as sprung our system of calculation, by the steps which are pointed out in the following articles. Suppose that there are two companies of horsemen, and a person wishes to know in which one of them is the greater number, and also to be able to recollect how many there are in each.

Suppose that while the first company passes by, he drops a pebble into a basket for each man whom he sees. There is no connection between the pebbles and the horsemen, but this. for every horse meant there is a pebble. That is, in common language, the number of pebbles and of horsemen is the same. Suppose that while the second company passes, he drops a pebble for each man into a second basket. He will then have two baskets of pebbles, by which he will be able to convey to any person a notion of how many horsemen there were in each company. When he wishes to know which company was the larger or contained most horsemen, he will take a pebble out of each basket and put them aside. He will go on doing this as often as he can, that is, until one of the baskets is emptied. Then, if he also find the other basket empty, he says that both companies contained the same number of horsemen. If the second basket still contains some pebbles, he can tell by them how many more were in the second that in the first. In this way, a primitive person could keep an account of any numbers in which he was interested. He could thus register his children, his cattle, or the number of summers and winters which he had seen by means of pebbles, or any other small objects which could be got in large numbers. Something of this sort is the practice of primitive person nations at this day, and it has in some places lasted even after the invention of better methods of reckoning. At Rome, in the time of the Republic, the Prater, one of the magistrates, used to go every year in Great Pomp and drive a nail into the door of the Temple of Jupiter, a of remembering the number of years which the city had been built, which probably took its rise for the introduction of writing. In process of time, names would be given to those collections of pebbles which are met with most frequently. But as long as small numbers only were required, the most convenient way of reckoning them would be by means of the fingers. Any person could make with his two hands the little calculations, which would be necessary for his purposes, and would name all the different collections of the fingers. He would thus get words in his own language, answering to 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. As his wants increased, he would find it necessary to give names to larger numbers. But here, he would be stopped by the immense quantity of words which he must have, in order to express all the numbers which he would be obliged to make use of. He must then, after giving a separate

name to a few of the first numbers, manage to express all other numbers by means of those names. I now proceed to explain the way in which these signs are made to represent other numbers. Suppose a man first to hold up one finger, then two, and so long, until he has held up every finger, and suppose a number of men to do the same thing. It is plain that we made thus distinguish one number from another, by causing two different sets of persons to hold up each a certain number of fingers, and that we may do this in many different ways. For example, the number 15 might be indicated either by 15 men each holding up one finger, or by four men each holding up two fingers and a fifth holding up seven and so on. The question is of all these contrivances for expressing the number which is the most convenient. In the choice which is made for the purpose consists what is called the method of numereration. I have used the foregoing explanation because it is very probable that our system of numeration, and almost every other which is used in the world, sprung from the practice of reckoning on the fingers, which children usually follow when they begin to count. The method which I have described is the rudest possible, but by a little alteration, a system may be formed which will enable us to express enormous numbers with great ease. Suppose that you are going to count some large number, for example, to measure a number of yards of cloth. Opposite to yourself, suppose a man to be placed, who keeps his eye upon you, and holds up a finger for every yard which cheesies you measure. When ten yards have been measured, he will have held up ten fingers and will not be able to count any further unless he began again. Holding up one finger at the eleventh yard, two at the twelfth, and so on.

But to know how many have been counted, you must know, not only how many fingers he holds up, but also how many times he has begun again. You may keep this in view by placing another man on the right of the former, who directs his eye towards his companion, and holds up one finger the moment he perceives him ready to begin again. That is, as soon as ten yards have been measured, each finger of the first man stands only for one yard, but each finger of the second stands for as many as all the fingers of the first together. That is, for ten. In this way, a hundred may be counted, because the first may now reckon his ten fingers once for each finger of the second man. That is, ten times in all, ten-ten is one hundred. Now, place a third man at the right of the second, who shall hold up a finger whenever he perceives the second ready to begin again. One finger of the third man counts as many as all the ten fingers of the second. That is, counts one hundred. In this way, we may proceed until the third has all his fingers extended, which will signify that ten hundred or one thousand have been counted. A fourth man would enable us to count as far as ten thousand. A fifth as far as one hundred thousand. A sixth as far as a million. And so on. Each new person placed himself towards your left in the rank opposite to you. Now rule columns as in the next page, and to the right of them all place in words the number which you wish to represent. In the first column on the right, place the number of fingers which the first man will be holding up when that number of yards has been measured. In the next column, place the fingers which the second man will then be holding up and so on. Section 2. Addition and subtraction. There is no process in arithmetic which does not consist entirely in the increase or diminution of numbers. There is then nothing which might not be done with collections of pebbles. Probably at first, either these or the fingers were used. Our word calculation is derived from the Latin word calculus, which means a pebble. Shorter ways of counting have been invented by which many calculations, which would require long and tedious reckoning if pebbles were used, are made at once with very little trouble. The four great methods are addition, subtraction, multiplication, and division, of which the last two are only ways of doing several of the first and second at once. When one number is increased by others, the number which is as large as all the numbers together is called their sum. The process of finding the sum of two or more numbers is called addition. And, as was said before, is denoted by placing a cross between the numbers which are to be added together. Multiplication I have said that all questions and arithmetic require nothing but addition and subtraction. I do not mean by this that no rule should ever be used except those given in the last section, but that all other rules only show shorter ways of finding what might be found. If we pleased, by the methods, they are deduced. Even the last two rules themselves are only short and convenient ways of doing what may be done with a number of pebbles or counters. I want to know the sum of 517s, or I ask the following question. There are five heaps of pebbles and 17 pebbles in each heap. How many are there in all? right 5,s in a column and make the addition, which gives 85. In this case, 85 is called the product of 5 and 17, and the process of finding the product is called multiplication, which gives nothing more than the addition

of a number of the same quantities. Here, 17 is called the multiplication and 5 is called

the multiplier. 17, 17, 17, 17, 85.

If no question harder than this were ever proposed, there would be no occasion for a shorter way than the one here followed. and if there were 1,367 heaps of pebbles and 429 in each heap. The whole number is then 1,367 times 429, or 429 multiplied by 1,367. I should have to write 429,1367 times, and then to make an addition of enormous length. Suppose I ask whether 156 can be divided into a number of parts each of which is 13, or how many 13s 156 contains. I propose a question. The solution of which is called division. In this case, 156 is called the dividend, 13, the divisor, and the number of parts required is the quotient. And when I find the quotient, I am said to divide 156 by 13.

The simplest method of doing this is to subtract 13 from 156 and then to subtract 13 from their remainder, and so on, or in common language to tell off 156 by 13. On the proportion of numbers, when two numbers are named in any problem, it is usually necessary in some way or another to compare the two. or another, to compare the two.

That is, by considering the two together, to establish some connection between them,

which may be useful in future operations.

The first method which suggests itself and the most simple is to observe which is the greater and by how much it differs from the other. The connection thus established between two numbers may also hold good of two other numbers. For example, eight differs from 19 by 11, and a hundred differs from 111 by the same number. In this point of view, eight stands to 19 in the same situation in which 100 stands to 111. The first of both couples differing in the same degree from the second. The four numbers thus noticed, 819, 100, 111 are said to be in ariththmatical proportion. When four numbers are thus placed, the first and last are called the extremes, and the second and third, the means. It is obvious that 111 plus 8 equals 100 plus 19. That is, the sum of the extremes is equal to the sum of the means. And this is not accidental, arising from the particular numbers we have taken, must be the case in every arithmetical proportion, for in 111 plus 8, by any diminution of 111 will not affect the sum, provided a corresponding increase be given to 8. by the definition just given, one mean is as much less than 111, as the other is greater than eight. A set or series of numbers is said to be in continued mathematical proportion. Or in arithmetical progression, when the difference between every two succeeding terms

of the series is the same.

This is the case in the following series. 1, 2, 3, 4, 5, 3, 6, 9, 12, 15, 1 and a half, 2 and a half, 3 and ½. The difference between two succeeding terms is called the common difference. In the three series just given, the common differences are 1, 3 and ½. If a certain number of terms of any arithmetical series be taken, the sum of the first and last terms is the same as that of any other two terms. Provided one is as distant from the beginning of the series as the other is from the end. For example, let there be seven terms and let them be A, B, C, D, E, F, G. Then, since by the nature of the series, B is as much above A as F is below G, A plus G equals B plus F. Again, since C is as much above B as E is below F, B plus F equals C plus E, but A plus G equals B plus F. Therefore, A plus G equals C plus E, and so on. Again, twice the middle term, or the term equally distant from the beginning and the end, which exists only when the number of terms is odd, is equal to the sum of the first and last terms. For since c is as much below d, as e is above it, we have c plus e equals d plus d equals 2d but C plus E equals A plus G. Therefore, A plus G equals 2D. This will give a short rule for finding the sum of any number of terms of an erythmatical series. But there be seven, those just given. Since A plus G, B plus F and C plus E are the same. There's some is three times A plus G, which with D, the middle term, or half a plus g, is 3 times and a half a plus g, or the sum of the first and last terms multiplied by 3 and a half, or 7 divided-1-1 or 7-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1 E and F. We know that A plus F, B plus E and C plus D are the same. Once the sum is three times A plus F or the sum of the first and last terms multiplied by half the number of terms as before. The rule then is to sum any number of terms of an earth-matical progression, multiply the sum of the first and last terms by half the number of terms. On permutations and combinations, if a number of counters distinguished by different letters be placed on the table and any number of them, say for, be taken away. The question is to determine in how many different ways this can be done. Each way of doing it gives what is called a combination of four. But which might, with more propriety, be called a selection of four. Two combinations or selections are called different, which differ in any way, whatever. Thus, ABCD and A, B, C, E are different, D being in one and E in the other. The remaining parts being the same. But there be six counters. ABCDE and F. The combinations of three, which can be made out of them, are 20 in number as follow. A, B, C, A, C, E, B, C, D, B, E, F, A, B. B-C-D B-E-F A-B-D A-C-F B-C-E C-D-E A-B-E-A-B-E

B, C, F, C, D, F, A, B, F A C D A E, F. the combinations of four are 15 in number, namely A, B, C, D. A, B, D, B. A, C, D, E, A, D, E, F, B B C D E B B C D E B B C D E B D C F A B C F A B E F A C E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E F, A, C, E, F, B, C, D, F, C, D, E, F.