Is a science which explains the properties of numbers, and shews the method or art of computing by them.

History of Arithmetic.

At what time this science was first introduced into the world, we can by no means determine. That some part of it, however, was coeval with the human race is absolutely certain. We cannot conceive how any man endowed with reason can be without some knowledge of numbers. We are indeed told of nations in America who have no word in their language to express a greater number than three; and this they call postararorinouraco: but that such nations should have no idea of a greater number than this, is absolutely incredible. Perhaps they may compute by threes, as we compute by tens; and this may have occasioned the notion that they have no greater number than three.

But though we cannot suppose any nation, or indeed any single person, ever to have been without some knowledge of the difference between greater and smaller numbers, it is possible that mankind may have subsisted for a considerable time without bringing this science to any perfection, or computing by any regular scale, as 10, 60, &c. That this, however, was very early introduced into the world, even before the flood, we may gather from the following expression in Enoch's prophecy, as mentioned by the Apostle Jude: "Behold, the Lord cometh with ten thousands of his saints." This shews, that even at that time men had ideas of numbers as high as we have at this day, and computed them also in the same manner, namely by tens. The directions also given to Noah concerning the dimensions of the ark, leave us no room to doubt that he had a knowledge of numbers, and of measures likewise. When Rebekah was sent away to Isaac, Abraham's son, her relations wished she might be the mother of thousands of millions; and if they were totally unacquainted with the rule of multiplication, it is difficult to see how such a wish could have been formed.

It is probable, therefore, that the four fundamental rules of Arithmetic have always been known to some nation or other. No doubt, as some nations, like the Europeans formerly, and the Africans and Americans now, have been immersed in the most abject and deplorable state of ignorance, they might remain for some time unacquainted with numbers, except such as they had immediate occasion for; and, when they came afterwards to improve, either from their own industry, or hints given by others, might fancy that they themselves, or those from whom they got the hints, had invented what was known long before. The Greeks were the first European nation among whom arithmetic arrived at any degree of perfection. M. Goguet is of opinion, that they first used pebbles in their calculations: a proof of which, he imagines, is, that the word πέτρα, which comes from πέτρα, a little stone or flint, among other things, signifies to calculate. The same, he thinks, is probable of the Romans; and derives the word calculation. calculation from the use of little stones (calculi) in their first arithmetical operations.

If this method, however, was at all made use of; it must have been but for a short time, since we find the Greeks very early made use of the letters of the alphabet to represent their numbers. The 24 letters of their alphabet, taken according to their order, at first denoted the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 100, 200, 300, 400, 500, 600, 700, and 800; to which they added the three, following (v, v, v), to represent 6, 90, and 900. The difficulty of performing arithmetical operations by such marks as these may easily be imagined, and is very conspicuous from Archimedes's treatise concerning the dimensions of a circle.

The Romans followed a like method; and besides characters for each rank of classes, they introduced others for five, fifty, and five hundred. Their method is still used for distinguishing the chapters of books, and some other purposes. Their numeral letters and values are the following:

I V X L C D M

One, five, ten, fifty, one hundred, five hundred, one thousand.

Any number, however great, may be represented by repeating and combining these according to the following rules:

1st, When the same letter is repeated twice, or oftener, its value is represented as often. Thus II signifies two; XXX thirty, CC two hundred.

2nd, When a numeral letter of lesser value is placed after one of greater, their values are added: thus XI signifies eleven, LXV sixty-five, MDCXXVIII one thousand six hundred and twenty-eight.

3rd, When a numeral letter of lesser value is placed before one of greater, the value of the lesser is taken from that of the greater: thus IV signifies four, XL forty, XC ninety, CD four hundred.

Sometimes IO is used instead of D for 500, and the value is increased ten times by annexing O to the right hand.

Thus IO signifies 500. Also CI is used for 1000

100 5000 CC 1000 for 10000

1000 50000 CCC 10000 for 100000

Sometimes thousands are represented by drawing a line over the top of the numeral, V being used for five thousand, L for fifty thousand, CC two hundred thousand.

About the year of Christ 200, a new kind of arithmetic, called sexagesimal, was invented, as is supposed, by Claudius Ptolemaeus. The design of it was to remedy the difficulties of the common method, especially with regard to fractions. In this kind of arithmetic, every unit was supposed to be divided into 60 parts, and each of these into 60 others, and so on: hence any number of such parts were called sexagesimal fractions; and to make the computation in whole numbers more easy, he made the progression in these also sexagesimal. Thus from one to 59 were marked in the common way: then 60 was called sexagesima prima, or first sexagesimal integer, and had one single dash over it; so 60 was expressed thus I'; and so on to 59 times 60, or 3540, which was thus expressed LIX'. He now proceeded to 60 times 60, which he called a sexagesima secunda, and was thus expressed I''. In like manner, twice 60 times 60, or 7200, was expressed by II''; and so on till he came to 60 times 3600, which was a third sexagesimal, and expressed thus, I'''. If any number less than 60 was joined with these sexagesimals, it was added in its proper character without any dash: thus IX represented 60 and 15, or 75; IVXXV is four times 60 and 25, or 265; X'II'XV, is 10 times 3600, twice 60 and 15, or 36135, &c. Sexagesimal fractions were marked by putting the dash at the foot, or on the left hand of the letter: thus I', or I', denoted \( \frac{1}{60} \); I'', or "I", \( \frac{1}{3600} \) &c.

The most perfect method of notation, which we now use, came into Europe from the Arabs, by the way of Spain. The Arabs, however, do not pretend to be the inventors of them, but acknowledge that they received them from the Indians. Some there are indeed, who contend that neither the Arabs nor the Indians were the inventors, but that they were found out by the Greeks. But this is by no means probable; as Maximus Planudes, who lived towards the close of the 13th century, is the first Greek who makes use of them; and he is plainly not the inventor; for Dr Wallis mentions an inscription on a chimney in the parsonage-house of Helendon in Northamptonshire, where the date is expressed by M9133, instead of 1133. Mr Luffkin furnishes a still earlier instance of their use, in the window of a house, part of which is a Roman wall, near the market-place in Colchester; where between two carved lions stands an escutcheon with the figures 1090. Dr Wallis is of opinion that these characters must have been used in England at least as long ago as the year 1050, if not in ordinary affairs, at least in mathematical ones, and in astronomical tables. How these characters came to be originally invented by the Indians we are entirely ignorant.

The introduction of the Arabian characters in notation did not immediately put an end to the sexagesimal arithmetic. As this had been used in all the astronomical tables, it was for their sakes retained for a considerable time. The sexagesimal integers went first out, but the fractions continued till the invention of decimals.

The oldest treatises extant upon the theory of arithmetic are the seventh, eighth, and ninth books of Euclid's Arithmetic, elements, where he treats of proportion and of prime and composite numbers; both of which have received improvements since his time, especially the former. The next of whom we know anything is Nicomachus the Pythagorean, who wrote a treatise of the theory of arithmetic, consisting chiefly of the divisions and divisions of numbers into classes, as plain, solid, triangular, quadrangular, and the rest of the figurate numbers as they are called, numbers odd and even, &c., with some of the more general properties of the several kinds. This author is, by some, said to have lived before the time of Euclid; by others, not long after. His arithmetic was published at Paris in 1538. The next remarkable writer on this subject is Boethius, who lived at Rome in the time of Theodoric the Goth. He is supposed to have copied most of his work from Nicomachus.

From this time no remarkable writer on arithmetic appeared till about the year 1200, when Jordanus of Namur wrote a treatise on this subject, which was published and demonstrated by Joannes Faber Stapulensis in the 15th century, soon after the invention of printing. The same author also wrote upon the new art of computation by the Arabic figures, and called this book *Algorithmus Demonstratus*. Dr Wallis says this manuscript is in the Savilian library at Oxford, but it hath never yet been printed. As learning advanced in Europe, so did the knowledge of numbers; and the writers on arithmetic soon became innumerable. About the year 1464, Regiomontanus, in his triangular tables, divided the Radius into 10,000 parts instead of 60,000; and thus tacitly expelled the sexagesimal arithmetic. Part of it, however, still remains in the division of time, as of an hour into 60 minutes, a minute into 60 seconds, &c. Ramus in his arithmetic, written about the year 1550, and published by Lazarus Schonerus in 1586, uses decimal periods in carrying on the square and cube roots to fractions. The same had been done before by our countrymen Buckley and Record; but the first who published an express treatise on decimals was Simon Stevinus, about the year 1582. As to the circulating decimals, Dr Wallis is the first who took much notice of them. He is also the author of the *arithmetic of infinities*, which has been very usefully applied to geometry. The greatest improvement, however, which the art of computation ever received, is the invention of logarithms. The honour of this invention is unquestionably due to Lord Napier baron of Merchiston in Scotland, about the end of the 16th or beginning of the 17th century. By these means arithmetic has advanced to a degree of perfection which the ancients could never have imagined possible, much less hoped to attain; and we believe it may now be reckoned one of those few sciences which have arrived at their utmost height, and which is in its nature capable of little further improvement.

**Chap. I. Notation and Numeration.**

The first elements of arithmetic are acquired during our infancy. The idea of one, though the simplest of any, and suggested by every single object, is perhaps rather of the negative kind, and consists partly in the exclusion of plurality, and is not attended to till that of number be acquired. Two is formed by placing one object near another; three, four, and every higher number, by adding one continually to the former collection. As we thus advance from lower numbers to higher, we soon perceive that there is no limit to this increasing operation; and that, whatever number of objects be collected together, more may be added, at least, in imagination; so that we can never reach the highest possible number, nor approach near it. As we are led to understand and add numbers by collecting objects, so we learn to diminish them by removing the objects collected; and, if we remove them one by one, the number decreases through all the steps by which it advanced, till only one remain, or none at all. When a child gathers as many stones together as suits his fancy, and then throws them away, he acquires the first elements of the two capital operations in arithmetic. The idea of numbers, which is first acquired by the observation of sensible objects, is afterwards extended to measures of space and time, affections of the mind, and other immaterial qualities.

Small numbers are most easily apprehended: a child soon knows what two and what three is; but has not any distinct notion of seventeen. Experience removes this difficulty in some degree; as we become accustomed to handle larger collections, we apprehend clearly the number of a dozen or a score; but, perhaps could hardly advance to an hundred without the aid of clasical arrangement, which is the art of forming so many units into a class, and so many of these classes into one of a higher kind, and thus advancing through as many ranks of classes as occasion requires. If a boy arrange a hundred stones in one row, he would be tired before he could reckon them; but if he place them in ten rows of ten stones each, he will reckon an hundred with ease; and if he collect ten such parcels, he will reckon a thousand. In this case, ten is the lowest class, an hundred is a class of the second rank, and a thousand is a class of the third rank.

There does not seem to be any number naturally adapted for constituting a class of the lowest, or any higher rank, to the exclusion of others. However, as ten has been universally used for this purpose by the Hebrews, Greeks, Romans, and Arabians, and by all nations who have cultivated this science, it is probably the most convenient for general use. Other scales, however, may be assumed, perhaps on some occasions, with superior advantage; and the principles of arithmetic will appear in their full extent, if the student can adapt them to any scale whatever: thus, if eight were the scale, 6 times 3 would be two classes and two units, and the number 18 would then be represented by 22. If 12 were the scale, 5 times 9 would be three classes and nine units, and 45 would be represented by 39, &c.

It is proper, whatever number of units constitutes a class of the lower rank, that the same number of each class should make one of the next higher. This is observed in our arithmetic, ten being the universal scale; but is not regarded in the various kinds of monies, weights, and the like, which do not advance by any universal measure; and much of the difficulty in the practice of arithmetic arises from that irregularity.

As higher numbers are somewhat difficult to apprehend, we naturally fall on contrivances to fix them in our minds, and render them familiar; but notwithstanding all the expedients we can fall upon, our ideas of high numbers are still imperfect, and generally far short of the reality; and though we can perform any computation with exactness, the answer we obtain is often incompletely apprehended.

It may not be amiss to illustrate, by a few examples, the extent of numbers which are frequently named without being attended to. If a person employed in telling money reckon an hundred pieces in a minute, and continue at work ten hours each day, he will take seventeen days to reckon a million; a thousand men would take 45 years to reckon a billion. If we suppose the whole earth to be as well peopled as Britain, and to have been so from the creation, and that the whole race of mankind had constantly spent their time in telling from a heap consisting of a quadrillion of pieces, they would hardly have yet reckoned the thousandth part of that quantity.

All numbers are represented by the ten following characters:

``` 1 2 3 4 5 6 7 8 9 0 ```

One, two, three, four, five, six, seven, eight, nine, cypher. The nine first are called significant figures, or digits; and Numeration and sometimes represent units, sometimes tens, hundreds, or higher classes. When placed singly, they denote the simple numbers subjoined to the characters. When several are placed together, the first or right-hand figure only is to be taken for its simple value; the second signifies so many tens, the third so many hundreds, and the others so many higher classes, according to the order they stand in. And as it may sometimes be required to express a number consisting of tens, hundreds, or higher classes, without any units or classes of a lower rank annexed; and as this can only be done by figures standing in the second, third, or higher place, while there are none to fill up the lower ones; therefore an additional character or cypher (o) is necessary, which has no signification when placed by itself, but serves to supply the vacant places, and bring the figures to their proper station.

The following table shews the names and divisions of the classes.

| Trillions | Billions | Millions | Thousands | Hundreds | Tens | |-----------|----------|----------|-----------|----------|------| | 8.4579825643 | 64738972645 | Units |

The first six figures from the right hand are called the unit period, the next six the million period, after which the trillion, quadrillion, quintillion, sextillion, septillion, octillion and nonillion periods follow in their order.

It is proper to divide any number, before we reckon it, into periods and half periods, by different marks. We then begin at the left hand, and read the figures in their order, with the names of their places, from the table. In writing any number, we must be careful to mark the figures in their proper places, and supply the vacant places with cyphers.

As there are no possible ways of changing numbers, except by enlarging or diminishing them according to some given rule, it follows, that the whole art of arithmetic is comprehended in two operations, Addition and Subtraction. However, as it is frequently required to add several equal numbers together, or to subtract several equal ones from a greater, till it be exhausted, proper methods have been invented for facilitating the operation in these cases, and distinguished by the names of Multiplication and Division; and these four rules are the foundation of all arithmetical operations whatever.

As the idea of number is acquired by observing several objects collected, so is that of fractions by observing an object divided into several parts. As we sometimes meet with objects broken into two, three, or more parts, we may consider any or all of these divisions promiscuously, which is done in the doctrine of vulgar fractions, for which a chapter will be allotted. However, since the practice of collecting units into parcels of tens has prevailed universally, it has been found convenient to follow a like method in the consideration of fractions, by dividing each unit into ten equal parts, and each of these into ten smaller parts; and so on. Numbers divided in this manner are called Decimal Fractions.

**Chap. II. ADDITION.**

Addition is that operation by which we find the amount of two or more numbers. The method of doing this in simple cases is obvious, as soon as the meaning of number is known, and admits of no illustration. A young learner will begin at one of the numbers and reckon up as many units separately as there are in the other, and practice will enable him to do it at once. It is impossible, strictly speaking, to add more than two numbers at a time. We must first find the sum of the first and second; then we add the third to that number; and so on. However, as the several sums obtained are easily retained in the memory, it is neither necessary nor usual to mark them down. When the numbers consist of more figures than one, we add the units together, the tens together; and so on. But, if the sum of the units exceed ten, or contain ten several times, we add the number of tens it contains to the next column, and only set down the number of units that are over. In like manner, we carry the tens of every column to the next higher. And the reason of this is obvious from the value of the places; since an unit, in any higher place, signifies the same thing as ten in the place immediately lower.

**Rule.** "Write the numbers distinctly, units under units, tens under tens; and so on. Then reckon the amount of the right-hand column. If it be under ten, mark it down. If it exceed ten, mark the units only, and carry the tens to the next place. In like manner, carry the tens of each column to the next, and mark down the full sum of the left-hand column."

**Example.**

``` 346863 + 876734 123467 314213 712316 438987 279654 3092234 ```

As it is of great consequence in business to perform addition readily and exactly, the learner ought to practise it till it become quite familiar. If the learner can readily add any two digits, he will soon add a digit to a higher number with equal ease. It is only to add the unit place of that number to the digit; and, if it exceed ten, it raises the amount accordingly. Thus, because 8 and 6 is 14, 48 and 6 is 54. It will be proper to mark down under the sums of each column, in a small hand, the figure that is carried to the next column. This prevents the trouble of going over the whole operation again, in case of interruption or mistake. If you want to keep the account clean, mark down the sum and figure you carry, on a separate paper, and, after revising them, transfer the sum only. After some practice, we ought to acquire the habit of adding two or more figures at one glance. This is particularly useful when two figures which amount to 10, as 6 and 4, or 7 and 3, stand together in the column.

Every operation in arithmetic ought to be revised, to prevent mistakes; and, as one is apt to fall into the same mistake if he revise it in the same manner he performed it, it is proper either to alter the order, or else to trace back the steps by which the operation advanced. Addition, which will lead us at last to the number we began with. Every method of proving accounts may be referred to one or other of these heads.

1st. Addition may be proven by any of the following methods: repeat the operation, beginning at the top of the column, if you began at the foot when you wrought it.

2nd. Divide the account into several parts; add these separately, and then add the sums together. If their amount correspond with the sum of the account, when added at once, it may be presumed right. This method is particularly proper when you want to know the sums of the parts, as well as that of the whole.

3rd. Subtract the numbers successively from the sum; if the account be right, you will exhaust it exactly, and find no remainder.

When the given number consists of articles of different value, as pounds, shillings, and pence, or the like, which are called different denominations, the operations in arithmetic must be regulated by the value of the articles. We shall give here a few of the most useful tables for the learners information.

I. Sterling Money.

| Articles | Equivalent | |-------------------|------------| | 4 Farthings | = 1 penny | | 12 Pence | = 1 shilling, s. | | 20 Shillings | = 1 pound, L. | | Alto, 6s. 8d. | = 1 noble | | 10s. | = 1 angel | | 13s. 4d. or two thirds of a pound | = 1 merk. |

Scots money is divided in the same manner as Sterling, and has one twelfth of its value. A pound Scots is equal to 13. 8d. Sterling, a shilling Scots to a penny Sterling, and a penny Scots to a twelfth part of a penny Sterling; a mark Scots is two thirds of a pound Scots, or 13½d. Sterling.

II. Averdupois weight.

| Articles | Equivalent | |-------------------|------------| | 16 Drams | = 1 ounce, oz. | | 16 Ounces | = 1 pound, lb. | | 28 Pound | = 1 quarter, qr. | | 4 Quart | = 1 hom.wght, C. | | 20 Flun.weight | = 1 ton, T. | | 10s. | = 1 angel | | 13s. 4d. or two thirds of a pound | = 1 merk. |

Examples in Sterling Money.

| L | 145 | 6 | 8 | L | 16 | 9 | 11½ | |-------|-----|---|---|-------|----|---|------| | 215 | 3 | 9 | | 169 | 16 | 10 | | | 172 | 18 | 4 | | 36 | 12 | 9½ | | | 645 | 7 | 7 | | 54 | 7 | 6 | | | 737 | 2 | 3 | | 30 | 1½ | | | | 35 | 3 | 9 | | 7 | 19 | 6 | | | 9 | — | 7 | | 727 | 19 | 11 | | | 1764 | 12 | 3 | | 14 | 14 | 4 | | | 780 | — | | | 84 | 18 | 8½ | | | 99 | 9 | 9 | | 125 | 3 | 7 | | | 150 | 10 | — | | 16 | 16 | 8½ | | | 844 | 8 | 7 | | 62 | 5 | 3 | |

III. Troy Weight.

| Articles | Equivalent | |-------------------|------------| | 20 Mites | = 1 grain, gr. | | 24 Grains | = 1 pen.wt., dwt. | | 20 Pennywts | = 1 ounce, oz. | | 12 Ounces | = 1 pound, lb. |

IV. Apothecaries Weight.

| Articles | Equivalent | |-------------------|------------| | 20 Grains | = scruple, 3 | | 3 Scruples | = dram, 3 | | 8 Drams | = ounce, 3 | | 12 Ounces | = pound, lb. |

V. English Dry Measure.

| Articles | Equivalent | |-------------------|------------| | 2 Pints | = quart | | 4 Quarts | = gallon | | 2 Gallons | = peck | | 4 Pecks | = bushel | | 8 Bushels | = quarter |

VI. Scots Dry Measure.

| Articles | Equivalent | |-------------------|------------| | 4 Lippics | = peck | | 4 Pecks | = firlot | | 4 Firlots | = boll | | 16 Bolls | = chalder |

VII. English Land Measure.

| Articles | Equivalent | |-------------------|------------| | 30½ Square yards | = pole or perch | | 40 Poles | = rood | | 4 Rods | = acre |

VIII. Scots Land Measure.

| Articles | Equivalent | |-------------------|------------| | 36 Square ells | = fall | | 40 Falls | = rood | | 4 Rods | = acre |

IX. Long Measure.

| Articles | Equivalent | |-------------------|------------| | 12 Inches | = foot | | 3 Feet | = yard | | 5½ Yards | = pole | | 40 Poles | = furlong | | 8 Furlongs | = mile | | 3 Miles | = league |

X. Time.

| Articles | Equivalent | |-------------------|------------| | 60 Seconds | = minute | | 60 Minutes | = hour | | 24 Hours | = day | | 7 Days | = week | | 365 Days | = year | | 52 Weeks & 1 day | = year |

Rule for compound Addition. "Arrange like quantities under like, and carry according to the value of the higher place."

Note 1. When you add a denomination, which contains more columns than one, and from which you carry to the higher by 20, 30, or any even number of tens, first add the units of that column, and mark down their sum, carrying the tens to the next column; then add the tens, and carry to the higher denomination, by the number of tens that it contains of the lower. For example, in adding shillings, carry by 10 from the units to the tens, and by 2 from the tens to the pounds.

Note 2. If you do not carry by an even number of tens, first find the complete sum of the lower denomination, then inquire how many of the higher that sum contains, and carry accordingly, and mark the remainder, if any, under the column. For example, if the sum of a column of pence be 43, which is three shillings and seven pence, mark 7 under the pence-column, and carry 3 to that of the shillings.

Note 3. Some add the lower denominations after the following method: when they have reckoned as many as amounts to one of the higher denomination, or upwards, they mark a dot, and begin again with the excess of the number reckoned above the value of the denomination. The number of dots shows how many are carried, and the last reckoned number is placed under the column.

Examples in Troy Weight.

| T. G. qr. lb. | T. G. qr. lb. | |---------------|---------------| | 1 19 3 26 | 3 15 2 22 | | — 14 1 16 | 6 3 — 19 | | 2 18 1 16 | 5 7 3 26 | | — 1 2 27 | 3 2 2 — | | 3 9 — 10 | 4 3 1 10 | | — 17 2 24 | — 18 1 12 | | — 15 3 18 | 1 1 1 1 | | 4 6 — 5 | 5 3 — 7 | | — 6 3 9 | 6 4 — 9 | | 6 4 — 4 | 4 6 — 5 | | 5 5 — 5 | 2 1 3 4 |

In Averdupois Weight.

| T. G. qr. lb. | T. G. qr. lb. | |---------------|---------------| | 1 19 3 26 | 3 15 2 22 | | — 14 1 16 | 6 3 — 19 | | 2 18 1 16 | 5 7 3 26 | | — 1 2 27 | 3 2 2 — | | 3 9 — 10 | 4 3 1 10 | | — 17 2 24 | — 18 1 12 | | — 15 3 18 | 1 1 1 1 | | 4 6 — 5 | 5 3 — 7 | | — 6 3 9 | 6 4 — 9 | | 6 4 — 4 | 4 6 — 5 | | 5 5 — 5 | 2 1 3 4 |

When one page will not contain the whole account, we add the articles it contains, and write against their sum, Carried forward; and we begin the next page with the sum of the foregoing, writing against it, Brought forward.

When the articles fill several pages, and their whole sum is known, which is the case in transferring accounts, it is best to proceed in the following manner: Add the pages, placing the sums on a separate paper; then add the sums, and if the amount of the whole be right, it only remains to find what numbers should be placed at Subtraction is the operation by which we take a lesser number from a greater, and find their differences. It is exactly opposite to addition, and is performed by learners in a like manner, beginning at the greater and reckoning downwards the units of the lesser. The greater is called the minuend, and the lesser the subtrahend.

If any figure of the subtrahend be greater than the corresponding figure of the minuend, we add ten to that of the minuend; and, having found and marked the difference, we add one to the next place of the subtrahend. This is called borrowing ten. The reason will appear, if we consider that, when two numbers are equally increased by adding the same to both, their difference will not be altered. When we proceed as directed above, we add ten to the minuend, and we likewise add one to the higher place of the subtrahend, which is equal to ten of the lower place.

Rule. "Subtract units from units, tens from tens, and so on. If any figure of the subtrahend be greater than the corresponding one of the minuend, borrow ten."

Example. Minuend 173694 738641 Subtrahend 21453 379235 Remainder 152241 359406

To prove subtraction, add the subtrahend and remainder together; if their sum be equal to the minuend, the account is right.

Or subtract the remainder from the minuend. If the difference be equal to the subtrahend, the account is right.

Then we transcribe L778:16s. at the foot of the first and top of the second pages, L1224:10s.5 at the foot of the second and top of the third; and so on.

Note 1. The reason for borrowing is the same as in simple subtraction. Thus, in subtracting pence, we add 12 pence when necessary to the minuend, and at the next step, we add one shilling to the subtrahend.

Note 2. When there are two places in the same denomination, if the next higher contain exactly so many tens, it is best to subtract the units first, borrowing ten when necessary; and then subtract the tens, borrowing, if there is occasion, according to the number of tens in the higher denomination.

Note 3. If the value of the higher denomination be not an even number of tens, subtract the units and tens at once, borrowing according to the value of the higher denomination.

Note 4. Some choose to subtract the place in the subtrahend, when it exceeds that of the minuend, from the value of the higher denomination, and add the minuend to the difference. This is only a different order of proceeding, and gives the same answer.

Note 5. As custom has established the method of placing the subtrahend under the minuend, we follow it when there is no reason for doing otherwise; but the minuend may be placed under the subtrahend with equal propriety; and the learner should be able to work it either way, with equal readiness, as this last is sometimes more convenient; of which instances will occur afterwards.

Note 6. The learner should also acquire the habit, when two numbers are marked down, of placing such a number under the lesser, that, when added together, the sum may be equal to the greater. The operation is the same as subtraction, though conceived in a different manner, and is useful in balancing accounts, and on other occasions.

It is often necessary to place the sums in different columns, in order to exhibit a clear view of what is required. For instance, if the values of several parcels of goods are to be added, and each parcel consists of several articles, the particular articles should be placed in an inner column, and the sum of each parcel extended to the outer column, and the total added there.

If any person be owing an account, and has made some partial payments, the payments must be placed in an inner column, and their sum extended under that of the account in the outer column, and subtracted there.

An example or two will make this plain.

1st.] 30 yards linen at 2s. L. 3 45 ditto at 1s. 6d. 3 7 6 L. 6 7 6 120 lb thread at 4s. L. 24 40 ditto at 3s. 6 30 ditto at 2s. 6d. 3 15 33 15 L. 40 2 6 24.] Ex. 76859 multiplied by 4, or 76859 added 4 times.

If the multiplier be 10, we annex a cypher to the multiplicand. If the multiplier be 100, we annex two cyphers; and so on. The reason is obvious, from the use of cyphers in notation.

If the multiplier be any digit, with one or more cyphers on the right hand, we multiply by the figure, and annex an equal number of cyphers to the product. Thus, if it be required to multiply by 50, we first multiply by 5, and then annex a cypher. It is the same thing as to add the multiplicand fifty times; and this might be done by writing the account at large, dividing the column into 10 parts of 5 lines, finding the sum of each part, and adding these ten sums together.

If the multiplier consist of several significant figures, we multiply separately by each, and add the products. It is the same as if we divided a long account of addition into parts corresponding to the figures of the multiplier.

Example. To multiply 7329 by 365.

\[ \begin{array}{cccc} 7329 & 7329 & 7329 & 36645 = 5 \text{ times.} \\ 5 & 60 & 300 & 439740 = 60 \text{ times.} \\ 36645 & 439740 & 2198700 & 2675085 = 365 \text{ times.} \end{array} \]

It is obvious that 5 times the multiplicand added to 60 times, and to 300 times, the same must amount to the product required. In practice, we place the products at once under each other; and, as the cyphers arising from the higher places of the multiplier are lost in the addition, we omit them. Hence may be inferred the following:

Rule. "Place the multiplier under the multiplicand, and multiply the latter successively by the significant figures of the former; placing the right-hand figure of each product under the figure of the multiplier from which it arises; then add the product."

Ex. 7329 42785 37846 93956

\[ \begin{array}{cccc} 365 & 91 & 235 & 8704 \\ 36645 & 42785 & 189230 & 375824 \\ 43974 & 385065 & 113538 & 657692 \\ 21987 & 3893431 & 75992 & 751648 \\ 2675085 & 8893810 & 817793024 \end{array} \]

A number which cannot be produced by the multiplication of two others is called a prime number; as 3, 5, 7, 11, and many others.

A number which may be produced by the multiplication of two or more smaller ones, is called a composite number. For example, 27, which arises from the multiplication of 9 by 3; and these numbers (9 and 3) are called the component parts of 27.

Contractions and Varieties in Multiplication.

First, If the multiplier be a composite number, we may multiply successively by the component parts. Because the second product is equal to five times the first, and the first is equal to nine times the multiplicand, it is obvious that the second product must be five times nine, or forty-five times as great as the multiplicand.

Secondly, If the multiplier be 5, which is the half of 10, we may annex a cypher and divide by 2. If it be 25, which is the fourth part of an 100, we may annex two cyphers, and divide by 4. Other contractions of the like kind will readily occur to the learner.

Thirdly, To multiply by 9, which is one less than 10, we may annex a cypher; and subtract the multiplicand from the number it composes. To multiply by 99, 999, or any number of 9's, annex as many cyphers, and subtract the multiplicand. The reason is obvious; and a like rule may be found, though the unit place be different from 9.

Fourthly, Sometimes a line of the product is more easily obtained from a former line of the same than from the multiplicand.

In the first example, instead of multiplying by 5, we may multiply 5488 by 2; and, in the second, instead of multiplying by 3, we may divide 8088 by 2.

Fifthly, Sometimes the product of two or more figures may be obtained at once, from the product of a figure already found.

In the second example, we multiply first by 4; then, because 12 times 4 is 48, we multiply the first line of the product by 12, instead of multiplying separately by 8 and 4; lastly, because twice 48 is 96, we multiply the second line of the product by 2, instead of multiplying separately by 6 and 9.

When we follow this method, we must be careful to place the right-hand figure of each product under the right-hand figure of that part of the multiplier which it is derived from.

It would answer equally well in all cases, to begin the work at the highest place of the multiplier; and contractions are sometimes obtained by following that order.

It is a matter of indifference which of the factors be used as the multiplier; for 4 multiplied by 3 gives the same product as 3 multiplied by 4; and the like holds universally true. To illustrate this, we may mark three rows of points, four in each row, placing the rows under each other; and we shall also have four rows, containing three points each, if we reckon the rows downwards.

Multiplication is proven by repeating the operation, using the multiplier for the multiplicand, and the multiplicand for the multiplier. It may also be proven by division, or by calling out the 9's; of which afterwards; and an account, wrought by any contraction, may be proven by performing the operation at large, or by a different contraction.

**Compound Multiplication.**

**Rule I.** "If the multiplier do not exceed 12, the operation is performed at once, beginning at the lowest place, and carrying according to the value of the higher place."

[Examples.]

| L. | 13 | 6 | 7 | 12 | 2 | 8 | 13 | 3 | 18 | 7 | 5 | 9 | |----|----|----|----|----|----|----|----|----|----|----|----|----| | | 9 | | | | | | | | | | | |

| L. | 119 | 19 | 3 | 62 | 3 | 12 | 83 | — | 28 | 89 | 5 | 8 | |----|-----|----|---|----|---|----|----|---|----|----|----|----|

**Rule II.** "If the multiplier be a composite number, whose component parts do not exceed 12, multiply first by one of these parts, then multiply the product by the other. Proceed in the same manner if there be more than two."

| Ex. | 1st. | L. | 15 | 3 | 8 by 32 = 8 × 4 | |-----|------|----|----|----|-----------------| | | | | | | | | | | | | | | | | | | | | |

| L. | 121 | 9 | 4 = 8 times | |----|-----|----|------------| | | | | |

| L. | 485 | 17 | 4 = 32 times | |----|-----|----|-------------| | | | | |

| 2nd. | L. | 17 | 3 | 8 by 75 = 5 × 5 × 3 | |------|----|----|----|-------------------| | | | | | | | | | | | |

| L. | 51 | 11 | = 3 times | |----|----|----|----------| | | | | |

| L. | 257 | 15 | = 15 times | |----|-----|----|-----------| | | | | |

| L. | 1288 | 15 | = 75 times | |----|-------|----|-----------| | | | | |

Note 1. Although the component parts will answer in any order, it is best, when it can be done, to take them in such order as may clear off some of the lower places at the first multiplication, as it done in Ex. 2nd.

Note Note 2. The operation may be proved, by taking the component parts in a different order, or dividing the multiplier in a different manner.

Rule III. "If the multiplier be a prime number, multiply first by the composite number next lower; then by the difference, and add the products."

L. 35 17 9 by 67 = 64 × 3 Here, because 8 times 8 = 64, we multiply twice by 8, which gives L. 2296

L. 287 2 — = 8 times. 16s. equal to 64 times the multiplicand; then we find the amount of 3 times the multiplicand, which is

L. 2296 16 — = 64 times. L. 107 13 3; and it is evident that these added, amount to 67, the multiplicand.

Rule IV. "If there be a composite number little above the multiplier, we may multiply by that number, and by the difference, and subtract the second product from the first."

L. 17 4 5 by 106 = 108 — 2 Here we multiply

L. 206 13 — = 9 times. 933 shillings in L. 46 13

L. 1859 17 — = 108 times. 11196 pence in L. 46 13

L. 1825 8 2 = 106 times.

Example. L. 34 8 2 by 3465

Rule V. "If the multiplier be large, multiply by 10, and multiply the product again by 10; by which means you obtain an hundred times the given number. If the multiplier exceed 1000, multiply by 10 again; and continue it farther if the multiplier require it; then multiply the given number by the unit-place of the multiplier; the first product by the ten-place, the second product by the hundred-place; and so on. Add the products thus obtained together."

L. 34 9 2 by 5 = L. 172 1 5 times

The use of multiplication is to compute the amount of any number of equal articles, either in respect of measure, weight, value, or any other consideration. The multiplicand expresses how much is to be reckoned for each article; and the multiplier expresses how many times that is to be reckoned. As the multiplier points out the number of articles to be added, it is always an abstract number, and has no reference to any value or measure whatever. It is therefore quite improper to attempt the multiplication of shillings by shillings, or to consider the multiplier as expressive of any denomination. The most common instances in which the practice of this operation is required, are, to find the amount of any number of parcels, to find the value of any number of articles, to find the weight or measure of a number of articles, &c.

This computation, for changing any sum of money, weight, or measure, into a different kind, is called Reduction. When the given quantity is expressed in different denominations, we reduce the highest to the next lower, and add thereto the given number of that denomination; and proceed in like manner till we have reduced it to the lowest denomination.

Example. To reduce L. 46 13 8½ to farthings.

L. 46 20 Or thus:

920 shillings in L. 46 13

933 shillings in L. 46 13

11196 pence in L. 46 13

11204 pence in L. 46 13

44816 farthings in L. 46 13

44819 farthings in L. 46 13 8½

It is easy to take in or add the higher denomination at the same time we multiply the lower.

Chap. V. Division.

In division, two numbers are given; and it is required to find how often the former contains the latter. Thus, it may be asked how often 21 contains 7, and the answer is exactly 3 times. The former given number (21) is called the Dividend; the latter (7) the Divisor; and the number required (3) the Quotient. It frequently happens that the division cannot be completed exactly without fractions. Thus it may be asked, how often 8 is contained in 19? the answer is twice, and a remainder of 3.

This operation consists in subtracting the divisor from the dividend, and again from the remainder, as often as it can be done, and reckoning the number of subtractions; as,

| Subtraction | Remainder | |-------------|-----------| | 21 | | | 7 first subtraction | 19 | | 7 second subtraction | 11 | | 7 third subtraction | 3 remainder |

As this operation, performed at large, would be very tedious, when the quotient is a high number, it is proper to shorten it by every convenient method; and, for this purpose, we may multiply the divisor by any any number whose product is not greater than the dividend, and so subtract it twice or thrice, or oftener, at the same time. The best way is to multiply it by the greatest number, that does not raise the product too high, and that number is also the quotient. For example, to divide 45 by 7, we inquire what is the greatest multiplier for 7, that does not give a product above 45; and we shall find that it is 6; and 6 times 7 is 42, which, subtracted from 45, leaves a remainder of 3. Therefore 7 may be subtracted 6 times from 45; or, which is the same thing, 45, divided by 7, gives a quotient of 6, and a remainder of 3.

If the divisor do not exceed 12, we readily find the highest multiplier that can be used from the multiplication table. If it exceed 12, we may try any multiplier that we think will answer. If the product be greater than the dividend, the multiplier is too great; and, if the remainder, after the product is subtracted from the dividend, be greater than the divisor, the multiplier is too small. In either of these cases, we must try another. But the attentive learner, after some practice, will generally hit on the right multiplier at first.

If the divisor be contained oftener than ten times in the dividend, the operation requires as many steps as there are figures in the quotient. For instance, if the quotient be greater than 100, but less than 1000, it requires 3 steps. We first inquire how many hundred times the divisor is contained in the dividend, and subtract the amount of these hundreds. Then we inquire how often it is contained ten times in the remainder, and subtract the amount of these tens. Lastly, we inquire how many single times it is contained in the remainder. The method of proceeding will appear from the following example:

To divide 5936 by 8.

From 5936 Take 5600 = 700 times 8

Rem. 336 From which take 320 = 40 times 8

Rem. 16 From which take 16 = 2 times 8

742 times 8 in all.

It is obvious, that as often as 8 is contained in 59, so many hundred times it will be contained in 5900, or in 5936; and, as often as it is contained in 33, so many ten times it will be contained in 330, or in 336; and thus the higher places of the quotient will be obtained with equal ease as the lower. The operation might be performed by subtracting 8 continually from the divided, which will lead to the same conclusion by a very tedious process. After 700 subtractions, the remainder would be 336; after 40 more, it would be 16; and after 2 more, the dividend would be entirely exhausted. In practice, we omit the cyphers, and proceed by the following rule:

Rule. 1st, "Assume as many figures on the left hand of the multiplier as contain the divisor once or oftener: find how many times they contain it, and place the answer as the highest figure of the quotient."

2nd, "Multiply the divisor by the figure you have found, and place the product under the part of the dividend from which it is obtained."

3rd, "Subtract the product from the figures above it."

4th, "Bring down the next figure of the dividend to the remainder, and divide the number it makes up, as before."

Example. 1st. 8)5936(742

| 56 | |----| | 33 |

2nd. 63)30114(478

| 32 | |----| | 16 |

3rd. 365)974932(2671

| 730 | |----| | 2449 |

Remainder 17

The numbers which we divide, as 59, 33, and 16, in the first example, are called individuals.

It is usual to mark a point under the figures of the dividend, as they are brought down, to prevent mistakes.

If there be a remainder, the division is completed by a vulgar fraction, whose numerator is the remainder, and its denominator the divisor. Thus, in Ex. 3, the quotient is 2671, and remainder 17; and the quotient completed is 2671 \(\frac{17}{365}\).

A number which divides another without a remainder is said to measure it; and the several numbers which measure another, are called its aliquot parts. Thus, 2, 4, 6, 8, and 12, are aliquot parts of 24. As it is often useful to discover numbers which measure others, we may observe,

1st, Every number ending with an even figure, that is, with 2, 4, 6, 8, or 0, is measured by 2.

2nd, Every number ending with 5, or 0, is measured by 5.

3rd, Every number, whose figures, when added, amount to an even number of 3's or 9's, is measured by 3 or 9, respectively.

Contractions and Varieties in Division.

First, When the divisor does not exceed 12, the whole computation may be performed without setting down any figures except the quotient.

Ex. 7)35868(5124 or 7)35868

Secondly, When the divisor is a composite number, and one of the component parts also measures the dividend, we may divide successively by the component parts.

Ex. This method might be also used, although the component parts of the divisor do not measure the dividend; but the learner will not understand how to manage the remainder till he be acquainted with the doctrine of vulgar fractions.

Thirdly, When there are cyphers annexed to the divisor, cut them off, and cut off an equal number of figures from the dividend; annex these figures to the remainder. Ex. To divide 378643 by 5200.

The reason will appear, by performing the operation at large, and comparing the steps.

To divide by 10, 100, 1000, or the like. Cut off as many figures on the right hand of the dividend as there are cyphers in the divisor. The figures which remain on the left hand compose the quotient, and the figures cut off compose the remainder.

Fourthly, When the divisor consists of several figures we may try them separately; by inquiring how often the first figure of the divisor is contained in the first figure of the dividend, and then considering whether the second and following figures of the divisor be contained as often in the corresponding ones of the dividend with the remainder (if any) prefixed. If not, we must begin again, and make trial of a lower number. When the remainder is nine, or upwards, we may be sure the division will hold through the lower places; and it is unnecessary to continue the trial farther.

Fifthly, We may make a table of the products of the divisor, multiplied by the nine digits, in order to discover more readily how often it is contained in each individual. This is convenient when the dividend is very long, or when it is required to divide frequently by the same divisor.

Rem. 8

Sixthly, To divide by 9, 99, 999, or any number of 9's, transcribe under the dividend part of the same, shifting the highest figure as many places to the right hand as there are 9's in the divisor. Transcribe it again, with the like change of place, as often as the length of the dividend admits; add these together, and cut off as many figures from the right hand of the sum as there are 9's in the divisor. The figures which remain on the left hand compose the quotient, and those cut off the remainder.

If there be any carriage to the unit-place of the quotient, add the number carried likewise to the remainder, as in Ex. 2.; and if the figures cut off be all 9's, add 1 to the quotient, and there is no remainder.

To explain the reason of this, we must recollect, that whatever number of hundreds any dividend contains, it contains an equal number of 99's, together with an equal number of units. In Ex. 1., the dividend contains 3241 hundreds, and a remainder of 23. It therefore contains 3241 times 99, and also 3241 besides the remainder already mentioned. Again, 3241 contains 32 hundreds, and a remainder of 51; it therefore contains 32 99's, and also 32, besides the remainder of 41. Consequently the dividend contains 99, altogether, 3241 times, and 32 times, that is 3273 times, and the remainder consists of 23, 41, and 32, added, which makes 96.

As multiplication supplies the place of frequent additions, and division of frequent subtractions, they are only repetitions and contractions of the simple rules, and when compared together, their tendency is exactly opposite. As numbers, increased by addition, are diminished and brought back to their original quantity by subtraction; in like manner, numbers compounded by multiplication are reduced by division to the parts from which they were compounded. The multiplier shows how many additions are necessary to produce the number; and the quotient shows how many subtractions are necessary to exhaust it. It follows that the product, divided by the multiplicand, will quote the multiplier; and, because either factor may be assumed for the multiplicand, therefore the product, divided by either factor, quotes the other. It follows, also, that the dividend is equal to the product of the divisor and quotient multiplied together; and hence these operations mutually prove each other.

To prove multiplication. Divide the product by either factor. If the operation be right, the quotient is the other factor, and there is no remainder.

To prove division. Multiply the divisor and quotient together; to the product add the remainder, if any; and, if the operation be right, it makes up the dividend. dividend. Otherwise divide the dividend (after subtracting the remainder, if any) by the quotient. If the operation be right, it will quote the divisor. The reason of all these rules may be collected from the last paragraph.

**Compound Division.**

**Rule I.** "When the dividend only consists of different denominations, divide the higher denomination, and reduce the remainder to the next lower, taking in (p. 659, Rule V.) the given number of that denomination, and continue the division."

**Examples.**

Divide L.465 : 12 : 8

by 72.

| L. s. d. | L. s. d. | |---------|---------| | 72)465 | 12 8 |

Divide 345 cwt. 1 q. 8 lb.

by 22.

| Cwt. q. lb. | Cwt. q. lb. | |-------------|-------------| | 22)345 | 1 8 |

Or we might divide by the component parts of 72, (as explained under Thirdly, p. 661).

| 22)484 | |--------|

**Rule II.** "When the divisor is in different denominations, reduce both divisor and dividend to the lowest denomination, and proceed as in simple division. The quotient is an abstract number."

To divide L.38 : 13 s.

by L.3 : 4 : 5.

| L. 3 4 5 | L. 38 13 | |----------|----------| | 20 20 | 4 4 |

To divide 96 Cwt. 1 q. 20 lb.

by 3 cwt. 2 q. 8 lb.

| Cwt. q. lb. | Cwt. q. lb. | |-------------|-------------| | 3 2 8 | 96 1 20 |

It is best not to reduce the terms lower than is necessary to render them equal. For instance, if each of them consists of an even number of sixpences, fourpence, or the like, we reduce them to sixpences, or fourpence, but not to pence.

The use of division is to find either of the factors by whose multiplication a given number is produced, when the other factor is given; and therefore is of two kinds, since either the multiplier or the multiplicand may be given. If the former be given, it discovers what that number is which is contained so many times in another. If the latter be given, it discovers how many times one number is contained in another. Thus, it answers the questions of an opposite kind to those mentioned under Rule IV. p. 659. as, To find the quantity of a single parcel or share; to find the value weight, or measure, of a single article; to find how much work is done, provisions consumed, interest incurred, or the like, in a single day, &c.

The last use of division is a kind of reduction exactly opposite to that described under Rule V. p. 659. The manner of conducting and arranging it, when there are several denominations in the question, will appear from the following examples.

1. To reduce 15783 pence to pounds, shillings, and pence.

2. To reduce 174865 grs. to lbs. oz. and dwt. Troy.

In the first example, we reduce 15783 pence to shillings, by dividing by 12, and obtain 1315 shillings, and a remainder of 3 pence. Then we reduce 1315 shillings to pounds, by dividing by 20, and obtain 65 pounds, and a remainder of 15 shillings. The divisions might have been contracted.

In the practice of arithmetic, questions often occur which require both multiplication and division to resolve. This happens in reduction, when the higher denomination does not contain an exact number of the lower.

**Rule for mixed reduction.** "Reduce the given denomination by multiplication to some lower one, which is an aliquot part of both; then reduce that by division to the denomination required."

Ex. Reduce L.31742 to guineas.

Here we multiply by 20, which reduces the pounds to shillings; and divide the product by 21, which reduces the shillings to guineas.

Answer, 30230 guineas and 10 shill. As Portuguese money frequently passes here in payments, we shall give a table of the pieces, and their value.

| Piece | Value | |----------------|-------| | A moitore | L.1 | | A half moitore | 7 | | A quarter moitore | 6 | | A double Joannes | 3 | | A Joannes | 12 | | A half ditto | 16 | | A quarter ditto| 9 | | An eighth ditto| 4 |

Note 1. Guineas may be reduced to pounds, by adding one twentieth part of the number.

2. Pounds may be reduced to merks by adding one half.

3. Merks may be reduced to pounds by subtracting one third.

4. Four moitores are equal to three Joannes; therefore moitores may be reduced to Joannes, by subtracting one fourth; and Joannes to moitores, by adding one third.

5. Five Joannes are equal to L.9. Hence it is easy to reduce Portuguese money to Sterling.

Another case, which requires both multiplication and division, is, when the value, weight, measure, or duration of any quantity is given, and the value, &c. of a different quantity required, we first find the value, &c. of a single article by division, and then the value, &c. of the quantity required, by multiplication.

Ex. If 3 yards cost 15 s. 9 d. what will 7 yards cost, at the same rate?

| s. d. | |-------| | 3 ) 15 9 Price of 3 yards. | | 5 3 Price of 1 yard by Rule IV. p. 659. | | 7 |

L.1 16 9 Price of 7 yards (by par. 1. p. 662. col. 2.)

Many other instances might be adduced, where the operation and the reason of it are equally obvious. These are generally, though unnecessarily, referred to the rule of proportion.

We shall now offer a general observation on all the operations in arithmetic. When a computation requires several steps, we obtain a just answer, whatever order we follow. Some arrangements may be preferable to others in point of ease, but all of them lead to the same conclusion. In addition, or subtraction, we may take the articles in any order, as is evident from the idea of number; or, we may collect them into several sums, and add or subtract these, either separately or together. When both the simple operations are required to be repeated, we may either complete one of them first, or may introduce them promiscuously; and the compound operations admit of the same variety.

When several numbers are to be multiplied together, we may take the factors in any order, or we may arrange them into several classes, find the product of each class, and then multiply the products together. When a number is to be divided by several others, we may take the divisors in any order, or we may multiply them into each other, and divide by the product; or we may multiply them into several parcels, and divide by the products successively. Lastly, when multiplication and division are both required, we may begin with either; and, when both are repeatedly necessary, we may collect the multipliers into one product, and the divisors into one another; or, we collect them into parcels, or use them singly, and that in any order. Still, we shall obtain the proper answer, if none of the terms be neglected.

When both multiplication and division are necessary to obtain the answer of a question, it is generally best to begin with the multiplication, as this order keeps the account as clear as possible from fraction. The example last given may be wrought accordingly as follows:

| s. d. | |-------| | 15 9 | | 7 |

Some accountants prove the operations of arithmetic by a method which they call casting out the 9's, depending on the following principles:

Firstly, if several numbers be divided by any divisor, (the remainders being always added to the next number), the sum of the quotients, and the last remainder, will be the same as those obtained when the sum of the numbers is divided by the same divisor. Thus, 19, 15, and 23, contain, together, as many 5's, as many 7's, &c. as their sum 57 does, and the remainders are the same; and, in this way, addition may be proven by division. It is from the correspondence of the remainders, that the proof, by casting out the 9's, is deduced.

Secondly, If any figure, with cyphers annexed, be divided by 9, the quotient consists entirely of that figure; and the remainder is also the same. Thus, 40, divided by 9, quotes 4, remainder 4; and 400, divided by 9, quotes 44, remainder 44. The same holds with all the digits; and the reason will be easily understood; every digit, with a cypher annexed, contains exactly so many tens; it must therefore contain an equal number of 9's, besides a remainder of an equal number of units.

Thirdly, If any number be divided by 9, the remainder is equal to the sum of the figures of the number, or to the remainder obtained, when that sum is divided by 9. For instance, 3765, divided by 9, leave a remainder of 3, and the sum of 3, 7, 6, and 5, is 21; which, divided by 9, leaves a remainder of 3. The reason of this will appear from the following illustration:

\[ \begin{array}{c} 3000 \text{ divided by } 9 \text{ quotes } 333; \text{ remainder } 3 \\ 700 \text{ quotes } 77; \text{ remainder } 7 \\ 60 \text{ quotes } 6; \text{ remainder } 6 \\ 5 \text{ quotes } 0; \text{ remainder } 5 \\ \hline 3765 \text{ Sum of rem. } 21 \end{array} \]

Again; 21 divided by 9 quotes 2; remainder 3

Wherefore, 3765 divid. by 9 quotes 418; remainder 3; for the reason given. Hence we may collect the following rules for practice:

To cast the 9's out of any number, or to find what remainder will be left when any number is divided by 9: Add the figures; and, when the sum exceeds 9, add the figures which would express it. Pass by the 9's; and, when the sum comes exactly to 9, neglect it, and begin anew. For example, if it be required to cast the 9's out of 3573294, we reckon thus: 3 and 4 O 2 5 is Division: 5 is 8, and 7 is 15; 1 and 5 is 6, and 3 is 9, which we neglect; 2 and (falling by 9), 4 is 6; which is the remainder or Result. If the article out of which the 9's are to be cast contains more denominations than one, we cast the 9's out of the higher, and multiply the result by the value of the lower, and carry on the product (casting out the 9's, if necessary), to the lower.

To prove addition, cast the 9's out of the several articles, carrying the results to the following articles; cast them also out of the sum. If the operation be right, the results will agree.

To prove subtraction, cast the 9's out of the minuend; cast them also out of the subtrahend and remainder together; and if you obtain the same result, the operation is presumed right.

To prove multiplication, cast the 9's out of the multiplicand, and also out of the multiplier, if above 9. Multiply the results together, and cast the 9's, if necessary, out of their product. Then cast the 9's out of the product, and observe if this result correspond with the former.

Ex. 1st.] 9276 ref. 6×8=48 ref. 3. 8 74208 ref. 3. 2nd.] 7898 ref. 5×3=15 ref. 6. 48 ref. 3 63184 31592 379104 ref. 6.

The reason of this will be evident, if we consider multiplication under the view of repeated addition. In the first example it is obviously the same. In the second, we may suppose the multiplicand repeated 48 times. If this be done, and the 9's cast out, the result, at the end of the 9th line, will be 0; for any number, repeated 9 times, and divided by 9, leaves no remainder. The same must happen at the end of the 18th, 27th, 36th, and 45th lines; and the last result will be the same as if the multiplicand had only been repeated 3 times. This is the reason for casting out the 9's from the multiplier as well as the multiplicand.

To prove division, cast the 9's out of the divisor, and also out of the quotient; multiply the results, and cast the 9's out of the product. If there be any remainder, add it to the result, casting out the 9's, if necessary. If the account be right, the last result will agree with that obtained from the dividend.

Ex. 42 ) 2490 ( 59 ref. 5×6=30 ref. 3. ref. 6 210 390 378

Rem. 12 ref. 3.

And the result of the dividend is 6.

This depends on the same reason as the last; for the dividend is equal to the product of the divisor and quotient added to the remainder.

We cannot recommend this method, as it lies under the following disadvantages:

First, If an error of 9, or any of its multiples, be committed, the results will nevertheless agree; and so the error will remain undiscovered. And this will always be the case, when a figure is placed or reckoned in a wrong column; which is one of the most frequent causes of error.

Secondly, When it appears by the disagreement of the results, that an error has been committed, the particular figure or figures in which the error lies are not pointed out; and, consequently, it is not easily corrected.

Chap. VI. RULE of PROPORTION.

Sect. i. Simple Proportion.

Quantities are reckoned proportional to each other, when they are connected in such a manner, that, if one of them be increased or diminished, the other increases or diminishes at the same time; and the degree of the alteration on each is a like part of its original measure; thus four numbers are in the same proportion, the first to the second as the third to the fourth, when the first contains the second, or any part of it, as often as the third contains the fourth, or the like part of it. In either of these cases, the quotient of the first, divided by the second, is equal to that of the third divided by the fourth; and this quotient may be called the measure of the proportion.

Proportionals are marked down in the following manner:

\[ \frac{6}{12} : \frac{3}{6} :: \frac{8}{9} : \frac{27}{24} : \frac{18}{10} : \frac{15}{10} \]

The rule of proportion directs us, when three numbers are given, how to find a fourth, to which the third may have the same proportion that the first has to the second. It is sometimes called the Rule of Three, from the three numbers given; and sometimes the Golden Rule, from its various and extensive utility.

Rule. "Multiply the second and third terms together, and divide the product by the first."

Ex. To find a fourth proportional to 18, 27, and 34.

\[ \frac{18}{27} : \frac{34}{51} \]

To explain the reason of this, we must observe, that, if two or more numbers be multiplied or divided alike, the products or quotients will have the same proportion.

\[ \frac{18}{27} \] Multiplied by 34, 612 : 918 Divided by 18, 34 : 51 The products 612, 918, and the quotients 34, 51, have therefore the same proportion to each other that 18 has to 27. In the course of this operation, the products of the first and third term is divided by the first; therefore the quotient is equal to the third.

The first and second terms must always be of the same kind; that is, either both monies, weights, measures, both abstract numbers, or the like. The fourth, or number sought, is of the same kind as the third.

When any of the terms is in more denominations than one, we may reduce them all to the lowest. But this is not always necessary. The first and second should not be reduced lower than directed p. 662, col. 1, par. ult.; and, when either the second or third is a simple number, the other, though in different denominations, may be multiplied without reduction.

\[ \frac{L}{s} : \frac{d}{s} = \frac{5}{7} : \frac{25}{11} \]

\[ \frac{L}{s} : \frac{d}{s} = \frac{7}{9} \]

Ex. 5 : 7 :: 25 : 11

The accountant must consider the nature of every question, and observe the circumstance which the proportion depends on; and common sense will direct him to this, if the terms of the question be understood. It is evident that the value, weight, and measure of any commodity is proportioned to its quantity; that the amount of work or consumption is proportioned to the time; that gain, loss, or interest, when the rate and time are fixed, is proportioned to the capital sum from which it arises; and that the effect produced by any cause is proportioned to the extent of the cause. In these, and many other cases, the proportion is direct, and the number sought increases or diminishes along with the term from which it is derived.

In some questions, the number sought becomes less, when the circumstances from which it is derived become greater. Thus, when the price of goods increases, the quantity which may be bought for a given sum is smaller. When the number of men employed at work is increased, the time in which they may complete it becomes shorter; and, when the activity of any cause is increased, the quantity necessary to produce a given effect is diminished. In these, and the like, the proportion is said to be inverse.

**General Rule for stating all questions, whether direct or inverse.** "Place that number for the third term which signifies the same kind of thing with what is sought, and consider whether the number sought will be greater or less. If greater, place the least of the other terms for the first; but, if less, place the greatest for the first."

Ex. 1st.] If 30 horses plough 12 acres, how many will 42 plough in the same time?

H. H. A.

\[ \frac{30}{42} : \frac{12}{12} \]

Here, because the thing sought is a number of acres, we place 12, the given number of acres, for the third term; and, because 42 horses will plough more than 12, we make the lesser number 30, the first term, and the greater number, 42, the second term.

2nd.] If 40 horses be maintained for a certain sum on hay, at 5 d. per stone, how many will be maintained on the same sum when the price of hay rises to 8 d.

\[ \frac{8}{5} : \frac{40}{40} \]

Here, because a number of horses is sought, we make the given number of horses, 40, the third term; and, because fewer will be maintained for the same money, when the price of hay is dearer, we make the greater price, 8 d., the first term; and the lesser price, 5 d., the second term.

The first of these examples is direct, the second inverse. Every question consists of a supposition and demand. In the first, the supposition is, that 30 horses plough 12 acres; and the demand, how many 42 will plough? and the first term of the proportion, 30, is found in the supposition, in this, and every other direct question. In the second, the supposition is, that 40 horses are maintained on hay at 5 d. and the demand, how many will be maintained on hay at 8 d.? and the first term of the proportion, 8, is found in the demand, in this and every other inverse question.

When an account is stated, if the first and second term, or first and third, be measured by the same number, we may divide them by that measure, and use the quotients in their stead.

Ex. If 36 yards cost 42 shillings, what will 27 cost?

Y. Y. s.

\[ \frac{36}{27} : \frac{42}{42} \]

Here 36 and 27 are both measured by 9, and we work with the quotients 4 and 3.

\[ \frac{4}{3} : \frac{126}{31} \]

Sect. ii. Compound Proportion.

Sometimes the proportion depends upon several circumstances. Thus, it may be asked, if 18 men consume 6 bolls corn in 40 days, how much will 24 men consume in 56 days? Here the quantity required depends partly on the number of men, partly on the time, and the question may be resolved into the two following ones:

1st, If 18 men consume 6 bolls in a certain time, how many will 24 men consume in the same time?

M. M. B. B.

\[ \frac{18}{24} : \frac{6}{8} : \frac{8}{6} \]

Answer. 24 men will consume 8 bolls in the same time.

2nd, If a certain number of men consume 8 bolls in 28 days, how many will they consume in 56 days?

D. D. B. B.

\[ \frac{28}{56} : \frac{8}{16} \]

Answer. The same number of men will consume 16 bolls in 56 days.

In the course of this operation, the original number of bolls, 6, is first multiplied into 24, then divided by 18, then multiplied into 8, then divided by 28. It would answer the same purpose to collect the multipliers into one product, and the divisors into another; and then to multiply the given number of bolls by the former, and divide the product by the latter. p. 663, col. 1, par. ult.

The above question may therefore be stated and wrought as follows: Here we multiply 18 into 40 for a divisor, and 6 into the product of 24 by 56, for a dividend.

If it be required, how many lb. averdupois will make 100 of Seville, the terms would have been placed in the different columns thus:

\[ \begin{array}{ccc} 104 & : & 112 \\ 89 & : & 100 \\ 117 & : & 110 \end{array} \]

Sect. iii. Distributive Proportion.

If it be required to divide a number into parts, which have the same proportion to each other that several other given numbers have, we add these numbers together, and state the following proportion: As the sum is to the particular numbers, so is the number required to be divided to the several parts sought.

Ex. 151. Four partners engage to trade in company; A's stock L.150, B's L.320, C's L.350, D's L.500, and they gain L.730; Required how much belongs to each, if the gain be divided among them in proportion to their stocks?

| Partner | Stock | Gain | |---------|-------|------| | A | L.150 | L.130 | | B | L.320 | L.110 | | C | L.350 | L.110 | | D | L.500 | L.110 |

This account is proved by adding the gains of the partners; the sum of which will be equal to the whole gain, if the operation be right; but, if there be remainders, they must be added, their sum divided by the common divisor, and the quotient carried to the lowest place.

Ex. 24. A bankrupt owes A L.145, B L.170, C L.45, D L.480, and E L.72; his whole effects are only L.342:7:6. How much should each have?

A's debt L.145:9:15 = L.342:7:6 = L.54:15 A's share. B's 170:9:15 = 342:7:6 = 65:15 B's share. C's 45:9:15 = 342:7:6 = 16:17 C's share. D's 480:9:15 = 342:7:6 = 180:15 D's share. E's 72:9:15 = 342:7:6 = 27:15 E's share.

This might also be calculated, by finding what composition the bankrupt was able to pay per pound; which is obtained by dividing the amount of his effects by the amount of his debts; and comes to 7s. 6d. and then finding by the rules of practice, how much each debt came to at that rate.

Chap. VII. Rules for Practice.

The operations explained in the foregoing chapters comprehend the whole system of arithmetic, and are sufficient for every computation. In many cases, however, the work may be contracted, by adverting to the particular circumstances of the question. We shall explain, in this chapter, the most useful methods which practice has suggested for rendering mercantile computations easy; in which, the four elementary rules of arithmetic are sometimes jointly, sometimes separately employed.

Sect. i. Computation of Prices.

The value of any number of articles, at a pound, a shilling, shilling, or a penny, is an equal number of pounds, shillings, or pence; and these two last are easily reduced to pounds. The value, at any other rate, may be calculated by easy methods, depending on some contraction already explained, or on one or more of the following principles.

1st. If the rate be an aliquot part of a pound, a shilling, or a penny, then an exact number of articles may be bought for a pound, a shilling, or a penny; and the value is found by dividing the given number accordingly. Thus, to find the price of so many yards at 2s. 6d. which is the eighth part of a pound, we divide the quantity by eight, because every eight yards cost L. 1.

2nd. If the rate be equal to the sum of two other rates which are easily calculated, the value may be found by computing these separately, and adding the sums obtained. Thus, the price of so many yards, at 9d., is found, by adding their prices, at 6d. and 3d. together.

3rd. If the rate be equal to the difference of two easy rates, they may be calculated separately, and the lesser subtracted from the greater. Thus, the value of so many articles at 1½d. is found, by subtracting their value at a penny from their value at a shilling. We may suppose that a shilling was paid for each article, and then a penny returned on each.

4th. If the rate be a composite number, the value may be found by calculating what it comes to at one of the component parts, and multiplying the same by the other.

CASE I. "When the rate is an aliquot part of a pound, divide the quantity by the number which may be bought for a pound."

Table of the aliquot parts of L. 1.

| Shillings | Pence | |-----------|-------| | 10 | 1 | | 6 s. 8 d. | 1 | | 5 s. | 1 | | 4 s. | 1 | | 3 s. 4 d. | 1 | | 2 s. 6 d. | 1 | | 2 s. | 1 | | 1 s. 8 d. | 1 |

Ex. 1st.] What is the value of 7463 yards, at 4s?

Ex. 2nd.] What is the value of 1773 yards at 3d.

In the first example we divide by 5, because 4s. is ¼ of a pound; the quotient 1492 shows how many pounds they amount to; besides which there remains three yards at 4s. and these come to 12s. In the second example, we divide by 80, as directed, and the quotient gives L. 22, and the remainder 13 yards, which at 3d. comes to 3s. and 3d.

This method can only be used in calculating for the particular prices specified in the table. The following 6 cases comprehend all possible rates, and will therefore exhibit different methods of solving the foregoing questions.

CASE II. "When the rate consists of shillings only, multiply the quantity by the number of shillings, and divide the product by 20; Or, if the number of shillings be even, multiply by half the number, and divide the product by 10.

The learner will easily perceive, that the method in which the second example is wrought, must give the same answer as if the quantity had been multiplied by 14, and divided by 20; and, as the division by 10 doubles the last figure for shillings, and continues all the rest unchanged for pounds, we may obtain the answer at once, by doubling the right-hand figure of the product before we set it down.

If the rate be the sum of two or more aliquot parts of a pound, we may calculate these as directed in Case I. and add them. If it be any odd number of shillings, we may calculate for the even number next lower, and add thereto the value at a shilling. If it be 19s., we may subtract the value at a shilling, from the value at pound.

CASE III. "When the rate consists of pence only."

Method 1. If the rate be an aliquot part of a shilling, divide the quantity accordingly, which gives the answer in shillings; if not, it may be divided into two or more aliquot parts: calculate these separately, and add the values; reduce the answer to pounds.

1 penny is ¼ of a shilling. 2 d. is ½ of ditto. 3 d. is ¾ of ditto. 4 d. is ¼ of ditto. 5 d. is the sum of 4 d. and 1 d. or of 2 d. and 3 d. 6 d. is the sum of 4 d. and 3 d. or of 6 d. and 1 d. 7 d. is the sum of 4 d. and 3 d. or of the double of 4 d. 8 d. is the sum of 6 d. and 2 d. or the double of 6 d. 9 d. is the sum of 6 d. and 3 d. 10 d. is the sum of 6 d. and 4 d. 11 d. is the sum of 6 d. 3 d. and 2 d.

Ex. 1st.] 7423 at 4d.

Ex. 2nd.] 1773 yards at 3d.

At 6d. = ¼ of 1s. 4893 At 3d. = ½ of 6d. 2446

At 9d. = ¾ of 1s. 4856 at 11d. 3d. and the added and reduced give the answer.

Here we calculate what the articles would cost at 6d. at 3d. and at 2d. and add the values.

It is sometimes easier to calculate at two rates, whose difference is the rate required, and subtract the lesser value from the greater. Thus, the last example may be wrought by subtracting the value at a penny from the value at a shilling. The remainder must be the va-