SIMPLIFIED ARITHMETIC.

--INTRODUCTION.--Mathematics is Based on Two Laws-First, Increase, and Second, Decrease. DEFINITIONS.

1. Quantity is (a) the amount or extent of anything, or (b) quantity is anything that is capable of increase, decrease or measure.

2. Mathematics is the science of quantity.

3. A Unit is a single thing, or simply one

4. A Number is (a) a unit, (b) a collection of units (c) one or more units.

5. An Int ger is a whole number.

6. Arithmeticis is (a) the science of numbers, (b) the art of computation.

7. Arithmetic, as a science, treats of the properties and relations of numbers or quantities, and as an art it teaches the methods of application to practical, every day uses or purposes.

8. A Rule is a direction for performing an operation.

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9. A Formula is a fixed method by which anything is to be done or arranged.

10. The Unit of a Number is one of the kind or name as the number itself, thus: The unit of 40 cows is 1 cow; of 25 cents, 1 cent etc.

11. An Abstract Number has no reference to any particular thing, as 3,8,7,5, etc., but should we say 3 horses, 8 cows, 7 sheep, 5 persons, etc, then the number becomes Concrete.

12. Now what is a Concrete number?

13. Denominate numbers--concrete number, according to some authors.

14. A Sign --a symbol or character indicating an operation to be performed.

15. An Operation --the process of finding, from given quantities others that are required.

16. The Processes used in arithmetic are (a) Notation, (b) Numeration, (c) Addition, (d) Subtraction, (e) Multiplication and (f) Division Note --addition,subtraction, multiplication and division being so closely related should be taught together or very closely connected, thus: Take for example a simple process;Addition--2 plus 2 plus 2 equal 6.Subtraction--6 minus 2 minus 2 minus 2 equal 0.Multiplication--2 times 3, or 2 threes equal 6.Division--6 divided by 2, or 6 divided by 3, etc.

In this way, pupils can be led to see the close relation existing among the four so called processes.

NOTATION.17. Notation is the process of representing numbers (a) by letters, (b) by figures. or (c) by words, 00043there are three methods of expressing numbers: First method or (a)--

One, I; five, V; ten, X; fifty, L; one hundred, C; five hundred, D; one thousand, M.

Second method or (b)--

One 1; two, 2; three, 3; four, 4; five, 5; six 6; seven, 7; eight, 8; nine, 9, etc

Third method or (c)--

One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen twenty, etc.

Hence we can express numbers either by letters, figures or words, and the words may be either written or spoken.

18. How to teach Roman Notation.1-1. V-5,X-10,L-50,C-100,D-500, M-100; I plus I equal 2; I plus I plus I equal 3; IV, one less than V equal 4; VI one more than V equal 6; VII, two more than V, equal 7; VIII, three more than V, equal 8; IX. one less than X, equal 9; XI, one more than X, equal 11; XII, two more than X, equal 12, etc.

Rule 1.--A less letter, or a letter of less value, placed before a larger letter, subtract their difference, as IV equal 4 because the difference between five and one is four. IX equal 9. Why? XX equal 20; but if we place I between the two X's, thus, XIX, its value will be 20- 1 equal 19. Why?

Rule 2--A letter of greater value placed before a letter of less value, denotes the addition of their values, thus, V plus II equal 7; X plus V equal 15; L plus X plus V equal 65, and so on with other letters.

Remark--By inspecting Roman Notation it will be seen that if any small letter be taken as the unit or starting point, the number (formed by placing a larger letter to the left or right) will increase or decrease, not in a tenfold ratio, as in the Arabic and 00054decimal systems, but in an irregular order; yet it may be represented by the pendulum which will be found on another page in this book. By making a careful investigation of this article, pupils can be taught, in a short time the why and wherefore of this much misunderstood system of notation. This is nothing but the rough foundation of a vast Roman structure which the author will leave for progressive teachers to complete. A dash(--) placed over a letter increases its value -a thousand times, X equals 10:X equals 10,000. etc.

ARABIC NOTATION.20. Arabic Notation was first made known through the Arabs who employ ten characters called figures, viz: 1, one; 2, two: 3, three; 4, four: 5, five; 6, six; 7, seven; 8, eight; 9 nine; 0, cipher or zero. Units increase from right to left in a ten fold ratio, 5555. The 5 on the right equals 5 units; the 5 next equals 50 which equals 5 times 10; the 3d 5 equals 10 times 50 equals 500, and the 5 on the left equals 500 times 10 equals 5000. Since units increase from right to left in a ten fold ratio, they decrease in reverse order.

Alphabet Of Arithmetic.1,2,3,4,5,6,7,8,9,0, or one, two, three, four, five, six, seven, eight nine, naught, zero or cipher.

Note to teachers:--Teach your pupils, first by objects, then associate the object with the character or figure that stands for the object, that is, "go from the known to the unknown," or plainer still, teach (a) the object, (b) the picture of the object (c) the character representing the object, and (d) the abstract idea.

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21. Numeration is reading numbers. Remarks 1--As soon as pupils understand objects and numbers as far as ten. give them all possible combinations in addition, subtraction, multiplication and division. A figure standing alone has its simple value, but when united with other figures it has its local or changeable value. In the number 628, 6 and 2 have their local value while 8 has its simple value. Remark 2--Simple and local values have been used so much that any other terms seem open to criticism, but I would prefer the terms stable and unstable values. The definitions would then be: The stable values of a figure is its value when standing alone, and its unstable value when combined with other figures.

22. Complete diagram of the increase and decrease of numbers, including the relation of decimals to whole numbers, as here shown by the use of the pendulum:Whole numbers.Common and Decimal Fractions.MillionTenthsH of ThousandsHundredthsTens of ThousandsThousandsThousandsTen ThousandthsHundredsH ThousandthsTensMillionthsExplanation--The pendulum when at rest equals 00076one unit. As it swings to the left the number increases in a tenfold ratio, units, thus, hundreds, etc. On its return the numbers decrease in a tenfold ratio. If this pendulum should continue in one direction, say from right to left, the numbers would still increase. Should it revolve and thus continue the numbers would become larger than any assignable quantity, hence it would equal infinity; but if revolved in an opposite direction it would become less than any assignable quantity or value, hence it would equal an infinitesimal.

Note--This diagram or a part of it should be drawn on the board and explained by the teacher; then the pupils should be required to draw it and explain. The most instructive method of explaining this diagram is as follows: Draw the horizontal part of it on the blackboard, attach a string to a ball of some kind and suspend it above by simply putting an ordinary pin through a small loop made in the upper end of the string: then have the pupils move it to and fro which will clearly demonstrate the relation of decimals to whole numbers, as well as increased and decrease of numbers. When allowed to come to rest, the pendulum will point to 1, showing clearly that 1 is the starting point as well as the basis of all calculation. Note.--If a string is not handy, suspend an ordinary ruler, or a lath instead and proceed as usual. Make use of only as much of this diagram as the advancement of the class will allow. Another simple method of teaching increase of numbers is this: Call one of the smallest pupils to the board and place figure 1 on the board about on a level with the shoulder; then 0008710 pace 2 just above the 1, 3 above the 2, 4 above the 9 3 etc until 9 is reached. The order of figures will8 then stand as they are represented in the left hand7 margin of this page. Now ask the pupil to place6 his finger on each figure beginning with one, two,5 three, etc. In this way pupils can be shown that4 the higher or larger the number, the more difficult3 it is to grasp, not only with his hand, but with hismind21

24. An Axiom is a self-evident truth, or it is a statement so plain that no reasoning can make it plainer.

AXIOMS.First. Only like things can be added or subtracted.

Second. If the same or equal numbers be added to equal numbers the sums will be equal.

Third. If the same or equal numbers be subtracted from equal numbers the remainders will be equal.

Fourth. Any number is equal to the sum of all its parts.

Fifth. If any two numbers be equally increased or diminished, their difference will not be changed.

Sixth. If the same number be both added to and subtracted from another number, its value will not be changed.

Seventh. If a number be both multiplied and divided by the same number, its value, will not be changed.

25. First. Write all the numbers from 1 to 10.

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Second. Begin at ten and write the numbers back to one.

Third. Begin at one and write every other number to 9.

Fourth. Begin at two and write every other number to ten

Fifth. + means add, - means take from, = is called equal or equal to, X means multiply. + means to separate or divide. Add 2, 1 and 1, add 3, 2, 1, add 2, 2, 2, 1, 2 plus 2 equals? 3 plus 2 equals? 4 plus 1 equals? 5 plus 2 equals? 7 plus 2 equals? 8 plus 2 equals? 9 plus 1 equals? 10 less 1 equals? 10 less 2 equals? 9 less 1 equals? 9 less 2 equals? 8 less 1 equals? 7 less 2 equals? 6 less 2 equals? 4 less 2 equals? 2 less 1 equals? 2 less 2 equals? 5 less 2 equals?

NOTE--The sign (?) should be read "what," thus 3 plus 2 equal? should should be read 3 plus 2 equals what

26. How many ones in 5? How many 2's in 6? How many wheels on two carts? How many wheels on two bicycles? How many half dollars in one dollar? How many in two dollars? How many nickels in a dime? How much money in one half dime?

Remark to Teachers.--Only a few examples are given. Add as many more as will best suit your class. You might cut sticks of different lengths say one inch, two inches, three inches, etc., having the pupils measure and compare them. For example, take the inch stick and compare it with the two inch stick and you will have 1/2; then compare it with the three-inch stick, etc., and you will have the fractions 1/8, 1/4, etc. In this way 00109you can teach pupils fractions the first day they enter school.

27. Special Method in Simple Addition.When pupils have been taught objects as far as ten make use of the following method: Nine and how many make Ten? Eight and how many make Ten?" Seven and what number make Ten? etc., until one is reached, then reverse the operation. When this is well understood give the examples promiscuously . Next take, up two numbers, thus 3 plus 3 lacks how many of being ten? 2 and 2 lack how many of being 10? 1 and 3 lack how many of being 10? etc., giving all possible combinations, after which take three numbers, etc. Use concrete numbers often, but make use of abstract numbers also. Have pupils answer in complete sentences, thus combining language and numbers.

Note--Teachers should make suitable exercises until both accuracy and rapidity are accomplished.

Special Methods in Simple Subtraction.29. Simple Subtraction is the process of finding the difference between two simple numbers.

CASE I30. When no figure in the subtrahend is larger than the corresponding figure in the minuend make use of the following

EXAMPLE.Operation--Six units from 8 units leave 2 units which we place under the unit column; 2 tens from 4 tens leave 2 tens which we place under the tens' column, hence the result is 22. Proof--The remainder plus the subtrahend equal the minuend.48 Minuend.26 Subtrahend--22 Remainder.

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31. The minuend is the number from which we subtract. The subtrahend is the number to be subtracted. The difference or remainder is the result after subtraction.

Note--The remainder is always a part of the minuend.

CASE II.32. When part of the figures in the subtrahend is larger than the corresponding figures in the minuend.

First Method--When any figure in the subtrahend is larger than the corresponding figure in the minuend, subtract from ten and add the remainder to the figure above. According to the above method, 9 being larger than 7 subtract 9 from 10, which equals 1, and 1 plus 7 (the figure above) equals 8, the first figure in the answer. Since we subtracted from, 10 (which was taken from the 27 above, or 20 plus 7) we may add a 10 to the 19, or 10 plus 9, which gives two 10's instead of one 10; hence two 10's from two 10's equal no 10's, therefore our second figure equals 0, or zero. Since we cannot take 7 from 3 we say 7 from 10 equals 3, and 3 plus 3 equals 6, the third figure, 3 from 4 leaves 1; 8 from 10 equals 2 plus 6 equals 8, and 6 from 8 equals 2, consequently we have as an answer 281608.864327582179------281608

33. Second Method--When any figure of the subtrahend is larger than the corresponding figure of the minuend, take their difference from ten and add one to the next higher figure of the subtrahend. It is obvious that 9 in the subtrahend is larger by 2 than the 7 above it in the minuend, hence, 10 minus 2 equals 8, the first figure of the answer. The second figure is obtained thus, 2 plus 1 from 5 equals 2, or 5 minus 1 minus 2 equals 2; the third figure is obtained like the first, thus, the difference 001211between 7 and 3 from 10 equals 6, etc., whence the result, 281628. For more advanced pupils it might be thus represented: 10 minus 9 plus 7 equals 8; 5 minus (1 plus 2) equals 2: 10 minus 7 plus 3 equals 6, or, 10 minus (9 minus 7) equals 8; 5 minus 1 minus 2 equals 2; 10 minus (7 minus 3) equals 6, etc.864357582729------281628

Remark--The advantages of these two methods are this. A pupil is not compelled to subtract from any number higher than ten. wherein by all other methods now in use the minuend figure runs as high as eighteen. Teachers should make several examples of their own and have pupils work them according to each method.

Note--It should be borne in mind that whenever any figure has been taken from ten, (in either method) one (ten) must be added to the next higher figure of the subtrahend, or taken from the next higher figure of the minuend, or, the next figure of the minuend may be read one less than it appears, for in reality it is one less, from the fact that 1 (ten) is taken from the minuend every time we subtract from 10. Any live teacher can illustrate this on the black board. It will be well for teachers to make use of the first method as soon as a pupil can add numbers as high as ten. With the advance of education and the and science of teaching in the last decade, all practical teachers are presumed to be familiar with Grube's method of teaching numbers. This being granted, teachers will naturally associate addition and subtraction as well as multiplication and division, but as there are so many arithmetics in the schools of our land that make use of the "borrowing process," and going as high as 18, the author of this little volume has tried to 001312make some improvements in the cumbrous methods in subtraction.

38. To multiply by 10 annex one cipher. Multiply 362 by 10. Process 362 times 10 equals 3620 Ans. Explanation. Since numbers increase from right to left in a ten fold ratio, we merely changed our figures one place to the left which increased the number tenfold, therefore the number is multiplied by 10, or it may be represented thus: h t uth h t u3 6 2 times 10 equals 3 6 2 0

The diagram shows that each figure has been removed one place to the left in order to make the necessary increase, as well as make room for the cipher which, as we know is used to fill vacant places.

In writing 10, how many ciphers do we use? How many in 100? 1000? etc, then if we annex the ciphers to any number, we have multiplied by that number which contained the ciphers.

Annexing two ciphers multiplies by 100, because there are two ciphers used in writing 100 it also changes the figures of the multiplicand two figures or orders to the left; also annexing 3 ciphers multiplies by 1000 etc.

39. To multiply by 11, write the number to be multiplied twice, skipping one figure either to the right or left and add, thus: multiply 275 by 11.Process or method--275or275275275---------3025 ans3025 ansNote--The teacher should add other examplesAnother method--Multiply 8642 by 11.Explanation--The units figure 2 will be the first figure of the required answer. Then add the units 001413and ten's figure which equal 6, the second figure in the answer. Next add the ten's and hundred's which equal 10. Write the naught and carry the ten. Now write the hundred's and thousand's which equal 14 plus 1 to carry equal 15, 8 plus 1 equals 9. As a result we have, 95,062 the required answer. By examining the first method of multiplying by 11, it will be seen that the latter method is similar to the first, the only difference being is, the former is written twice, while the latter is written only once. Ex. 623 times 11. First method--623Second method623623 plus 11 equals 6853----68533 equals 3; 3 plus 2 equals 5; 2 plus 6 equals 8; 6 equals 6. Result equals 6853.

Rule for the latter method--First write the unit's figure for the first figure of the answer; 2nd, add the unit's and ten's figure for the second figure of the answer, etc., and so on until all have been added.

40. To multiply by 12, 13, 14, 15, 16, 17, 18 and 19, use the following method:

Multiply 1862 by 12. Operation, 2 times 2 equal 4, which place under the multiplication sign; 2 times 6 equal 12; 2 times 8 equal 16 plus 1 equal 17; 2 times 1 equal 2 plus 1 equal 3; draw a line and add as in the model.1862 X 123724----23344

Note--Multiply only by the unit's figure.41. To multiply by 20, 30, 40, 50, 60, 70, 80, 90; Since 20 equals 2 times 10, 30 equals 3 times 10, 40 equals 4 times 10, etc., if we first multiply by 10 and that result by 2 for 20, 3 for 30, 4 for 40, etc, we will have the required answer. Ex. 462 times 20 equals 001514What? Annexing one cipher we have 4620; then 2 times 4620 equals 9240, ans.

Rule--Annex one cipher and multiply the result by the ten's or significant figure, or for 30, annex a cipher and multiply the result by 3; for 40, the same, only multiply by 4, and so on for 50, 60, 70, 80, 90.

42. To multiply by 21, 31, 41, 51, 61, 71, 81, 91. Since 21, 31, 41, etc. are 12, 13, 14, etc. reversed, all we have to do is reverse our work, or skip one figure to the left instead of to the right. Compare the two examples, 8625 times 12, and 8625 times 21.8625 times 128625 times 21Solution 1725017250---------------103,500--result by 12.181,125--result by 21.

Rule--Begin one figure to the left, multiplying by the ten's figure instead of the unit. Add the result to the multiplicand.

43. To multiply by two or more figures. Multiply 653 by 25.

First method, 5 times 3 equals 15, write the 5 units under units and carry the 10.5 times 5 plus 1 (ten) equals 26, write the 6 (ten) under tens and carry the 2 (hundreds; 5 times 6 plus 2 (hundreds) equals 32; write the 32 in the answer. Next take up the 2 tens and say, 2 times 3 equals 6 (tens), which write under tens, etc. until each figure of the multiplicand has been multiplied by 2 (tens). Add the partial products. The second method is the same as the first only we multiply by the 2 tens first, and next by the 5 units, which causes us to skip to the right, instead of to the left. Multiply 876 by 38, using both methods.65365325 or25----------326513061306 3265----------1632516325

44 To multiply by 22,33,44,55,66,77,88,99, 001615also 222,333, 444, etc., or by any number of like figures, multiply by only one of the figures and then use the 11 method, because 2 times 11 equals 22, 3 times 11 equals 33, 4 times 11 equals 44, etc. Ex. Multiply 863 by 55, also by 555.863 8638635 equals 15 5------------ ------431543154315 equals 5 x 86343154315 4315equals 50 x 863------ 43154315equals 500 x 86347465 equals-------------5 x 11478935 equals478965 equals 555 x 8635 x 111

General Rule--Multiply by one of the figures of the multiplier, and re-write the result of the first figure as many times as there are figures in the multiplier, skipping each one of them one figure either to the right or left, then add the partial products as in the examples above.

45. To multiply by 102, 103, etc. to 109, also by 1002, 1003, 1005, etc., to 1009.

Rule--Multiply as you would by 12, 13, etc., beginning as many places to the right less one, as there are figures in the multiplier.

Note--The same is true of 201, 301, 401, etc., also 2001, 3001, 4001, etc., but in reverse order. Multiply 6862 by 104, also by 1004. Method--6862 times 1046862 times 100427448 27448---------- ----------712,6486,889,448Multiply 6862 by 301, by 3001.Method--6862 times 3016862 times 30012058620586--------------------2,065,462 20,592,862

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Note--These two methods can be used whenever there are any number of ciphers between two significant figures; provided, however, that either the right or left figure is 1. The other figure may be any number from 1 to 9 inclusive.

55. COMPOUND NUMBERS. Concrete numbers expressed in more than one denomination equal compound numbers, as, 2 dollars and 20 cents, 8 pounds and 6 ounces, etc. The scale of simple numbers increase and decrease by tens. The scale of compound numbers (except U.S. money and the Metric System) do not increase and decrease by tens, but each particular weight or measure has an irregular scale. U.S. money and the Metric System might be treated as simple numbers and decimals combined, and at the proper time it will be so used in this work. Reduction ascending and descending may be taught by making use of the pendulum on page 5, of this book. Swing the pendulum to the left and you represent reduction ascending. Swing it to the right and you have reduction descending, that is, the scale in compound numbers increase from right to left, and decrease from left to right but in an irregular order.

56. General Rule for Reduction.First. Ascending equals Division or divide.Second. Descending equals Multiplication, or multiply.

Note--This rule applies to every table in compound numbers.

Remark--Every teacher is supposed to teach the different tables by making use of measures and before leaving each table have the pupils commit it to 001817memory, then make use of the following simple method.

57. Dry Measure is used in measuring grain, salt, fruit, etc.

Table--2 pints equal one quart, 8 quarts equal 1 peck, 4 pecks equal 1 bushel. Abbreviations, pt. equals pint, qt. equals quart, pk. equals peck, bu. equals bushel, 1 bu. equals 4 pk. 32 qts. equal 64 pt. 1 pk. equal 8 qt. equals 16 pt. 1 qt. equals 2 pt.

58. Reduction Descending equals multiplication. Reduce 8 bu. 2 pk. 3 qt. 1 pt. to pt.Formula--1 bu.4 pk.8 qt.2 pt.-------------------------------------------------------8:2 plus:3 plus:1 plus4....----..32 plus 2 equal 34..8.-----.272 plus 3 equal 275.2------550 plus 1=551 A.

As this example is from a higher to a lower denomination, it is Reduction descending and therefore equals multiplication.

By examining the formula, it will be seen that the process runs from the left towards the right showing plainly that it is reduction descending and proving our former statement that reduction descending equals multiplication. Have pupils write the table on the board as in the formula letting it remain there until the recitation is over. It is very important to make use of the dotted lines which 001918show what figure of the table is brought down and used as the multiplier and also shows the name of the denomination, thus serving a two fold purpose. For instance the first dotted line in the above example shows that the 32 equals pks. because the line points to 32 and also to pks. above. The second dotted line points to qts. and the third to pts.

Special remark--Always add the number that is just to the right of the dottedline. Examine the above formula carefully and you will see the different additions. The different steps are as follows: First, write bu. under bu. pk. under pk. etc. filling vacant places with ciphers. Second, draw a dotted line from the first figure of the table to the right using that figure of the table for the multiplier. Third, add the figure that stands next to the right of the dotted line. Continue the work until you reach the required denomination which will be the result sought, or the answer.

Reduce 420 quarts to bushels. Now this example is somewhat difficult for pupils to determine whether to multiply or divide, but the formula here given will remove the difficulty and insure the correct modus operandi. Write the table thus:1 bu. 4 pk. 8 qt.2 pt.and2 pt)..------equal..8 qt)420.8)420equal 4.equal 4-----.------4)524 pk)52equal 0equal 0------ ----131 bu)13

Then place 420 qts. under qts. in each table as in the above. Now ask the pupils if they must multiply or divide in order to change 420 qts to bu., 002019and without a doubt the immediate answer will be divide! The reason is very plain because should they make one multiplication the change would be from qts to pts, instead of (a) pks and (b) bushels. Each line points toward the required denomination. Inspect examples above. Observe everything. Look at nothing carelessly; search for small things.

68. To measure off one square acre of land. (a) Measure off a square whose sides equal 8 rods. (b) Run a line 4 rods long from each corner of the inner square as shown in the accompanying diagram. (c) Join the corners which will enclose exactly one square acre.

Explanation--The inner square equals 64 sq. rods: then we have four right angled triangles, each 12 times 4 rods which equal 2 rectangles, each 12 times 4 rods equal 48 square rods or 96 square rods in the four triangles, hence we have 96 square rods plus 64 square rods which is equal to 160 square rods, equal to one acre.Note.--Should it be desirable to have the acre laid off due north and south the inner square must be laid off at an angle as in the diagram which equals about 18 degrees.

69--Cubic Measure is used in measuring all bodies that have length, breadth and thickness. Such bodies are called cubes. To find the contents of a cube multiply together the three numbers which stand for its three sides. Example. A body is 3 ft. long, 2 ft. wide and 4 ft. high; what are its solid 002120contents? 3 times 2 times 4 equal 24 cubic ft. ans.

Note--One wine gallon of pure water weighs 10 lbs.

One cubic foot of water weighs 62 1/2 lbs. sea water 64 3-10 lbs. oakwood 55 lbs. coal from 50 lbs. to 54 lbs. red pine 42 lbs. white pine 30 lbs.; but a cubic foot of lead weighs 708 8/4 lbs. while gold is still heavier, a cubic foot weighing 1207 3-10 lbs. and platinum 1312 1/2 lbs. Water is heaviest at 39 degrees above zero Fahrenheit.

73. Time Measure is used in computing time. (a) Time equals duration, (b) Time means how long anything lasts or continues, and (c) time is the regulator of the world, sun, moon and planets, etc., that is, time is the chronometer of the universe.

TABLE.60 seconds (sec.) equals 1 minute, ab. min.60 minutes equals 1 hour, ab. hr.24 hours equals 1 day, ab. da.365 days equals 1 year, ab. yr100 years equals 1 century, ab. c.7 days equals 1 week, ab. wk.4 weeks equals 1 month, ab. mon.12 months equals 1 year, ab. yr.

Note--365 da. 5 hr. 48 min. 46 sec. equals 1 exact or solar year, but in calculations we make use of 365 1/4 da. 6 hr.

366 days equals 1 leap year which occurs every fourth year, except some of the centuries which have only 365 days.

January, Jan; March, Mar.; May--; July, Jul.; August, Aug.; October, Oct.; December, Dec.; 31 days each.

April, Apr.; June, Jun., September, Sept.; November, Nov.; 30 days each.

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February, Feb.; 28 da. for 3 yrs. then 29 da. the fourth year.

A solar year is the time from the sun's "crossing the line" or equinox to its return to the same again which as has already been stated equals 365 da. 5 hr. 48 min. 46 sec.

A sidereal year is the time in which the earth revolves around the sun which equals 365 da. 6 hr. 9 min. 9 sec. hence a sidereal year is longer by 20 min. and 23 sec. than a solar year.

Note carefully that degrees, minutes and seconds do not mean time. 10 degrees, 20 min equal 10 20-60 or 10 1/3 degrees, which generally has reference to the space or distance passed over by a body of some kind, or it means the size of an angle. From East to South is called 90 degrees or one quadrant, and so for the other points of the compass. From South to West equals 90 degrees; then 90 degrees plus 90 degrees equals 180 degrees, but from east to west is a straight line, hence 180 degrees equal a straight line, Q.E.D

To Teachers--Now since the above statement is true, explain to your pupils that 90 degrees equal a square corner, and that 90 degrees taken on each side of a vertical line equal a straight line which will be horizontal. Almost any child knows the meaning of a square corner, because it is a familiar name or term at home, but when you mention a right angle, he is bewildered. When the pupils understand that the two square corners equal a straight line (this can be shown by putting the end of two crayon boxes together, the bottom or top will be the straight line and the two ends together with the bottom equal the two sq. corners), it will be an 002322easy matter to teach them that 90 degrees plus 90 degrees or 180 degrees equal all the degrees that can be formed on one side of as traight line. When they understand this, call their attention to the other side of the line which being the same equal 180 degrees, then when they add 180 degrees and 180 degrees the number of degrees on both sides of the line, they will know why there are 360 degrees in a circle. "Go from the known to the unknown," is a psychological principle which can be applied here, therefore, granting that the pupils are familiar with the term square corner," associate it with the term "right angle," and by the association of ideas," (another psychological principle) he will grasp the right ideal and forever remember it.

Note--Clocks and watches are made on the principle that the minute-hand moves twelve times as fast as the hour hand, then since they both move, the hour-hand will pass over one space while the minute-hand passes over 12. 12 minus 1 equal 11 spaces gained by the minute-hand. Then in order to solve examples concerning the relation between the two hands of a clock or a watch, multiply or divide by 12-11 according to the nature of the example.

80 United States money is the money of the U.S. which consists of gold, silver, paper, nickel and cooper.

Note--The value of the different coins are: 1 cent, 2 cents, 3 cents, 5 cents, 10 cts., 25c, 50c, 1 dollar $2 1/2, $3, $5, $10, $20. $ equals dollars whch is placed before the number representing dollars. A dot is placed between dollars and cents but not between cents and mills.

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The highest denomination of money in the United States is the legal tender notes which equal $10,000. As U.S. money increases and decreases in a ten fold ratio, the same as simple numbers, it may be treated as such.

90. LONGITUDE and TIME. Longitude is distance east or west from an established meridian. There are two standard meridians adopted (a) at Greenwich (pronounced Gren'ej) near London, England and (b) at Washington City, U.S.A. Meridian lines run north and south from pole to pole. Each meridian starts and ends from one point are widest at the center, that is, at the equator, a degree of longitude equals 69 1-6 miles and becomes less either north or south of the equator until the poles are reached where there is no longitude, because the lines come together. To teach Longitude and Time draw the following diagram on the blackboard so that the horizontal line will point toward the east, and the vertical line down or toward the south:It is supposed that at 6 a.m., the sun has just 002524risen and consequently due east and horizontal to the observer. One hour latter the sun will have passed over 15 degrees of space or distance and its distance above the horizontal line will be represented by the line marked 7, which squals 7 o'clock. At 8 o' clock it will have passed 15 degrees farther or higher in the sky and so on to 9,10,11 and 12 M. The angle made by the sun from 6 a.m., to 12m equals a square corner; (children's home language) which we call a right angle it being equal to 90 degrees or 1 quadrant.

Note--"Go from the known to the unknown," Associate what the pupils know with what he is about to make his own and he will surely remember it thus:( (1) Degrees,( (2) Right angle,A Square Corner.( (3) Quadrant.( (4) One fourth of a circle

Have the pupils point out all of the square corners he may see in the room, and have them make sentences using the different terms as given in the diagram. Now teach them that nothing can pass over space or go any distance without taking time. It takes time to think, nothing can be done without consuming some portion of time. Degrees' "equal distance or space. Hr min. sec. equal time.

Method--D equal the first letter of the word Degrees and also the first letter of Distance, and Division: then we have D. equal Degrees and Distance, D equal Division, therefore when the given example is (°) divide by 15° according to the Formula for Division of Compound Numbers, the result 002625will be hr. min. sec. Again hr. min. sec. equal time; and time or times equal multiplication or (X), hence when "time" is given (hr. min. sec.) multiply by 15° according to the Formula for Multiplication of Compound Numbers: the result will be (°' ")Note--One 15 degree space in the Diagram represents 1 hour of time and vice versa I.E. 15° 1 equal hour.

91. Why 15° equal 1 hour of time.

Since the earth rotates from west to east and makes one rotation every 24 hours, the sun appears to move at the same rate, but in an opposite direction: that is the sun appears to pass around the earth once every 24 hours, now as every circle equal 360° it follows that the sun will appear to pass over 360° +24 in one hour or. 15°

Again, 1 hr. equal 60 min; this divided by 15 equal 4 min., the time of the sun's passing over 1°, and further; the sun passes over 1/4 of a degree every minute of time, and 1-240 of a degree every second of time; hence if the time be reduced to seconds and the result be divided by 240, the answer will be degrees.

Note--When it is noon at any place, it is past noon east of it, but not noon west of it, therefore let E stand for east and W. for west; Now, for places east we add and for places west we subtract. Since plus stands for addition and minus stands for places west we subtract. Since plus stands for addition and minus stands for subtraction, we have the Formula. E. equals plus and W. equals minus.

Note--Longitude can never exceed 180; this being half way around the globe, any distance beyond, in 002726one direction, will be nearer in the opposite direction. Suppose a pupil starts around the school house; when he has gone half way, that will be as far from the starting point as he can get, because if he continues, he will be getting nearer in the direction in which he is going. This being true, any number of degrees above 180 should be subtracted from 360. Suppose a man starts arounds the world and has gone 225°, he will be 360° minus 225° from where he started which equals 135°.

FACTORING.92 Factoring is the process of separating quantities into their factors.

93. All whole numbers are either prime or composite.

94. A prime number is a number which can not be exactly divided by any other number except itself and 1. The only even prime number is 2.

95. A composite number is a number which can be exactly divided by some number besides itself.

66. Even numbers end in 0,2,4,6,8. Odd numbers end in 1,3,5,7,9.

97. The reciprocal of a number is one divided by that number; thus the reciprocal of 3 is 1/3, etc.

98. All even numbers are divisible by 2.

99. All numbers ending in 5 or 0 are divisible by 5.

100. Every number the sum of whose figures 3 or 9 will exactly divide is divisible by 3 or 9.

101. Take two ones two twos two threes etc. up to and including two nines and place two ciphers between them. Each can be exactly divided by either 7, 11 or 13.

102 Take any number and multiply it by 9; the sum of the figures in the product will equal 9 or some multiple of 9.

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103. Take any number in which the sum of the figures in the odd places equal the sum of the figures in the even places and the number is exactly divisible by 11. 86471 divided by 11 equal 7861. The sum of the even figures beginning at the unit's side, are 7 plus 6 equal 13; the odd; 1 plus 4 plus 8 equal 13. Each equal the same, therefore 11 will exactly divide the number. Again, if their difference can be exactly divided by 11, the number itself can be exactly divided by 11.

Greatest Common Divisor.106 Method.--Let G.C.D. equal Prime. By this it is meant that this operation must be continued until the several quotients are "prime" to each other; then the product of the several divisors will be the G.C.D.

Find the G.C.D. of 18, 36 and 45.DEVISORS.FORMULA.3 X 3 equal 9G.C.D. 3) 18 36 45 18 equal 3 X 3X 2 equal 2----------36 equal 3 X 3X 2 X 2 equal 4 3) 6 12 15 or45 equal 3 X 3X 5 equal 52 4 5Common.Not Common.

Two, four and five are "prime" to each other, therefore our division ends here. We have 3 and 3 as divisors common to each number; these, and only these, the product of which equal 9.

Note--Teach the "method" thoroughly and as soon as a pupil sees or hears G.C.D. he will think of "prime" also, which will remind him of the process to be pursued, hence the value of this method.

For very large numbers use the following rule:(a) Divide the larger number by the less. (b) Divide 002928the less by the first remainder, and (c) continue to divide the last divisor by the last remainder until nothing remains. The last remainder will equal the G.C.D. Should there be more than two numbers find the G.C.D. of two of them; then use that G.C.D. as a divisor with the next number, etc., until all have been used. The last G.C.D. will be the G.C.D. of all of them.

107. A Multiple is any number that is divisable by another number without a remainder.

108. A Common Multiple is a number that is exactly divisible by two or more given numbers.

109. The Least Common Multiple (L.C.M.) is the least number or dividend exactly divisible by two or more given numbers.

110 Method Let L.C.M. equal 1. By this it is meant that the operation must be continued until the several quotients equal ; then the product of the several divisors will be the L.C.M.

Note--I am of the opinion that pupils will do better work and understand both G.C.D. and L.C.M. better if they make use of the two Methods G.C.D. equal Prime and L.C.M. equal 1. Both operations are identical, the only difference is, the former ceases dividing as soon as the numbers become prime to each other, while the latter continues until every quotient becomes unity. When the numbers prime to each other are reached, if we should multiply these prime numbers and the several divisors together, the result would be L.C.M. that is, the product of the divisors equal the G.C.D and the product of the prime factors equals the L.C.M.

148. The Metric System is a decimal systems of 003029measures and weights with the meter as the base that is, it is based on the meter which is 39.37 inches in length. A pendulum of this length at sea level is supposed to vibrate once every second of time, or 60 times per minute, equal to 3,600 times per hour.

Meter equal unit of length, Ar. equal square measure, Stere equals cubic measure (for solids), Liter equals cubic measure (for liquids), Gram equals unit of weight. The French or originated this system in 1795.

To the right of the decimal point or pendulum the prefixes are deci equal 1-10, centi equal 1-100, milli equal 1-1000. To the left of the point they are deka equal 10, Hekto equal 100, kilo equal 1000, Myria equal 10000.

Note to teachers--Have your pupils make a meter measure if the school is without one. Dress off a common lath and measure off 3 ft, 3 3/8 inches scant. Almost any carpenter's square contains eights of an inch. Then divide the measure into 10 equal parts: each part will be very nearly 4 inches long. Measure your room with the meter and also with the yard measure and compare them. Get about 100 double 00 shot from the hardware store and you can weight several things. A few nickels will be helpful. Get a liter measure too. Hold the pupils to this subject till they know it thoroughly.

Note--All numbers in the Metric System increase and decrease in a ten fold ratio, the same as the Arabic system of notation.

150. To change meters, liters and grams from 003130one denomination to another use the following form:Mx Kx Hx Dx x dx cx mxFormula 10. 10. 10. 10. 10. 10. 10. 10.-------------------------------3.64 2

Example--Change 3642 m. to kilometers. Write the unit figure under x and each remaining figure under a ten toward the left and place a decimal point to the right of the denomination required in the answer. See example above. Annex or prefix ciphers as the example many require.

Note to teachers--The formula on this page applies to all of the tables (except the table of values) in the Metric System. Have the pupils place the formula on the background or on slates, paper, etc. Explain the use of x As has already been said, x stands for gram, meter or liter. N2 equals square meter, X3 equals cubic meter.

(1) A 5 cent nickel weighs 5 grams,

(2) From the center of a nickel to the edge equals 1 centimeter.

(3) Number 00 shot weighs 1 gram.

INTERESTS.$320x530.08 339.20Formula----------equals-------equals $37.68 8-9 A3 x 4 x 5 x 69

167. Case I known terms equals P. T. R. Unknown 003231term equal I. P. equal principal, R. equal rate per cent. T. equal time; I interest.

Rule--(1) Change the time to days. (2) place the dollars, days and rate per cent above the line as in the Formula above with 3 time 4 time 5 time 6 below the line, (3) apply cancellation: the result will be the required interest. If the amount is required add the interest to the principal

Note--In New York, 365 days equal interest year; therefore to find interest for a year of that kind, use 5 times 73 as factors instead of 3 times 4 times 5 times 6.

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NOTICE.The design of this little work is to give teachers a clearer idea of arithmetical knowledge as well as a shorter way of performing examples. It is thought unnecessary to use forty or fifty figures when fifteen or twenty will answer the same purpose. Simplicity and brevity characterize this work from beginning to end. Take the following example: Multiply 1875 by 333 1/2:

Old Method--One-third of 1875 equals 625; 624375 plus 625 equals 625000. Ans

Ordinary Method--Change 333 1/2 to thirds which equal 1000 thirds; which equal 100 thirds, then 1875 times 100 divided by 3 equal 65200. ans

Latest Method.--RULE--Annex three ciphers and divide the result by three. Solution--187500 divided by 3 equals 625000. Ans.1875333 1/3------562556255625--------624375

The first method requires 50 figures for its solution; the second 32 figures and the third, only 14 figures! Yet the latter method is just as easy to teach as either of the other two and at the same time the example can be solved in about one fifth of the time.

This little book is Section one of the original M. S. which will make a book of about two hundred pages. Other sections will follow.

Hoping this little work may be kindly received by the public, especially by the teachers, and may it be the means of simplifying arithmetic is the earnest wish of the author.Carrollton. Mo.,June 22, 1896.