Studies in Each of the Subjects on Which Examinations Are Based.
 
The following lessons are based on previous examinations. The subjects are somewhat broad in scope in order to carry the student over every possible contingency. Careful study will enable the competitor to meet all the requirements.
Spelling.
 
Confederacy, Vogue,
Deity, Squirrel, a small animal.
Chirography, Pippin,
Worthy, Yoke, a connecting frame
Paltry, for draft cattle.
Electioneer, Aspirant, one who seeks
Anvil, earnestly; a candidate.
Rumor, Terminus,
Gravity, Brutal,
Ancient, Cholera,
Chiropody, Glimmer,
[53]
Chirp, Delightful,
Ere, Inaugurate,
Intuition, Freight,
Niche, Earnest,
Granary, Quadrille,
Copartner, Lullaby,
Autocrat, Usury,
Inconstancy, Audacious,
Officiate, Though,
Delicacy, Equitable,
Ninetieth, Bivouac,
Credulous, Integrity,
Fiftieth, Asthma,
Tincture, Maniac,
Wigwam, Dissolve,
Eyelet, Admittance,
Tyranny, Occupy,
Undulate, Constituency,
Committee, Irritable,
Conservatory, Advertisement,
Literary, Halibut,
Legislature, Strength,
Anomalous, Melodious,
Desirous, Wheelbarrow,
Radiant, Curtain,
Jamb, Senate,
Chilblain, Superscribe,
[54]
Convertible, Familiar,
Adversary, Mammoth,
Illuminate, Drawee,
Circuit, Motor,
Remnant, Presumption,
Stencil, Monosyllable,
Degradation, Apprentice,
Claret, Alcohol,
Ludicrous, Charity,
Idea, Plantain,
Saucy, Stampede,
Recollect, Demonstrate,
Cupola, Longitude.
ARITHMETIC.
Lessons in Decimals.
 
The paper on arithmetic in second grade examinations usually contains one, sometimes two, problems in common or decimal fractions. These are no more difficult to solve when one understands the rules governing them, than any simple test in addition, division, etc. In whole numbers, as 57, 563, 4278, the various units increase on a scale of ten to the left (or decrease on the same scale of ten to the right). Thus in the last number we say 8 units, 7 tens, 2 hundreds,[55] and 4 thousands or four thousand two hundred seventy-eight.
Decimals also decrease on a scale of ten to the right (or increase on the same scale of ten to the left). In writing decimals, we first write the decimal point, which is the same mark we use at the close of a sentence and is called a period. Then the first figure to the right is called “tenths” and is written thus .6, meaning six tenths. The second figure stands for hundredths as .06, six hundredths; .006 for six thousandths; .0006 for six ten-thousandths; .00006 for six hundred-thousandths; .000006 for six millionths, etc. When a whole number, previously mentioned, and decimals are written together as 47.328, it is called a mixed number.
The only distinction between reading whole numbers and decimals is made by adding this to the ending of decimals, and the denomination of the right-hand figure must be expressed to give the proper value to decimal parts. For instance, .12, is twelve hundredths; .007, is seven thousandths; .062, is sixty-two thousandths; .201, is two hundred one thousandths; .5562, is five thousand five hundred sixty-two ten-thousandths; .24371, is twenty-four thousand three hundred seventy-one hundred-thousandths; .893254, is eight hundred ninety-three thousand two hundred fifty-four millionths, etc. Remember that in decimals the[56] first figure stands for, tenths; the second, hundredths; the third, thousandths; the fourth, ten-thousandths; the fifth, hundred-thousandths; the sixth, millionths, and that in reading decimals we add the denomination of the right-hand figure. When reading a mixed number the word “and” is used, and then only, to indicate the decimal point. Thus 45.304 should be read forty-five AND three hundred four thousandths.
Addition and subtraction of decimals differ from similar operations of whole numbers only in the placing of the figures. In whole numbers units come under units, tens under tens, etc. To illustrate:
What is the sum of 260, 4398, 305, 2, 29?
The figures are placed thus:
   260
4,398
   305
       2
     29
———
4,994
[57]
Now let us take the same figures expressed decimally: .260, .4398, .305, .2, .29.
  .260
  .4398
  .305
  .2
  .29
———
1.4948
In subtraction of whole numbers or decimals the figures are placed as in addition.
Examples—Subtract .204 from .4723.
.4723
.204
——–
.2683
Subtract 5.346 from .937.
5.346
  .937
——–
4.409
Subtract .753 from 18. (Note that the point or period is placed to the left of “753” indicating decimals, but in connection with the number “18,” a dot is placed to the right as a mark of punctuation merely, thus showing that “18” is a whole number.)
[58]
Now from the whole number “18,” which is the minuend because it is the number to be subtracted from, we are to subtract .753, and it is done in this way:
Minuend 18.000
Subtrahend    .753
———
17.247
The three ciphers are added to the minuend to correspond to the decimal places in the subtrahend. It is not necessary to put the ciphers down, but beginners are apt to get confused if there is nothing there to correspond to the decimals below. Annex as many ciphers to the minuend as there are decimals in the subtrahend, and place in the remainder a decimal point under those of the numbers subtracted.
Multiplication of decimals differs somewhat from the previous operations mentioned for the reason that we do not necessarily place the decimal points directly under each other. The right-hand figure of the multiplier usually goes under the right-hand figure of the multiplicand and the problem is then worked out as in multiplying whole numbers. When the product is obtained we point off as many decimal places in it as there are in both the multiplier and the multiplicand.
Let us take as an example: Multiply 2.648 by 2.35
[59]
Multiplicand    2.648
Multiplier      2.35
———–
   13240
   7944
 5296
———–
Product 6.22280
It will be seen that there are three decimals in the multiplicand and 2 decimals in the multiplier, hence we point off five decimals in the product.
In the operation of division of decimals the decimal point is not considered until the result is obtained. If the number of decimal places in the dividend is less than the number of decimal places in the divisor ciphers must be annexed or added to make up the deficiency, and the decimal point is then suppressed, thus reducing the operation to the division of two whole numbers. If there is no remainder, the quotient is a whole number, if there is a remainder, add a cipher to the right of it and place a decimal point to the right of the quotient obtained, then continue the division as far as desirable by adding ciphers to the right of the successive remainders, for each of which a new decimal will be obtained in the quotient.
Divide 460 by .5.
[60]
.5) 460 (92
45
—
10
10
—
0
Fractions are reduced to decimals by annexing ciphers to the numerator and then dividing by the denominator.
For instance—5/8 equals what decimal?
8) 5.000 (.625 = 5/8
4?8
—
.20
?16
?—
.40
?40
Lessons by Prof. Jean P. Genthon, C.E., Member Society of Municipal Engineers and Author of “The Assistant Engineer,” “The Chief’s” Text Book on Civil Engineering.
In solving problems the process should be not merely indicated, but all the figures necessary in solving each[61] problem should be given in full. The answers to each problem should be indicated by writing “Ans.” after it.
Arithmetic is the science of numbers.
A Number is the result of the comparison (also called measurement) of a magnitude or quantity with another magnitude or quantity of the same kind supposed to be known.
A Concrete Number is one the nature of the unit of which is known.
Denominate Number.—A concrete number the standard of which is fixed by law or established by long usage.
An Abstract Number is one of which the nature of the unit is unknown.
How to Read Numbers.—The right way to read 101,274, etc., is one hundred one, two hundred seventy-four, etc.
The Decimal Point.—A period, called decimal point, is placed in a mixed number between the integral part and the decimal portion which follows. It should never be omitted.
Roman Numbers.—I stands for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M for 1,000.
Abbreviations.—A smaller unit, written to the left of a greater one, is subtracted from the latter, as: IV = 4[62] (IV is marked IIII on clock and watch dials); IX = 9; XC = 90; CD = 400, etc. Sometimes a Roman number is surmounted by a dash or vinculum; it then expresses thousands, as IX = 9,000.
Addition.
 
Addition.—Operation which consists in taking in any order all the units and portions of units of several numbers and forming with them a single number called their Sum or Total.
Addition of Long Columns of Numbers.—When long columns of numbers are to be added, the student should endeavor to add more than one figure at a time. He may pick those which aggregate 10, 15, 20, etc., and add the intermediate figures when convenient.
Sign of Addition.—The sign of addition is the horizontal-vertical or Roman cross + placed between all the numbers to be added; it is read Plus.
To Prove an Addition.—The shortest way to prove an addition is to do it over again from bottom to top.
Sign of Equality.—The sign of equality is two short equal horizontal parallels =; it reads Equal.
Subtraction.
 
Subtraction.—An operation which consists in taking from a number called minuend (m) all the units and parts of units contained in another number called subtrahend (s).[63] The result is called the difference (d) of the two numbers or the remainder of their subtraction.
Sign of Subtraction.—The sign of subtraction is a horizontal dash - placed between the minuend, written first, and the subtrahend. Thus: 84 - 38 = d; 84 - 38 = 46. Generally m - s = d.
To Prove a Subtraction.—Add from bottom to top the difference and the subtrahend; the sum must equal the minuend.
Multiplication.
 
Multiplication.—An operation which consists in repeating a number called multiplicand (M) as many times as there are units in another column called multiplier (m); the result is called the product (p) of the numbers, and the numbers themselves are called factors of the product. This definition may be extended to the case where the factors are not whole numbers.
Sign of Multiplication.—The sign of multiplication is the oblique or St. Andrew’s cross ×, called multiplied by, and placed between the factors written one after the other.
Thus: 35 × 7 = p; 35 × 7 = 245. Generally M × m = p.
To Prove a Multiplication.—Multiplication may be proved by a second multiplication in which the factors are inverted.
[64]
This is the surest but the longest method.
Another Proof of the Multiplication.—Find the residue of the multiplicand and multiplier. Multiply them and find the residue of their product; this is equal to the residue of the product of the multiplication.
64327  4 Residue of the multiplicand.
781  7 Residue of the multiplier.
———— —
28 1   Residue of the product of the residues
64327
  514616
450289
————
50239387 1   Residue of the product of multiplication.
Proof Not Absolute.—Practically a proof is not absolute, because an error may be committed in its use, and also it may not work well in all cases.
Power of a Number.—When the factors of a product are equal, the product is called a power of the factor.
Square of a Number.—A power is a square when it is the product of two (2) equal factors, as 7 × 7 = 49, in which 49 is the square of 7. The term square is derived from the fact that the area of a square is obtained by multiplying the length of its side by itself, or taking it twice as a factor.
Cube of a Number.—A power is a cube when it is[65] the product of three (3) equal factors, as 5 × 5 × 5 = 125, in which 125 is the cube of 5.
The term cube is derived from the fact that the volume of a cube is obtained by multiplying the length of its side by itself and again by itself, or by taking it three times as a factor.
A product, for instance, of 4, 9, etc., equal factors would be called the 4th or the 9th, etc., power of that number.
Division.
 
Division.—An operation by means of which we find one of two factors of a product when that product and the other factor are given. The given product is called Dividend (D) of the division; the known factor is called the Divisor (d), and the unknown factor which is sought is called Quotient (q). We know that a quotient is seldom exact and that there is generally a Remainder (r) or Residue.
Sign of Division.—The sign of division is a small dash with a point above and one below ÷; it is read divided by, is placed after the dividend, and is followed by the divisor. For instance, to indicate the division of 72 by 8, which we know gives the quotient 9, we write 72 ÷ 8 = 9; generally D ÷ d = q.
Other Sign of Division.—In the study of fractions[66] it is shown that a fraction expresses the quotient of its numerator by its denominator, so that the preceding identity may be written 
72
8
 = 9, or more generally 
D
d
 = q, and another sign of division is a horizontal line separating the dividend written above it from the divisor written below it.
Proof of the Division.—We prove a division by multiplying the divisor by the quotient and adding the remainder, if there is any; the result thus obtained must equal the dividend. When there is a remainder, the formula of division is D = dq + r.
By 2.—A number is divisible by 2 when it is an even number, that is to say when it ends with 0, 2, 4, 6 or 8, as 70,836.
By 3.—A number is divisible by 3 when its residue is zero or is divisible by 3.
By 4.—A number is divisible by 4 when the number formed by the last two figures to the right is divisible by 4; 7528 is divisible by 4 because 28 is divisible by 4.
By 5.—A number is divisible by 5 when it ends with 0 or 5, as 75,270.
By 6.—A number is divisible by 6 when it is divisible by 2 and 3, as 474, because when a number is divisible by several others it is divisible by their product.
By 8.—A number is divisible by 8 when the number formed by the last three figures to the right is divisible[67] by 8; 37104 is divisible by 8 because 104 is divisible by 8.
By 9.—A number is divisible by 9 when its residue is 9 or 0.
By 10.—A number is divisible by 10 when the last figure to the right is 0.
By 100.—A number is divisible by 100 when the last two figures to the right are 00.
By 11.—A number is divisible by 11 when the sum of the figures of even rank subtracted from the sum of the figure............