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Degree, with one or several unknown quantities; Inequalities; Ratio
and Proijortion; Involution, including the Binomial Formula for an
entire and positive Exponent; Evolution; the Reduction of Radicals;
Equations of the Second Degree; Progressions; Permutations and Com-
binations; the Method of Indeterminate Coefficients; Fundamental Prop-
erties of Logarithms; Compound Interest.

Geometry : Plane, Solid, and Spherical, including Fundamental Notions
of Symmetry, and Examples of Loci and Maxima and Minima of Plane
Figures.

Trigonomietry : Including the Analytical Theory of the Trigonometrical
Functions and the usual Formula}; the Construction and Use of Trigo-
nometrical Tables, and the Solution of Plane Triangles, — so much, for
example, as is contained in Newcomb's Trigonometry to Art. 71), or in
Wheeler's or Richards's Plane Trigonometry.

Latin : Simple exercises in translating English into Latin. (Smith's Prin-
cipia Latina, Part I., and the First and Second Latin Books of the Ahn-
Henn Latin series, are named as indicating the nature and extent of this
requirement.) Ca;sar, Six Books of the Gallic War.

History of the United States.

Geography.

English: Inc-luding Grammar, Spelling, and Composition. In Grammar,
Whitney's " Essentials of English Grammar," or an equivalent.



EISTTEAJSTCE
EXAMIJ^ATIOJN^ PAPEES.

(Sheffield Scientific School.)

ARITHMETIC.
July, 1880.

1. (a) Select the prime numbers between 1 and 50.
{b) Find the prime factors of 6902.

41 _ 91

2. Find the vahie of -^^ ^^, in its simplest form.

3. Divide 0.10724 by 0.003125.

4. How many stones 10 in. long, 9 in. broad, and 4 in.
thick, would it require to build a wall 80 ft. long, 20 ft.
high, and 2^ ft. thick, without mortar?

5. The population of a certain town has gained 25 per
cent within the last five years. It is now 6575 ; what was it
five years ago?

6. Extract the square root of 3369 to three places of
decimals.

7. Give the approximate value of the meter in inches ; of
the gram in grains ; of the kilogram in pountls avoirdupois ;
of the liter in liquid quarts.

8. What is the weight of a liter of pure water at its maxi-
mum density?



I"i0 ARITHMETIC.

September, 1880.

1. Reduce y|-^ and ^ /^ ^- to their least common denomina-
tor ; add the results, and express the sum decimally to four
places.

2. If 8 horses consume 3|^ t. of hay iu 30 d^s., how long
will 4^ t. last 10 horses?

3. A buys 9 per cent stocks at 25 per cent premium, and
B buys 6 per cent stocks at 25 per cent discount ; supposing
dividends to be paid promptly, what rates of interest will
they receive on their investments ?

4. Calculate the square root of 2.064 to two places of
decimals.

5. Calculate the cube root of 3,3 to two places of decimals.

6. How man}' hectoliters of grain will a l)in hold whose
interior length, width, and depth, are each G ft. G in. ?

June, 1881.

1. Ascertain whether the fraction ^W^ is in its lowest
terms or not, and explain the process you employ.

2. (a) Divide ^ir ^J tIt' ^^'■^ explain the process.
(b) Can the quotient be exactl}' expressed by a termi-
nating decimal ? Give a reason for 30ur answer.

3. Divide 0.00309824 by 0.0376, and explain the position
of the decimal point in the quotient.

4. If a body revolves uniformly in the circumference of a
circle at the rate of 12° 15' 25" per minute, how long is it
in performing a. complete revolution?

5. If 6 men, working uniformly at a certain rate, do a
certain piece of work in 17 dys. of 9 hrs. each, how many



ARITHMETIC. 121

days of 8\ hrs. each will 24 men, working unifornil}- at the
same rate, require to do 20 sucii pieces?

G. $5674.83 is 10;") per cent of what sum?

7. Extract the square root of 2.26 to three places of deci-
mals. Show how you can derive from the square root of
this number that of 0.0226.

H. Find the weight in grams of a rectangular >)ar of gold
jdcm long, 21'='" wide, and 2'"" thick, assuming the bar to be
19 times as heavy as its owu volume of pure water at its
maximum density.

September, 1881.

1. Find the least common multiple of 1011, 1685, and
2359.

2. A man bought 16 horses and 19 cows for $1855. He
paid upon the average -^-^ as much for a cow as he did for a
horse. What was the average price he paid per head for
the horses?

3. Divide 0.006102 b}- 2.034, and explain the position of
the decimal point.

4. Reduce 18,216 ft. to miles and decimals of a mile.

5. A compan}-, w'hose capital is $275,000, has $15,125
from its earnings to divide. What per cent dividend can it
declare upon the basis of this sum?

6. If a block of granite 8 ft. long, 2 ft. wide, and 1 ft. 6 in.
thick, weigh 920 lbs., hoAV nuich will a block of the same kind
of granite weigh which is 12 ft. long, 3 ft. wide, and 2 ft.
thick ?

7. Exti'act the cube root of 1.025 to three jihices of deci-
mal.



122 ARITHMETIC.

8. (a) In 2.15'*"" how man}' cubic millimeters?
Reduce approximately
(6) GOO""" to miles, and
(c) 50^ to grains.

June, 1882.

1. Find the greatest common divisor of 17,640 and 1(S,37.').

2. Find the least common multiple of the foregoing num-
bers.

3. Give the table of troj- weights ; also the table of metric
weights. Express the gram in grains, and the kilogram in
pounds avoirdupois.

321

4. Express ^^ decimally to three places.

5. Of an alloy containing 21 i)arts copper and 4 parts
nickel, what per cent is copper and what per cent nickel?

6. For what amount must a note, pa^'able in GO days, be
given to a bank discounting at 6 per cent to obtain $500?

7. If 16 men build 18 rds. of wall in 12 dj^s., how many
men will be needed to build 72 rds. in 8 dys. ?

8. Extract the square root of .001601 to four places.

September, 1882.

1. (a) Which of the numbers 293, 371, 385, 440, 524,
017, and 713 are prime?

(b) Separate 1836 into its pi-jme factors.

2. Divide A of 12.', by -i of 8|.

1 f ' 2

3. Divide .000744 by .62, and t'xphvin the position of the
/lecimal point in the quotient.



ARITHMETIC. 123

4. Ill 80,9.'37,HG4 sq. in. how many acres?

5. Q and Y barter. Q makes of 10 cts. 12|^ cts., Y makes
of 15 cts. 19 cts. ; which makes the most per cent, and how
much?

6. Three men harvested and thrashed a fiekl of grain on
shares, A furnishing 4 hands 5 dys., B G hands 4 dys., and
C hands 8 d3's. The whole crop was G30 bu., of which
they had one-fifth ; how much did each receive ?

7. Extract the cube root of 81^ to three decimal places.

8. Bought 30'" of clotli at $2.50 per metre ; at what price
per yard must it be sold to gain $ 25 ?



June, 1883.

1. Find the greatest common divisor of 36,864 and 20,736.

2. Multiply I ofg by I of i-

3. (a) Give the table of metric weights.

(&) A cubical cistern holds 1331*'^ of water; what is
the length of an inner edge ?

4. Divide 67.56785 by 0.035, and multiply the result by |.
Explain the position of the decimal point after division.

5. How much money should be received on a note of $1000,
payable in 4 months, discounting at a bank where the interest
is 6 per cent?

6. If a man travel 117 miles in 15 days, employing only
9 hours a day, how far would he go in 20 days, travelling 12
hours a day ?

7. Extract the square root of 10 to five places.



124 ARITHMETIC.

September, 1883.

1. (a) Select the prime numbers between 50 and 100.

(6) What is the least number that can be exactly
divided by by J^, 21, 5, 6^, and j\?

2. Reduce 0.00096 to its simplest equivalent common
fraction.

3. 7465 is dS^ per cent of what number?

4. A broker bought 84 shares of railroad stock at 19 per
cent discount. He sold 35 shares at 27| per cent discount,
and the balance at 8 per cent discount. Did he gain or lose,
and how much?

5. Calculate the cube root of 3.7 to five decimal places.

6. Give the approximate value of the meter in feet; of
the kilogram in pounds avoirdupois.

7. Find the weight in kilos of 15 gallons of water.



GEOMETRY. 125

GEOMETRY.
July, 1880.

Note 1. Candidates who present themselves for the whole examination
may omit questions 2, 3, and 5. Candidates who present themselves for
the partial examination will confine themselves to the questions in Plane
Geometry.

Note 2. State what text-hook you have studied, and to what extent.

I. — Plane Geometry.

1. (a) Define the symmetry of a figure with respect to au
axis aud with respect to a point.

(b) Prove that if a figiu-e is symmetrical with respect
to two axes perpendicular to eacli other, it is also symmetri-
cal with respect to the intersection of these axes.

2. An angle formed by a tangent and a chord is meas-
ured by one-half the intercepted arc.

3. To bisect a given arc or angle.

4. (a) If a perpendicular be drawn from the vertex of the
right angle to the hypotheuuse of a right triangle, the two
triangles thus foriped are similar to each other and to the
whole triangle.

(b) What can you say of the perpendicular as com-
pared with the segments of the hypothenuse? Why?

(c) What of either side about the right angle ? Why ?

5. On a given straight line to construct a polygon similar
to a given polygon ?

6. The circumferences of two circles are to each other as
their radii, and their areas are to each other as the squares
of their radii.



126 GEOMETRY.

II. — Solid and Spherical Gkometuy.

7. If a straight lino and a plane are paralU'l, the intersec-
tion of the plane with planes passed through the line arc
parallel to that lino and to each other.

8. Dofiue a piisni. 'l\vo prisms are eqnnl. if three faces
including a tiii'dral angle of the one are resi)ectivoly equal
to three faces similarly placed including a triedral angle of
the other.

9. P^very section of a sphere made by a plane is a circle.

10. Between what two limits does the sum of the angles
of a spherical triangle lie ? Write expressions for the sur-
face and volume of the cylinder, cone, and sphere.

September, 1880.

[State what text-book you have studied and to what extent.]

1. To draw a connnon tangent to two given circles.

2. The bisector of an angle of a triangle divides the op-
posite side into segments which are proportional to the adja-
cent sides.

3. The area of a parallelogram is equal to the product of
its base and altitude.

4. How do you find the area of a trapezoid? The areas
of similar polygons are to each other in what ratio ? Of all
plane figures having the same area what one has the least
perimeter ?

!). If a straight line is perpendicular to each of two
straight lines at their point of intersection, it is perpendicular
to the plane of those lines.

6. A triangular pyramid is one-third of a triangular prism
of the same base and idtitude.



GEOMETRY. 127

7. Define the terms, spherical excess and trl-rectangidar
triangle. The area of a spherical triangle is equal to its
spherical excess (tlie ri SruEiucAL Geometry.

6. The sum of any two face angles of a triedral angle is
greater than the third.

7. If the base of a cone is a circle, every section of the
cone made by a plane parallel to the base is a circle.



128 GEoMiyriiY.

8. Calculate the area iu square feet of a spherical triangle
oil a sphere whose radius is 10 ft., the angles of the triangle
being 70°, 80°, and 120°.

9. Calculate the area of a zone on the sphere whose radius
is 30 ft. , the altitude of the zone being 3 ft.

10. AVrite expressions for the sui'face and volume of a
cone of revolution.

September, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1 . Through any three points not in the same straight line
a circumference can be made to pass, and but one.

2. If three or more straight lines drawn through the same
point intersect two parallels, the corresponding segments of
the parallels are proportional.

3. To find the locus of all the points whose distances from
two given points are in a given ratio,

4. Define the term lunit. Prove that the circumference
of a circle is the limit to which the perimeters of the in-
scribed and circumscribed regular polygons approach when
the number of their sides is increased indefinitely.

5. (a) AVhen is a straight line said to be perpendicuhir to
a plane? (&) How do you measure the dicdral angles in-
cluded between two intersecting planes?

(c) Prove that if a straight line is perpendicular to a
l)hine, every plane passing through the line is also perpen-
dicular to that plane.

G. The volume of an}' parallelopiped is equal to the prod-
uct of its base b}' its altitude.



GEOMETRY.



129



7. If two triangles on the same sphere are mutually equi-
angular, they are also mutually equilateral ; and are either
equal or syuuuetrical.

8. Tlif lateral area of a frustum of a cone of revolution is
equal to the half sum of the circumfereuces of its bases mul-
tiplied by its slant height.

June, 1882.

Note 1. Candidates for examination in this subject as a whole, should
take the whole of this paper; those for the first year's partial examination,
the first part of it; those for the second year's partial examination, the
second part.

Note 2. State at the head of your pajjer what text-book you have stud-
ied on the subject, and to what extent.

1. (a) Define and illustrate the symmetry of a figure with
reference to an axis and with reference to a point.

{b) Define the difterent classes and species of quadri-
laterals.

2. The thi-ee medial lines of a triangle meet in a point
whicli is at two-thirds tiu; distance from the vertex of each
angle to the middle of the opposite side.

3. To inscribe a circle in a given triangle. Define escribed
circles relati\-e to a triangle.

4. The bisector of an interior angle of a triangle divides
the opposite side into segments proportional to the adjacent
sides.

5. When is a variable magnitude said to have a limit?
Give an example to illustrate the definition.

G. If two sides of a triangle be given, its area v'ill l)e a
maximum when these sides are at riiJ-ht ambles.



7. If a straiglit line is perpendicular to each of two straight
lines at their point of intersection, it is pcrpendicuhu to the
plane of these lines.



130 GEOMETRY,

8. Dofine symmetricnl polyhedral angles. Illustrate the
definition l)y a figure.

9. Define the poles of a circle of a sphere. Prove that all
j)oints on tlie circumference of a circle of a sphere are equally
distant from each of its poles.

10. Tiie angle between two great circles is equal to the
diedral angle between their planes.

September, 1882.

[State what text-book j-ou have studied, and to what extent.]

1 . An inscribed angle is measured b}'^ one-half its inter-
cepted arc.

2. From a given point without a circle to draw a tangent
to the circle.

3. To construct a triangle which shall be equal in area to
a given quadrilateral.

4. The diameters of two concentric circles being 10 and
G feet, required the area of the ring contained between their
circumferences.

o. The sum of any two face angles of a triedral angle is
greater than the third.

G. The lateral area of a frustum of a regular pyramid is
equal to its slant heiglit into half the sum of the perimeters
of its bases.

7. AVrite expressions for the volume of the sphere and
right C3'linder. Show that the volume of a si)lunv is to that
of its circumscribed cyliuder as 2 to :?.

5. Required the area of a splierical tiinugle described on
a sphere, whose diameter is 30 feet, the angles being 140°,
92% and 08°.



GEOMETRY. 131

June, 1883.

Note 1. — Candidates for examination on the whole of this subject
should take the wliolo of tliis paper. Candidates for the first year's parti.al
examination sliould take the first part; those for the second year's partial
examination, the second part.

Note 2. — State what text-book you have studied, and to what extent.

I. — Plane Geometry.

1. (a) Define the symmetry of a figure with respect to a
poiut and with respect to an axis.

(6) Determine the symmetrical figure of a triangle
with respect to a poiut lying within and with respect to an
axis cutting the triangle.

2. The tlu'ee perpendiculars from the vertices of a triangle
to the opposite sides meet in the same poiut.

3. To construct a polygon similar to a given polygon, the
ratio of similitude of the two polygons being given.

4. (a) What is the area of a circle circumscribed about a
square whose side is a?

(b) What must be the diameter of a circle in order
that the length of its circumference may be 100 feet?

5. To find the locus of the point from which a given line
subtends a right angle.

6. Of all triangles having the same base and equal areas,
that which is isosceles has the minimum perimeter.

II. — Solid and Spherical Geometry.

7. Between two lines not in the same plane, one, and
onl}' one, common perpendicular can be drawn.

8. All parallel()pii)ods having the same base and equal
altitudes are equal in volume.

9. Show that opposite spherical triangles are symmetrical.
If two symmetrical spherical triangles are isosceles, they are
identically equal.



132 GEOMETRY.

10. Write expressions for the lateral area and volume of
the prism and regular pj-ramid : also for the entire surface
and volume of the right cylinder and sphere.

September, 1883.

[State what text-book you have studied, and to what extent.]

1. Every point in the bisector of an angle is equally
distant from the sides of the angle ; and every point within
the angle, but not on the bisector, is nearer that side toward
which it lies.

2. If the sum of two opposite angles of a quadrilateral is
equal to two right angles, the vertices of the four angles lie
on a circle.

3. (a) Write expressions for the area of the triangle,
parallelogram, trapezoid, and circle.

(5) The areas of similar figures are to each other in
what ratio ?

(c) How would you find the area of any quadrilateral ?

4. (a) When is a variable magnitude said to have a
limit f Give an example to illustrate the definition.

(6) Apply the theory of limits to show that the area
of a circle is equal to half the product of its circumference
b}' its radius.

5. If from any point perpendiculars be dropped upon two
intersecting i)lanes, the angle between these perpendiculars
will be equal to the dihedral angle between the planes, ad-
jacent to the angle in wliich the i)oint is situated.

6. The area of a sphericnl triangle is proportional to its
spherical excess.

7. Required the entire surface and volume of a right cone
whose altitude is 12 feet, and the diameter of its base 15
feet.

8. The radius of a sphere is bisected at riglit angles by a
plane. What is the ratio of the two parts into which the
plane divides the spherical surface?



ALGEBRA. 133

ALGEBRA.
July, 1880.

[State what text-book you have studied, and to what extent.]

I.

1. Resolve the following expressions into factors,

1 G a^ b- — 24 a? bmx -f J) m" x' ; 9 rr 6" — 1 cr c- ; (V^ — 8 h'^.

2. Given 1 ^— = and ^ = ;

„ , 1 a4-b a — b a — b « + o a—b a-j-o

nncl X and y.

3. Given Va — cc -f V6 — x= , to find a;.

4. (a) From 2V72«- take Vl62tt^

(&) Find the value of V2 x a/3 X a/5.
(c) Divide 8 a — & by 2a^ — b\

5. Given ^' + 4 ;« - 8 > 3 and Gx + i^^Ejzl^ < 18, to find

2 3

a superior and inferior limit of x.

II.

6. Given '-^ '■ ~ = 4, to find the values of x.

x-2 x+2 "

7. Given x" -{- xy = ay- -f- xy = Zy, to find .f and y.

8. Expand — '^^^ — ^ into a series b}' the method of inde-

1 — .X — X-

terminate coefficients.



134



ALGEBRA.



9. The number of permutations of n things, 3 together, is
6 times the number of combinations, 4 together ; (nul n.

Note. — The first division of the paper extends as far as Quadratic
Equations, and Avill be all that is required of candidates who propose to
pass the partial exaiuiiialion only.



September, 1880.

1. Divide — ^, by — ■

x - 2bx-{-b- -^ x — b

2. Given = and 5 ax — 2 by = c, to find x

and y.

3



3. VU — x-i-Vll — x= , to find a;.

Vl 1 - a;



, — , — ^ a -\- b la — b

4. Simphfy V24 + V54 - VG ; ,7:r^>J^^^ ?

5. Extract the square root of

4 x' + 1 (> (/•' - 1 2 cu%^ — 24 a'^x + 25 a- x-.

6. Solve the equation 3.x - + 2.i; — 9 = 76.

7. If —' = — = —'•••=-!'=: /•, show that

61 />^ &.; &„

n, -I- r>, -f g,. + ••• +a„ ^ _^.

/,, .|_ 6,+ /'; 4- •■■ +b"

8. Having tlu; lirst term (a), the ratio (r), .ind llio Last
term (7), of a geometric pi-ogression, find the sum (.s), and
tlic miiuher (11) of tile tei'ius.

b

9. Expand - into a series bv the liinomial formula.

Var + u^

10. .Solve the eqiiiilions .r" = y^ and .r" = ?/".



ALGEBRA.



June, 1881.



135



Note 1. — Candidates for examination in this subject, as a whole, should
take the whole of this paper; those for the first year's partial examination,
the lirst part of it; those for the second year's partial examination, the
second part.

Note 2. — State at the head of your paper what text-book you have
studied on the subject, and to what extent.

I.

1. Find the greatest common divisor of

ixi'-i-lOa'b-^oa^b'^ and a^ b -}- 2 a- b' -{- 2 ab' -\- bK

- ^. 13m— 7x Am — x m+p , , ^ -,

2. Given 1 = kx, to find x.

m -\- 1) m — 2) ni — 2>

3. Two masons, A and B, propose to build a wall. If
both work together, they can finish it in 12 dys. ; but if A
work 2 dys. and B 3 dys., thej' will complete only one-fifth
of the job. How long will it take each of them separately
to do the work?

4. (a) Reduce Va, 'Vb, V(?, to the same index.
(b) Simplify the following expressions:



(1) Vsa'Va*"'^;

(2) 4V48-fVl47-4VT2;
(2)^(9)^(3)^



(3)



(3)«(4)^(2)^



12a; + 4



o. Given 2V2x~+2 +^/lirf2 ^ ^ T^_ , to find x.

V8a; + 8

II.

oG—x
T). Given l.r = 4^, to find x.

X

7. Determine ]}y inspection the roots of the equation,
X- — (a + b)x + (a + c) {b — a) = 0,
and state the jn'inciple upon which you do so.



13G ALGEBRA.

8. Given ary -\- xy- = a and x'-y — xy- = />, to find x and y.

9. Kxpiuul {ir-\-Ir)^ to four terms l)y the binoiniul fur-
iniilii.

10. Show that the nioduUis of any system of loi>arithnis
is eqnal to the reciprocal of the Naperian logarithm of the
base of the system ; and also that it is eqnal to the loga-
rithm of the Naperian base taken in the system.



September, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1. Resolve a'^ — U' into fonr factors.

7 4 12

2 . Given — = -\ -^ = 4 and — = -| — -= = 1 , to fi nd x and y.

V;c Wy -\/x Vy

3. Simplify the following expressions :

(a) i^; {h) («5)^; (c) a/^? ;

4. Resolve the trinomial .r-)-2;v — 120 into its binomial
factors.



^, . V'J XT — 1 4- VS — or o , ,. 1
;). Given — :;:^^;^:=^ =-^. to inid x.

V8.r-1-V3-.T- '>

G. rJivcn ").;■- + :^?/- = 22 and 3 .r'-' —.")_?/-= 7 , to find x
and //.

7. A 1):U1 rolls down an ineluied plane, describing I ft. the
first second, '.'> 11. Uie second, and so ou. IIow far will it go
in 10 sec, and liow far in the tenth second?



ALGEBRA. 137

8. Jf the poi)ul:iti()ii of a certain city is now 10. 000, and it
increases at the annual rate of 10 per cent for tlie next 10
yi's., what will it l)e at the end of that time? [Given (1.1)'"
= 2.59^7+.]

9. Expand " ^ ^ ' into a series by the method of indeter-

4 + 5 .X-

minate coefficients.

10. Find the number of combinations of 10 things taken
4 together, and also taken G together.



June, 1882.

Note 1. — Candidates for examination in this subject, as a whole, should


1 2 3 4 5 7 9 10

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