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Full text of "Mensuration for beginners with numerous examples .."

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Irving Stringliarn 




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.VIENSURATION FOK BEGINNERS, 



MENSURATION FOR BEGINNERS 




/A 

// WITH NUMEROUS EXAMPLES. 




BY 



I. TODHUNTER, M.A., F.R.S. 

HONOBART FELLOW OP S. JOHN'S COLLEGE, CAMBRIDGE. 



NEW EDITION. 



Hontron: 
MACMILLAN AND CO, 

i88r 

[The Right of Translation is reserved.] 






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figSth. D©0^* 



PRINTED BY C. J. CLAY, ILA, 
AT THE UNI\"EESITY PRESS- 



PEEFACE. 

The applications of Mensuration are numerous and im- 
portant, extending from practical questions of every day 
life to the highest investigations of science. Mensuration 
has therefore justly obtained a prominent position in 
teaching and in examinations. The value of Mensuration 
is increased by its close and necessary connection with 
Arithmetic and Geometij. The operations of Arithmetic 
are here illustrated by a new class of examples ; and some of 
these operations, such as the extraction of the square root 
and of the cube root, thus gain their true significance and 
interest. Many of the principal facts of Geometry are in- 
troduced and applied, so as to furnish a good introduction 
to the study of Euclid's Elements, or some substitute for 
it, to those who have not the opportunity for that study. 

The subjects included in the present work are those 
which have usually found a place in Elementary Treatises 
on Mensuration. The mode of treatment has been deter- 
mined by the fact that the work is intended for the use of 
beginners. Accordingly it is divided into short indepen- 
dent chapters, which are followed by appropriate examples. 
A knowledge of the elements of Arithmetic is all that is 
assumed; and in connection with most of the Rules of 
Mensuration it has been found practicable to give such 
explanations and illustrations as will supply the place of 
formal niathcmatical demonstrations, which would have been 
unsuitable to the character of the work. 

The Examples amount on tlie whole to nearly twelve 
hundred in number; some of them are taken from printed 

Hi 41 ) 1 1^ 



yi 



PREFACE. 



examination papers, but most of tliera are original, and 
have been expressly constructed with reference to the 
most important points, and to the usual difficulties of 
beginners. The miscellaneous collection of Examples at the 
end of the book is arranged in sets of ten in each set. 

Although great care has been taken to ensure accuracy, 
it can hardly be hoped that a book involving so large an 
amount of numerical calculation will be free from error. 
Any corrections or remarks relating to the book will be 
most thankfully received. 



I. TODHUNTER. 



CAMBEID3E, 

April, 1869. 



CONTENTS. 

PAGE 

Inteoduction 1 

FIRST SECTION. GEOMETRY. 

I. Definitions 2 

IL Theorems 8 

ni. Problems 15 

SECOND SECTION. LENGTHS. 

IV. Tables of Lineal Measure 23 

V. Eiglit-angled Triangle 24 

YI. Similar Figures Bl 

Vn. Chords of a Circle 37 

YIII. Cii'cumference of a Circle 45 

IX. Arc of a Circle 50 

THIRD SECTION. AREAS. 

X, Table of Square Measure 55 

XI. Rectangle 56 

XII. Parallelogram 70 

XIIL Triangle 73 

XIV. Quadrilaterals 80 

XV. Rectilineal Figure 88 

XVL Cucle 91 

XVII. Sector of a Circle and Segment of a Circle... 100 

XVin. Simpson's Rule 108 

XIX. SimHar Figures 114 

FOURTH SECTION. VOLUMES. 

XX. Definitions 121 

XXI. Solid Measure 128 

XXII. Rectangular Parallelepiped 129 

XXin. Parallelepiped, Prism, Cylinder 138 

XXIV. Segments of a Right Cii'cular Cylinder. Ring 146 



viii CONTENTS. 

PAGE 

XXV. Pyramid. Cone 149 

XXVI. Frustum of a Pyramid or Cone 155 

XXVn. Wedge 159 

XXVIII. Prismoid 164 

XXIX. Sphere 169 

XXX. Zone and Segment of a Sphere 174 

XXXI. Irregular SoUds 178 

XXXII. SimHarSonds ISO 

FIFTH SECTION. AREAS OF THE SURFACES 

OF SOLIDS. 

XXXIII. PLane Surfaces 185 

XXXIV. Eight Circular Cylinder 192 

XXXV. Segments of a Eight Circular Cylinder. Ring 198 

XXXVI. Eight Circular Cone 201 

XXXVII. Frustum of a Eight Circular Cone 207 

XXXVni. Sphere 212 

XXXIX. Zone of a Sphere. Segment of a Sphere ... 215 

SIXTH SECTION. PRACTICAL APPLICATIONS. 

XL. Introduction 221 

XLL Artificers' V\^ork 224 

XLII. Tknber Measure 228 

XLin. Gauging ... 232 

SEVENTH SECTION. LAND SURVEYING. 

XLIV. Use of the Chain 237 

XLV. Perpendiculars 240 

XLVL The Field-hook 244 

XLVn. Problems 255 

XLVin. Duodecimals 257 

XLIX. Metrical System 264 

Miscellaneous Examples 267 

ANSWERS 285 



h 



MENSURATION. 



INTRODUCTION. 

Mensuration gives rules for estimating lengths, areas, 
and colwmes. 

We shall assume that the beginner in Mensuration is 
familiar with the elements of Arithmetic, including the 
process for the extraction of the square root of a number. 

We shall also assume that he is familiar with the use of 
certain convenient symbols, namely that + denotes addi- 
tion, — denotes subtraction, x denotes multiplication, 
-i- denotes division, and J denotes the square root. 

Some knowledge of Geometry is also necessary; and 
accordingly the first three Chapters of the book treat 
of that subject. The beginner should at once read care- 
fully the first Chapter, in which various terms are defined, 
which we shall have to employ hereafter; he will how- 
ever probably find that he is already familiar with the 
meaning of many of these terms from the common use of 
them. He can then proceed to the fourth and the follow- 
ing Chapters, referring to the second and the third as 
occasion may require. 

T. M. 1 



DEFINITIONS. 



FIRST SECTION. GEOMETRY, 



1. DEFINITIONS. 

1. The words point and line are too well known to 
require any definition ; but a caution must be given with 
respect to the strict sense in which these words are used 
in Geometry. 

A point is represented in a printed book by a spot of 
ink, which may be very small, but still has some size ; w^e 
must not however suppose that a point in Geometry has 
any size. 

Lines may be straight or curved. A line is represented 
in a printed book by a band of ink, w^hich may be very 
narrow, but still has some breadth ; we must not however 
suppose that a line in Geometry has any breadth. 

2. The word surface is also in common use. Surfaces 
may be flat or curved. A flat surface is usually called a 
2jlane surface in Geometry. We must not suppose that a 
sm-face in Geometry has any thickness. 

3. Thus we may say that a point has neither length, 
breadth, nor thickness ; a line has only length ; a surface 
has length and breadth. A solid body has length, breadth, 
and thickness. "We shall not consider solid bodies imtil 
we arrive at the Fourth Section of the book; in the first 
Three Sections "we shall consider only lines and figures 
on a plane surface. 

4. An angle is the inclination of two straight lines to 
one another which meet together, but are not in the same 
straight line. 



DEFINITIONS. 




Thus the two straight lines 
AOy BO, which meet at 0, form 
an angle there. The angle is 
not altered by altering the 
lengths of the straight lines 
which form it: thus CO and DO 
form the same angle as AO 
and BO. 

5. If there is only one angle formed at a point, the 
angle may be denoted by a single letter placed at the 
angular point : thus, for example, the angle in the pre- 
ceding Article may be called the angle 0. 

If several angles are formed 
at the same point each angle is 
denoted by three letters, the 
middle letter being at the angu- 
lar point, and the other two 
letters being on the straight lines 
which form the angle, namely 
one letter on each straight line. 
Thus FEG denotes the angle 
formed by i^^ and GE ; GEH 
denotes the angle formed by GE 
and HE; FEH denotes the angle formed by FE and HE. 

6. If one angle can be so placed on another that the 
straight lines whicli form the one angle fall exactly on the 
straight lines which form the other angle, the two angles 
are said to be equal. 

Thus, if we place BG on EF so that B is on E, and 
then find that BA falls on ED, the angle ABC is said to 






be equal to the angle DEF. It is necessary to have this 
distinct notion of what is meant by the equality of angles; 
and the beginner may actually try whether two assigned 

1—2 



4 DEFINITIONS. 

angles are equal by cutting them out in paper and placing 
one on the other. But in the theory of Geometry methods 
are devised by which angles can be shewn to be equal 
without actually placing one on the other. 

7. Suppose in the diagram to Art. 5 that the angle 
FEG is equal to the angle GEH; then the whole angle 
FEH is twice the angle FEG. Similarly it is easy to un- 
derstand what is meant by the statement that a certain 
angle is three times another angle, or four tiines another 
angle ; and so on. 

8. When a straight line standing on another straight 
line makes the adjacent angles ecjual to one another, each 
of the angles is called a right angle, and the straight line 
which stands on the other is called a perpendicular to it. 



Thus, in the figure, if the angle 
ABC 19, equal to the angle ABD each 
of them is a right angle, and AB is 
perpendicular to DG. 



D- 



B 



An obtuse angle is an angle which 
is greater than a right angle. 



An acute angle is an angle which 
is less than a right angle. 



9. Parallel straight lines are such 
as are in the same plane, and which 
being produced ever so far both ways 
do not meet. 



10. A rectilineal figure is a figure which is bounded 
by straight lines ; the boundaries of the figure are called 
sides. 

A triangle is a rectilineal figure with three sides. 
A quadrilateral is a rectilineal figure with four sides. 



DEFINITIONS. 6 

Any rectilineal figure with more than four sides is 
called a polygon ; if it has five sides it is called a pen- 
tagon, if it has six sides it is called a hexagon, and so on. 

A regular polygon is one which has all its sides equal, 
and all its angles equal. 

11. The following names are used for various kinds of 
triangles : 



An equilateral triangie is one 
wliich has all its sides equal. 




An isosceles triangle is one which 
has two of its sides equal. 




A right-angled triangle is one 



which has a right angle. 




In a right-angled triangle it is found convenient to use 
the word sides only for the two straight lines which include 
the right angle: the straight line which is opposite the 
right angle is called the hypotenuse. 



An obtuse-angled triangle is one 
which has an obtuse angle. 




An acute-angled triangle is one 
which has three acute angles. 




6 



DEFINITIONS. 



12, The following names are used for various kinds of 
quadrilaterals : 



A parallelogram has its opposite 
sides parallel and equal. 




A rectangle is a i)aranelograra 
with all its angles right angles. 



A square is a rectangle with all 
its sides equal. 



A rhombus is a parallelogram 
which has all its sides equal, but its 
angles are not right angles. 



/ 



allel. 



A trapezoid has two sides par- 




13. Any side of a triangle may be called the base; 
then the height of the triangle is the perpendicular drawn 
from the opposite angular point on the base. 

Any side of a parallelogram niay be called the base; 
then the height of the parallelogram is the perpendicular 
drawn from any point in the opposite side on the base. 

14. A diagonal of a quadrilateral is a straight line 
joining two opposite angles. A straight line joining two 
angles of a polygon which are not adjacent is also called a 
diagonal of the polygon. 



DEFINITIONS. 

15. A circle is a plane figure 
bounded by one line which is called 
the circumference, and is such that 
all straight lines drawn from a cer- 
tain point within the figure to the 
circumference are equal to one an- 
other : this point is called the centre 
of the circle. 

A radius of a circle is a straight 
line drawn from the centre to the circumference. 

A diameter of a circle is a straight line drawn through 
the centre and terminated both ways by tlie circumference. 

An arc of a circle is any part of the circumference. 

A chord of a circle is the straight line which joins tlse 
ends of an arc. 




A segment of a circle is the 
figure bounded by a chord and tlie 
arc it cuts off. 




A sector of a circle is the figuie 
bounded by two radii and the arc 
between them. 




The angle formed by the two radii is called the angle 
of the sector. 



A zone of a circle is the portion 
of the circle contained between two 
parallel chords. 




8 THEOREMS. 



II. THEOREMS. 

16. We shall now state souie important geometrical 
facts which it will be advantageous to remember.^ These 
facts, with many others, ai'C demonstrated in Euclid's Ele- 
ment's of Geometry, that is tliey are shewn by strict reason- 
ing to be necessarily true : and all who have the opportunity 
should study Euclid's demonstrations. But the present 
work nicvy be used by those who have not yet applied them- 
selves to demonstrative Geometry ; and so they may for 
some time be satisfied with understanding the meaning of 
the statements which we shall make and committing them 
to memory. 

When the demonstration is of a very simple character 
we shall however give the substance of it, and an attempt 
to master this, even if at first not entirely successful, will 
prove very beneficial. Some of the statements for which 
we give no demonstration will appear almost self-evident ; 
others may be verified by repeated practical measurement ; 
so that at last a confidence in the truth of all may be 
gained, approaching to absolute conviction. 

17. We have selected only a few of the most important 
geometrical facts out of the large collection which has been 
formed by the investigations of mathematicians ; but these 
specimens will be sufficient to suggest some idea of the 
extent and variety of the results which follow by strict 
connexion from a few elementary principles, and may 
tempt the student to increase his knowledge of the subject 
hereafter. 

It will be found that Arts. 18... 21 relate to angles. 
Arts. 22... 27 relate to triangles, Arts. 28... 30 relate to the 
equivalence of areas, Arts. 31... 33 relate to properties of 
the circle, and Arts. 34... 38 to similar triangles. 

18. Let the str^iight line AB make with the straight 
line CD on one side of it the angles ABC and ABD : 
these angles will he together equal to ttco right angles. 

For let BE be at right angles to DG. Then the angle 
ABD is the su n of the angles ABE and EBB ; so that the 



THEOREMS. 



9 




B 




sum of the two angles ABG 
and ABD is equal to the sum 
of the three angles ABG, 
ABE, and EBD. 

But the angle EBD is a 
right angle; and the sum of 
the angles ABG and ABE 
is EBGy which is also a right j5" 
angle. 

Therefore the angles ABG and ABD are together 
equal to two right angles. 

19. Let two straight lines AB and CD cut one another 
at E ; the angle AEC will be equal to the angle BED, and 
the angle AED will he equal to the angle BEC. 

For the angles AEG and 
GEB are together equal to 
two right angles, by Art. 18 ; 
and so also are the angles 
GEB and BED. Thus the 
angles AEG and GEB are 
together equal to the angles 
GEB and BED. Therefore the angle AEG must be equal 
to the angle BED. 

In a similar manner we can shew that the angle AED 
is equal to the angle BEG. 

The angles AEG and BED are called verticallij oppo- 
site angles; and so also are the angles AED and BEG. 

20. Let the straight line 
EF cut the parallel straight 
linesAB,Qiy. theangleY.GB 
icill he equal to the angle 
GHD, and the ttoo angles 
BGH and GHD will be^ to- 
gether equal to two right 
angles. 

21. Since the angle EGB is equal to the angle AGH 
by Art. 19, it follows from the first part of the ])rcceding 
Article that the angle AGHis equal to the angle GHD : 
these angles are called alternate angles. 

So also the angle BGH is eq\ral to the alternate angle 
GHG. 





10 THEOREMS. 

22. Let BC a side of the triangle ABC he produced 
to T> ; the exterior angle A CD will he equal to the two in- 
terior and opposite angles. 

For suppose CE to be 
parallel to BA. Then the ^ j, 

angle ECD is equal to the 
angle ABC by Art. 20; 
and the angle AGE is 

equal to the angle BAG- ^ ^ 

by Art. 21. Thus the whole ^ u i; 

angle AGD is equal to the sum of the two angles ^4 5(7 and 
BAG. 

23. TJte three angles of any triangle are together 
equal, to two right angles^ 

For by Art. 22 the sum of the angles -45(7 and BAGh 
equal to the angle AGD. Thus the sum of the three angles 
ABG, BAG, and AGB is equal to the sum of the two 
angles AGD and AGB; that is to two right angles, 
by Art. 18. 

24. If ttco sides of a triangle are equal the angles 
opposite to them zcill also he equal. 

25. If two angles of a triangle ai^e equal the sides 
opposite to them icill also he equal. 

26. If two sides of one tinangle are equal to ttco sides 
of another triangle, each to each, and the angle contained 
hy the two sides of the one equal to the angle contained hy 
the two sides of the other, the triangles will he equal in all 
respects. 

27. If two angles of one triangle are equal to tico 
angles of another triangle, each to each, afid the side ad- 
jacent to the tioo angles of the one equal to the side ad- 
jacent to the tico angles of the other, the triangles icill he 
equal in all respects. 

28. A parallelogram, is equivalent to a rectangle on 
the same hase and hetween the same parallels. 

Let ABGD be a parallelogram, and ABEF a rect- 
angle, on the same base A B, and between the same paral- 



THEOREMS. 



11 



F D 


JG C 


/ 


/ 


A 


■8 



lels AB aud i^6'; the paral- 
lelogram is equivalent to the 
rectangle, that is the two 
figures are of the same size. 

It is in fact easy to 
admit that the triangle BEG 
is equal to the triangle 
AFD; and hence it fol- 
lows that A BCD is equiva- 
lent to ABEF. 

Instead of saying that the parallelogram and the rect- 
angle are between the same parallels, we may say that they 
have the same height: see Art. 13. 

29. A triangle is equivalent to half a rectangle ha dug 
the same base and height. 

Let ABC be a triangle, and 
ABDE a rectangle, on tiic saaie 
base AB, and having the same 
height: the triangle is equiva- 
lent to half the rectangle. 

Let CFhQ theperpendicr.lar 
from C on yi 5. It is easy to ad- 
mit that the triangle BFC is equal to the triangle CDB, ai.d 
that the triangle AFC is equal to the triangle .CEA ; and 
hence it follows that ABC is equivalent to half ABDE. 

Hence two triangles which have the same base and 
equal heights areequivalevit. 

30. Ill any right-angled triangle the square described 
on the hypotenuse is equal to the sum of the squares 
described on the sides. 

The figure represents a 
right-angled triangle having 
squares described on its hy- 
potenuse and its sides : the 
largest square is equal in 
size to the sum of the other 
two. This statement is one 
of the most imx)ortant in 
Geometry ; and we will shew 
how its truth may be illus- 
trated. 





12 



THEOREMS. 



Let BCDEFG be a fi- 
gure composed of two squares 
placed side by side : take GH 
equal to BC, and draw the 
straight lines CH and FH. 
Cut the whole figure out in 
paper or cardboard, and then 
divide it into the three pieces 
marked 1, 2, and 3. Fit the 
pieces together in the man- 
ner indicated by the figure 
HGAF. It will be ^ found 
tiiat a single square is thus 
obtained, each side being 
equal to FH. 

Hence we see that the 
square described on FH is 
equal to the sum of the 
squares described on FG 
and GH. 





31. If a straight line 
touch a circle the radius 
drawn to the point of con- 
tact will he perpendicular 
to the straight line. 




32. Let BAD and JiKD 

he angles in tJte same seg- 
ment BAED of a circle : 
these angles icill he equal. 




THEOREMS. 



13 



33. Let AB he a dior 
meter of a circle, and C any 
point on the circumference; 
draio the straight lines AC 
andl^Q'. then the angle ACB 
will he a right angle. 




34. Let ABO and DEF he two triangles such that 
the angle A is equal to the angle D, the angle B to the 





angle E, and the angle C to the angle F : then the sides 
opposite the equal angles will he proportionals. 

That is, if EF be double of BC, then FD is doable of 
CA, and DE is double oi AB ; if EF be three times BC, 
then FD is throe times CA, and DE is three times AB -, 
and so on. The two triangles are said to be similar. 

35. Let ABC and DEF 6^ similar triangles, the an- 
gles C a?2<i F heing corresponding angles ; leti^Qi and ¥11 




A Ct 




6^ perpendiculars from C awc^ F o^z ^^^ opposite sides : 
then CG will he to AB a* FH is to DE. 



14 



THEOREMS. 




36. Lei ABC he a tri- 
angle, and DE a straight 
line parallel to the side BC, 
and meeting the other sides 
AB and AG : the7i the tri- 
angles ABC and ADE will 
he similar. 

See Art. 20. 

37. Let ABC he a right- 
angled triangle, and let AD 
he the perpendicular from 
the right angle on the hypo- 
tenuse: then the triangles 
DBA and DAC u:ill he si- 
milar to the triangle ABC. 

38. Let AB and CD he two chords of a circle; Jet 
thein meet, produced if necessary, at E ; join BC 'and 





AD : then the triangles AED and BEC will he similar, 
the angles EAD and ECB heing equal, and the angles 
EDA and EBC heing equal. 

See Art. 32 



PROBLEMS. 



15 



III. PROBLEMS. 

39. "We shall now give the solutions of a few problems 
which occur in practice when it is necessary to draw figures 
accurately. We suppose that a ruler and compasses are 
employed; these instruments will be sufficient for our pur- 
pose. Other instruments are often useful, such as a square 
and parallel rulers ; but they are not absolutely necessary. 

The solutions which we shall give of the problems de- 
pend mainly on the principles stated in Chapter II ; there 
will not be much difficulty in verifying practically the cor- 
rectness of the results, and those who make themselves 
acquainted with the elements of demonstrative Geometiy 
Avill perceive the rigorous exactness of the processes. 



40. 



To divide a given straight line into two equal 



Let AB be the given straight 
line. From the centres A and B 
wdth a radius greater than half 
AB describe arcs cutting each 
other at D and E. Join DE, 
cutting AB at C. Then ^C will 
be equal to CB. The straight 
line DE will be at right angles to 
AB, so that we see how to draw 
a straight line which shall be at 
right angles to a given straight 
line and shall also divide it into 
two equal parts. 



A 



t 



C 



\ 



B 



16 



PROBLEMS. 



41. To divide a given angle into two equal parts. 

Let ABC be the given angle. 
From the centre B with any radius 
describe an arc, cutting BA at Z>, 
and BC at E. From D and E as 
centres with any suflScient radius 
describe arcs cutting each other 
at F. Join BF. Then the angle 
ABF will be equal to the angle 
CBF. 




42. To draw a straight line parallel to a given straight 
Ime, and at a given distance from it. 

Let AB ha the given straight line, and let C be equal 
to the given distance. From any two points Z> and E in 



Gr 



JD 



JS B 



AB as centres, with a radius equal to C, describe arcs. 
Draw a straight line FG touching these arcs. Then FG 
will be parallel to AB, and at a distance from it equal 
to G. 



PROBLEMS. 



17 



43. To make a triangle having its sides equal to three 
given straight lines. 

Let A, B, and C be the given straight lines. 







J3 



A 




Draw a straight line DE equal to one of the given 
straight lines, A. From the centre D, with a radius equal 
to B, describe an arc ; and from the centre E, with a radius 
equal to (7, describe another arc. Let these arcs cut each 
other at F. Join DF and EF. Then DEF will be the 
triangle required. 



44. Through a given point to draw a straight line 
parallel to a given straight line. 



A 



1^ 



Let A be the given 
point,and^Cthe given 
straight line. Take any 
point D in BG. From 
the centre D, with the 
radiusD^, describe an 
arc, cutting BC ^t E, 
and draw the chord 
AE. From the centre 
A, with the radius ^Z>, describe an arc, and draw the chord 
DF equal to the chord AE. Join AF. Then AF will be 
parallel to BC. 

T. M. 2 




18 



PROBLEMS. 



F 




45. To draw a straight line at right angles to a given 
straight line from a given point in it. 

Let AB be the given straight Hue, and G the given 
point in it. 

From any point D without 
the straight line as centre, with 
radius DC, describe a circle, 
cutting the given straight line 
at E. Join ED, and produce 
it to meet the circumference 
again at F. Join OF. Then CF 
will be at right angles to AB. 

46. To draw a perpendicular to a given straight line 
from a given point without it. 

Let ^^ be the given straight line, and G the given 
point without it. 

Take any two 

points D and E in 
AB. From the cen- 
tre D, with the ra- 
dius DG, describe 
ail arc ; and from 
the centre E, with 
the radius EG, de- 
scribe another arc. 
Let these arcs cut each other again at F. 
GF will be perpendicular to AB. 

47. To divide a given straight line into any nmnher 
of equal parts. 

Let ^^ be the given 
straight line. Suppose it 
is to be divided into fve 
equal parts. 

From A draw any 
straight line AC, and from 
B draw the straight line 
BD parallel to AG. Set off 
along AG four lengths, all 
equal, and mark the points 
of division 1, 2, 3, 4. Set off 



A D 



E 






"B 



JoinCi^. Then 




PROBLEMS. 



19 



along BD four lengths all equal to the former lengths, and 
mark the points of division 1, 2, 3, 4. Draw straight lines 
joining 1 to 4, 2 to 3, 3 to 2, and 4 to 1. These straight 
lines will divide AB into five equal parts. 

We should proceed in a similar way whatever be the 
number of equal parts into which yl^ is to be divided. 

4S. To find the centre of a given circle. 

Draw any chord AB, and 
divide it into two equal parts 
by the straight line DE at 
right angles to it. Then the 
centre of the circle is in DE. 

Draw anv other chord BG, 
and divide it into two equal 
parts by the straight line FG 
at right angles to it. Then 
the centre of the circle is in 
FG. 

Thus the centre of the circle is at 0, the point of 
intersection of DE and FG. 

This process also shews how to describe a circle which 
shall pass through three given points A, ^, G. 




49. Any polygon being given it is required to con- 
sti^uct an equivalent polygon with one side fewer. 

Let ABGDEF be a poly- 
gon. Join BD. Through G 
draw a straight line parallel 
to BD meeting A B produced 
at//. Join DH. 

Then the triangle BCD 
is equivalent to the triangle 
BHD,hj Art. 29 ; and there- 
fore the polygon ABGDEF 
j is equivalent to the polygon 
I AHDEF^ that is to a poly- 
I gon with one side fewer. 

By repeated operations of this kind we can construct a 
triangle equivalent to any given polygon. 

2—2 




20 PROBLEMS. 

60. To construct and use a decimal diagonal scale. 

Take a straight line AB of any convenient length ; 
produce it until the whole straight line is ten times AB, 
and take BG, CD, ... all equal to AB. Draw any straight 
line parallel to this, and in it set ofl" ten distances ad, be, 
cd, ..all equal to AB. Draw Aa, Bh, Cc, Dd, ... Divide 
Aa into ten equal parts, and through the points of division 
draw straight lines parallel to AB. Divide BA into ten 
equal parts, and mark the points of division by the numbers 
1, 2, 3, 4, 5, 6, 7, 8, 9. Divide ha into ten equal parts. 

Join the f)oinfcs^of division of BA and ha by diagonal 
straight lines ; that is join B with the point adjacent to h, 
join 1 with the next point in ha ; join 2 with the next 
point ; and so on. 

Thus the scale is constructed. 

It will be seen that our figure does not represent all 
the scale, the part beyond Dd being omitted for want of 
space. It is usual, but not absolutely necessary, to make 
Aa perpendicular to AB. 

The scale is used for finding straight lines of given 
lengths, and for finding the lengths of given straight hues. 

Suppose, for example, that AB represents one inch, and 
let it be required to find the straight line w^hich represents 
2o7 inches. Place one foot of a pair of compasses at D, and 
open the compasses until the other foot comes to 5 ; thus v>e 
have the length 2'5, which is nearly what is required. Now 
move one foot of the compasses along Dd, and the other foot 
along the diagonal straight line which begins at 5, the com- 
passes being at the same time spread to the necessary extent : 
when both feet are on the seventh straight line parallel to 
AD the distance between the feet will represent 2*57 inches. 

If instead of representing one inch, AB represents 10 
inches, then the distance just found will represent 257 
inches ; and if ^5 represents 100 inches, then the distance 
just found will represent 257 inches. 

Next suppose we have to find the length of a given 
straight line. Open the compasses so that the feet may 
fall on the ends of the given straight line. Move one foot 
of the compasses along one of the direct straight lines Bh, 
Cc, Dd, ... , and the other foot along one of the diagonal 
straight lines, until by trial it is found that the two feet 
fall on two points of division on one of the straight lines 



PROBLEMS. 



21 



parallel to AD : then the length will be known. For ex- 
ample ; suppose one foot of the compasses on Cc, and the 
other on the diagonal straight line which begins at 9, and 
both feet on the fifth straight line parallel to AB ; then 
the required length is 1*95 times the length of AB. 



12 






















O 










































« 






■ ' 


— 


I 




— 


— - 


— 




H 
CO 

00 


— 





















22 PROBLEMS. 

51. In a similar manner we can construct a duodecimal 
diagonal scale ; we shall only have to change the number 
ten, which occurs in the process already given, to the num- 
ber twelve. Then by the same method as we found a 
straight line of length 2*57 we should, with the aid of the 
duodecimal diagonal scale, find a straight line of length 

5 7 
2 + r7; + r7T; so that if u45 represents one foot we should 
12 144 

find a straight line representing the length 2 feet b^^ inches. 

Similarly the length of a distauce determined as in the 

second example of Art. 50 will now be to the length of 

9 5 
AB as 1 +r^ + T—r is to 1 ; so that if AB represents one 
12 144 

foot, the distance represents 1 foot 9/^ inches. 



23 



SECOND SECTION. LENGTHS. 



lY. TABLES OF LINEAL MEASURE. 

52. The student is probably already acquainted with 
the Table of Measures of Length ; but for convenience we 
will give it here: 

12 inches make 1 foot. 

3 feet make 1 yard, 

6 feet make 1 fathom. 
16^ feet or 5^ yards make 1 rod or pole. 
40 poles make 1 furlong. 

8 furlongs make 1 mile. 

Hence we obtain the following results : 

Mile. 



Inches. 


Feet. 


Yards. 


Poles. 1 


^'urlon 


12 


1 








36 


3 


1 






198 


16^ 


5i 


1 




7920 


660 


220 


40 


1 


63360 


5280 


1760 


320 


8 



53. In measuring land a chain is used, called Gunter's 
Chain^ which is 22 yards long, and consists of 100 equal 
links; each link is therefore "22 of a yard long, that is 
7*92 inches. Thus 25 hnks make a pole, 10 chains or 1000 
links make a furlong, and 80 chains or 8000 links make a 
mile. 



24 RTGHT-ANGLED TRIANGLE. 



V. RIGHT-ANGLED TRIANGLE. 

54. When we know the lengths of two of the three 
straight lines which form a right-angled triangle, we can 
calculate the length of the third straight line. We shall 
now give the rules for this purpose, which depend on the 
theorem of Art. 30, as will be more clearly seen hereafter. 
See Art. 138. 

55. The sides of a right-angled triangle being given, 
to find the hypotenuse. 

Rule. Add the squares of the sides and extract the 
square root of the sum. 

56. Examples : 

(1) One side is 8 feet, and the other side is 6 feet. 

The square of 8 is Qi, and the square of 6 is 36 ; the 
sum of 64 and 36 is 100; the square root of 100 is 10. 
Tims the hypotenuse is 10 feet. 

(2) One side is 2 feet, and the other side is 10 inches. 

2 feet are 24 inches ; the scfuare of 24 is 576, and the 
square of 10 is 100; the sum of 576 and 100 is 676: the 
square root of 676 is 26. Thus the hypotenuse is 26 
inches. 

57. In the Example just solved one side was given 
expressed in feet^ and the other side expressed in inches ; 
before we a,pplied the rule for finding the hypotenuse we 
turned the feet into inches, so that both the sides might 
be expressed in the same denomination. In like manner 
before using any rule in mensuration, it is necessary to 
express all the given lengths in the same denomination. 
"We may work with all the lengths expressed in inches, or 
with all expressed in feet, or with all expressed in yards, 
or with all expressed in any other denomination ; but we 
must not work with some of the lengths expressed in one 
denominntion, and some expressed in nnother. 



RIGHT-ANGLED TRIANGLE. 25 

58. In the two Examples solved in Art. 56, the square 
root could be found exactly, and so the length of the 
hypotenuse was determined accurately. But it may happen 
that the square root cannot be found exactly; in such a 
case Ave can continue the process for extracting the square 
root to as many decimal places as we think necessary. 

59. Examples : 

(1) One side is 3 feet 4 inches, and the other side is 
2 feet 8 inches. 

3 feet 4 inches = 40 inches, 2 feet 8 inches = 32 inches. 

• .32 40 2624-0606 (,51-22 
_32 40 25 

64 1600 lOU 124 

_JL^_ 1024 101 

1024 2624 1022>' 2300 

2044 

10242^1 25600 
20484 

5116 

Thus if we proceed to two decimal places we find that 
the hypotenuse is approximately 51 '22 inches. 

(2) One side is 2*4 feet, and the other side is 1*2 yards. 
1*2 yards = 3-6 feet. 



2-4 


3-6 


1 8-7206 U'32 


2-4 


3-6 


16 


96 


21 6 


83^* 272 


48 


108 


249 


5-7 6 


12-9 6 


862y' 2300 




5-7 6 


17-24 



1 8-7 2 576 

Thus if we proceed to two decimal places we find that 
the hypotenuse is approximately 4"32 feet ; or, taking the 
nearest figure, we may say that it is 4'33 feet. 



26 RIGHT-ANGLED TRIANGLE. 

60. The hypotenuse and one side of a right-angled 
triangle being given, to find the other side. 

Rule. From the square of the hypotenuse subtract 
the square of the given side, and extract the square root of 
the remainder. 

Or, Midtiply the sum, of the hypotenuse an I the side 
by their difference^ and extract the square root of the 
product. 

61. Examples : 

(1) The hypotenuse is 10 feet, and one side is 8 feet. 

The square of 10 is 100, and the pquare of 8 is 64 ; take 
64 from 100, and the remainder is 36 ; the square root of 
.36 is 6. Thus the other side is 6 feet. 

Or thus : the sum of the hypotenuse and the given side 
is 18 ; their diflerence is 2 ; the product of 18 and 2 is 36 : 
the square root of 36 is 6. 

(2) The hypotenuse is 26 inches, and one side is 10 
inches. 

The square of 26 is 676, and the square of 10 is 100 ; 
take 100 from 676, and the rensainder is 576 ; the square 
root of 576 is 24. Thus the other side is 24 inches. 

Or thus : the sum of the hypotenuse and tlie given side 
is 36 ; their difference is 16 ; the product of 36 and 16 is 
576 : the square root of 576 is 24. 

62. We have given two forms of the Rule in Art. 60 ; 
the first form is more obviously connected with the Rule 
in Art. 55 ; the second form is generally more convenient 
in practice, as requiring less work. 

63. In the two Examples solved in Art. 61 the square 
root could be foimd exactly, aiid so the length of the side 
was determined accurately. But it may happen that the 
square root cannot be found exactly ; in such a case we can 
continue the process for extracting the square root to as 
many decimal places as we think necessary. 



RIGHT-ANGLED TRIANGLE. 27 

64. Examples : 

(1) The hypotenuse is 1 foot 9 inches, and one side is 
14 inches. 

1 foot 9 inches = 21 inches. ^45-0600 (15-65 

21 + 14 = 35, 1 

21-14 = 7, 25ylT45~ 

35x7 = 245. 125 

306^* 2000 
1836 

3125>' 16400 
15625 

775 
Tims if we proceed to two decimal places, we find that 
the required side is approximately 15'65 inches. 

(2) The hypotenuse is 2*7 yards, and one side is 3'4 
feet. 

54-0506 (7-35 
49 

143y) 505 

429 



2-7 yards = 


8-1 feet 


8-1 + . "•4 = 


= 11-5, 


8-1-3-4 = 


= 4-7. 


1 1-5 




4-7 




805 




460 





1465yl 7600 
7325 

275 
5 4-0 5 
Thus if we proceed to two decimal places, we find that 
the required side va approximately 7-35 feet. 

65. We will now solve some exercises which depend on 
the E.ules already given. 

(1) One side of a right-angled triangle is 408 feet ; the 
sum of the hypotenuse and the other side is 578 feet : 
required the hypotenuse ai.d the other xide. 

By Art. 60 the square of 408 is equal to the product of 
the sum of the hypotenuFe and the other side into their 
difference ; therefore if the square of 408 be divided by 
578, the quotient will be the diS'erence of the hypotenuse 
and the other side. In this way we find that the diSer 
ence of the hypotenuse and the other side is 288. 



28 



RIGHT-ANGLED TRIANGLE. 



Thus the sum of the hypotenuse and the other side is 
578, and their difference is 2S8. Add and divide by 2; 
thus we obtain 433, which is the hypotenuse. Subtract 433 
from 578 and we obtain 145, which is the other side. 

(2) Each side of an equilateral triangle is 1 foot : 
required the height of the triangle. 

Let ABG be the triangle; 
Gp the height. GD will di- 
vide AB into two equal parts ; 



thus AD = \ 

Ad 



"We shall find 



GD by the second form of the 
rule in Art. 60, 




1 3 

^"2 = 2' 



1-1 = 1 

2 2' 



3 1 
2^2^ 



3 



3 1 

The square root of - = - of the square root of 3. 



The 



square root of 3 cannot be found exactly ; if we proceed 
to three decimal places we obtain r732, and half of this 
is -866. Thus the height is '866 feet approximately. 

(3) The base of a triangle is 56 feet, the height is 15 
feet, and one side is 25 feet : required the other side. 

Let^i? = .56, (7Z) = 15, BG-- 

We first find BD bv Art. 60. 
25 + 15 = 40, 25 -"15 = 10, 
40 X 10 = 400 : the square root 
of 400 is 20. Thus DB = 20. 

Therefore^Z) = 56-20-36. 

Then we find ^ C by Art. 55. The square of 36 is 1296, 
and the square of 15 is 225 ; the sum of 1296 and 225 is 
1521 : the square root of 1521 is 39. Thus AG= 39 feet. 




i 



EXAMPLES. V, 29 



EXAMPLES. V. 

Determine the hypotenuse from the given sides in the 
following right-angled triangles : 

1. 532 feet, 165 feet. 2. 7584 feet, 3937 feet. 

3. 278 feet 8 inches, 262 feet 6 inches. 

4. Half a mile, 345 yards 1 foot. 

Determine in feet, as far as two decimal places, the" 
hypotenuse from the given sides in the following right- 
angled triangles : 

5. 437 feet, 342 feet. 6. 4395 feet, 3874 feet. 

7. 314 feet 3 inches, 228 feet 9 inches. 

8. A quarter of a mile, 427 yards 2 feet. 

Determine the other side from the given hypotenuse 
and side in the following right-angled triangles : 

9. 725 feet, 644 feet. 

10. 16417 feet, 14208 feet. 

11. 269 feet 5 inches, 250 feet 8 inches. 

12. 340 yards 1 foot, 1 furlong. 

Determine in feet, as far as two decimal places, the 
other side from the given hypotenuse and side, in the 
following right-angled triangles : 

13. 647 feet, 431 feet. 14. 4987 feet, 3765 feet. 

15. 424 feet 3 inches, 276 feet 6 inches. 

16. 5 furlongs, 916 yards 2 feet. 

17. The sides of a triangle are 22620 feet and 12815 
feet, and the height is 11484 feet : find the base. 

18. One side of a right-angled triangle is 3925 feet ; 
the difference between the hypotenuse and the other side is 
625 feet : find the hypotenuse and the other side. 

19. A ladder 25 feet long stands upright against a 
wall : find how far the bottom of the ladder must be pulled 
out from the wall so as to lower the top one foot. 



30 EXAMPLES. V. 

20. A ladder 40 feet long is placed so as to reach a 
window 24 feet high on one side of a street, and on turning 
the ladder over to the other side of the street it reaches a 
window 32 feet high : find the breadth of the street. 

21. The bottom of a ladder is phiced at a point 14 feet 
from a house, and the top of the ladder rests against the 
liouse at 48 feet from the ground; and on turning the 
hidder over to the other side of the street its top rests at 
40 feet from the ground : find the breadth of the street. 

22. Find to ten decimal places the diagonal of a square 
of which the side is one inch. 

23. A side of a square is 110 feet : find the diagonal. 

24. The radius of a circle is 82*66 feet ; the perpendi- 
cular drawn from the centre on the chord is Tl'l feet : find 
the chord. 

25. A footpath goes along two adjacent sides of a 
rectangle; one side is 196 yards, pud the other is 147 
yards : find the saving in distance made by p: oceeding 
along the diagonal instead of along the two sides. 

26. The span of a roof is 28 feet ; each of its slopes 
measures 17 feet from the ridge to the eaves: find the 
height of the ridge above the eaves. 

27. The side of a square is 8 feet : find the radius of 
the circle described round the square. 

28. The radius of a circle is 6 feet : find the side of a 
square inscribed in the circle. 

29. The radius of a circle is 7 feet : find the perpen- 
dicular from the centre on a chord 8 feet long. 

30. The radius of a circle is 17 inches ; the perpendicu- 
lar from the centre on a chord is 13 inches: find the chord. 

31. The radius of a circle is divided into six equal 
parts, and at the five points of division straight lines are 
drawn at right angles to the radius to meet the circum- 
ference: find the lengths of these straight lines, in inclies to 
three decimal places, that of tlie radius being one foot. 

32. The radius of a circle is 7 feet ; from a point at the 
distance of 12 feet from the centre a straight line is drawn 
to touch the circle : find the length of this straight line. 



i. 



SIMILAR FIGURES. 31 



VI. SIMILAR FIGURES. 

66. Let ABC and DEF be two similar triangles. Then 
AB is to BC as DE is to EF ; see Art. 34. 





Thus if two sides of one triangle be given, and one of 

the corresponding sides of a similar triangle, the other 

corresponding side of the second triangle can be found. 

The process will be that which is known in Arithmetic as 

j Proportion or the Rule of Three. 

67. Examples : 

(1) Suppose AB = 5, BG= 6, DE= 7, 

5 : 6 :: 7 : EF 

Thu3^F=^=-f = 8f 
5 5^ 

(2) Suppose AB = 5,AC=-4,I)E=^1, 

5 : 4 :: 7 : BF. 

Thus DF=^ = ^ = 51 
5 5^ 

68. Similar triangles frequently present them elves in 
the theory and practice of mathematics. 

For example, we found in Art. 65 that if the side of an 
equilateral triangle be 1 foot the height is "See... feet. 
Now this proportion will always hold between the side 
and the height of an equilateral triangle ; so that if the 
side of an equilateral triangle be 7 feet the height will be 
7 X -866 . . feet. 



32 



SIMILAR FIGURES. 



Again, we have said that the triangles A ED aud BEG 
ill the diagram of Art. 38 are similar ; so that EA is to ED 
as EC is to EB. Hence it follows by the usual theory of 
Proportion that the product of EA into EB is equal to the 
product of EC into ED ; this is a very remarkable and very 
important property of the circle. 

69. By the aid of similar triangles we can determine 
the height of an object when we have measured the length 
of its shadow. 

For example, suppose that a stick is fixed upright in 
the ground, and that the height of the portion above the 
ground is 3 feet and the length of the shadow 4 feet. Also 
suppose we find at the same time that the length of the 
shadow of a certain tree is 52 feet. Then w^e determine 
the height of the tree by the proportion 

4 : 3 :: 52 : the height. 

52x3 
Thus the height is -^ — feet, that is 39 feet. 

70. From similar triangles we pass naturally to the 
consideration of similar rectilineal figures. 

Similar rectilineal figures are those which have their 
several angles equal, each to each, and the sides about the 
equal angles proportionals. 

71. Take for example two five-sided figures ABCDE 
and abcde ; these are similar if the angles at yl, ^, C, D, E 




A 33 a 13 

are equal to the angles at a, h, c, d, e respectively, and the 



SIMILAR FIGURES. 33 

sides about the equal angles proportional, that is, AB to 
BG as ah to he, and BG to GD as he to cd, and so on. 

72. Thus to ensure the similarity of rectilineal figures 
we must have two properties, namely, equality of angles 

i and projyortionality of sides. Theory demonstrates that if 
two triangles have one of these properties they will neces- 
sarily have the other ; and it is easy to test this practically. 
For example, let two triangles be drawn on paper, such 
that the sides of one are twice or three times as long as the 
sides of the other; cut the triangles out, and apply one 
triangle on the other ; it "will be found that the corre- 
sponding angles are equal. But in the case of rectilineal 
figures having more than three sides, either of the proper- 
ties may exist singly without the other. For example, 
take a square and any rectangle which is not a square ; 
here the angles of one figure are respectively equal to the 
angles of the other, but the sides are not proportional. 
Again, take a square and a rhombus ; here the sides of one 
figure are all in the same proportion to the sides of the 
other figure, but the angles of one figure are not equal to 
the angles of the other figure. 

73. Similar rectilineal figures can always be divided 
into the same number of similar triangles. TLus, for 
example, by drawing the straight lines GA, DA, ca, da, 




we can divide the five-sided figures of Art. 71 into three 
pairs of similar triangles. 

T. M. 3 



34 SIMILAR FIGURES. 

74. The statement made in Art. 66 with respect to 
similar triangles holds for any two similar rectilineal 
figures ; that is, if two straight lines situated in one figure 
be given, and a straight line corresponding to one of them 
in a similar fi^'ure, the straight line corresponding to the 
other can be to and by Proportion. 

75. Similar figures may occur which are bounded by 
curved lines as well as those which are bounded by straight 
lines. Thus, two maps of diiferent sizes may represent the 
same country ; the two maps will then be similar. For 
exam|)le, one map may be on the scale of an inch to a mile, 
and the other map on the scale of half an inch to a mile : 
then any line drawn on the first map will be twice as long 
as the corresponding line drawn on the second map. 

76. A good notion of similar figures may be conveyed, 
by saying that they are exactly alike in form although they 
may differ in size. 

All circles are similar figures. 

77. We will now solve some exercises which depend on 
the similarity of figures. 

(!) In the diagram of Art. 73 suppose AE=2 inches, 
^16^=4^ inches, a6 — \\ inches : find ac. 

2 : 4\ :: l^ : ac, 
4^ X l¥ _4o_oi.o 

Thus ac^2]% inches. 

(2) In the preceding exercise find the proportion of 
AD to ad. 

Since AE=2 and ae=l^, any straight line as AD 
bears to the corresponding straight line ad the proportion 
of 2 to li, that is of 2 to |, that is of | to 2, that is of 
8 to 5. 



SIMILAR FIGURES, 35 

(3) In the diagram of Art. 37 if BC=- 15, and BA = 12, 
find BD. 

The triangles ABC and DBA are similar : thus 

BG : BA :: BA . BD; 

that is, 15 : 12 :: 12 : BD. 

Thus BDJ^' = '4^ = *^ = n- 

(4) ABCD is a trapezoid. The elistance of the paral- 
lel sides AB and CD is 3 feet; .4^= 10 feet ; 2X7= 6 feet. 
Let AD and BG produced meet at 0. It is required to 
find the perpendicular distance of from DG. 




A OH B 



Draw Z>6r perpendicular to AB, and i)i/ parallel to 
BG. IhenBH^DG: thus ^i7 = 10-6=4. Also 2X9^ = 3. 
Now the triangles ADH and i>06' are similar. Therefore 
by Art. 35, 

AH : GD :: DG : required distance. 
Thus the required distance — —— = « = 4^. 

4 ^ 



3—2 



36 EXAMPLES. VI. 

EXAMPLES. VI. 

1. In the diagram of Art. 36 HAD — 5 inches, Z>^=4, 
and ^5-7, find ^a 

2. The side of an equilateral triangle is 2 feet 6 inches: 
find the height. 

3. The shadow of a man 6 feet high standing upright 
was measured and found to be 8 feet 6 inches : the shadow 
of a flag-staff, measured at the same time, was found to be 
56 feet 8 inches: determine the height of the flag-staff. 

4. A stick 3 feet long is placed upright on the ground, 
and its shadow is found to be 4 feet 6 inches long : find 
the length of the shadow of a pole which is 45 feet high. 

5. A country is 500 miles l<mg: find the length of a 
map which represents the country on the scale of one-eighih 
of an inch to a mile. 

6. The distance between two towns is 31 miles, and 
the distance between their places on a map is 7| inches : 
find the scale on which the map is drawn. 

7. The distance between two towns is 54 miles, and 
the distance between their places on a map is 6f inches : 
find the distance between two other towns if the distance 
between their places on the map is 8^ inches. 

8. In the diagram of Art. 3G if ^C= 20 inches, DE= 16, 
andi?Z> = 6, find £A. 

9. In the diagram of Art. 36 i^ AD = 8 inches, DE= 7, 
and^i> = 3, find^a 

10. In the diagram of Art. 36 if Z)^= 7 iuches, BC^ 10, 
imdBD = 2, ^ndDA. 

11. The parallel sides of a trapezoid are respectively 
16 and 20 feet, and the perpendicular distance between 
them is 5 feet; the otlier two sides are produced to meet: 
find the perpendicular distance of the point of intersection 
from the longer of the two parallel sides. 

12. The parallel sides of a trapezoid are respectively 
8 feet and 14 feet; two straight lines are drawn across the 
figure parallel to these so that the four are equidistant : 
find the lengths of the straight lines. 



CHORDS OF A CIRCLE. 



87 



VII. CHORDS OF A CIRCLE. 



tre of 




78. Let AB be any chord of a circle, G the ceu 
the circle; suppose CD drawn 
perpendicular to AB, and pro- 
duced to meet the circumfer- 
ence at E. 

Then D is the middle point 
of the chord ADB, and E is 
the middle point of the arc 
AEB. AB is the chord of 
the arc, and AE, or EB, is 
the chord of half the arc. 
DE is the height of the arc. 



79. Produce EC to meet the circumference at F. The 
angle EAF is a right angle, by Art. 33. Hence, by 
Art. 37, the triangles EAF and EDA are similar, so that 
ED is to EA as EA is to EF Therefore, 

EDxEF=EAxEA. 

Also, by Art. 68, 

ED X DF= AD X DB. 

The present Chapter consists of applications of these 
two important results. We shall put the applications in 
the form of Rules for the sake of convenient reference, but 
any person who masters these two results will find it un- 
necessary to commit the Rules to memory. 

80. Hacing given the height of an arc and the cJwrd 
of half the arc, to find the diameter of the circle. 

Rule. Divide the square of the chord of half the arc 
by the height of the arc, and the quotie^it will he the dia- 
meter of tJiC circle. 



88 CHORDS OF A CIRCLE. 

81. Examples. 

(1) The height of an arc is 4 inches, and the chord of 
lialf the arc is 12 inches. 

12 X 12 

— : = 36 : thus the diameter is 3*) inches. 



(2) The height of an arc is 1 foot 4 inches, and the 
chord of half the arc is 4 feet. 

4x4 

— — — = 4x4x| = 12: thus the diameter is 12 feet, 

82. Hamng aiven the chord o^ half (tn ^'^'c ^'^d the 
diameter of the circle, tofnd the height of the arc. 

Rule. Divide the square of the chord of half the arc 
hy the diameter of the circle, and the quotient will he the 
height of the arc 

83. Examples. 

(1) The chord of half an arc is 12 inches, and the dia- 
meter of the circle is 36 inches. 

12 X 12 

— n^-- = 4 : thus the height of the arc is 4 inches. 

(2) The chord of half an arc is 4 feet, and the diameter 
of the circle is 12 feet. 

4x4 4 
.^ = - =13^: thus the height of the arc is 1 J feet. 

84. Hamng given the height of an arc and the dia- 
meter of the circle, to find the chord of hat f the arc. 

Rule. Multiply the diameter of the circle hy the 
height of the arc; the square root of the product will 
he the chord of half the arc. 



CHORDS OF A CIRCLE, 39 

85. Examples. 

(1) The height of an arc 18 4 inches, and tho diameter 
of the circle is 36 inches. 

36x4 = 144; the square root of 144 is 12: thus the 
chord of half the arc i.s 12 inches. 

(2) The height of an arc is 1^ feet, and the diameter 
of the circle is 12 feet. 

4 

^ X 12 = 16 ; the square root of 16 is 4 : thus the chord 

of half the arc is 4 feet. 

86. HaTirig gireii the chord of an arc and the height 
of the arc, to Jin d the diameter of the circle. 

Rule Divide the square of half the chord, hy the 
height,, and the quotient trill he the remaining 'part of 
the diameter ; so that the smn of the quotient and the 
given height trill he the diameter. 

87. Examples. 

(1) The chord of an arc is 8 feet, and the height of the 
arc is 2 feet. 

4x4 

— — =8 ; thus the remaining part of the diameter iv^^ 

8 feet : thereun-e the diameter is 10 feet. 

(2) The chord of an arc is 21 feet, and the height of 
the arc is 4 feet. 

10-5 X 10-5 ri0-2r) ^„ ^^^- J.1 ^1 • • * 

=- = 27'562o : thus the remaining part 

4 4 

of the diameter is 27 '562.5 feet: therefore the diameter i^. 

31 '0025 feet. 

88. By tlie aid of the Rules given in the present 
Chapter, and in Chapter V., we can solve various other 
exercises which may be proposed with respect to the di;v 
gram of Art. 78, as we will now shew. 



40 



CHORDS OF A CIRCLE, 




89 . Having given the height of an arc and the dia- 
meter of the circle, to find the 

chord of the arc. y 

Here we know ED and 
EF ; therefore we know DF\ 
and then we can find AD by 
A.rt. 79. 

90. Examples. 

(1) The height of an arc is 
9 feet, and the diameter of the 
circle is 25 feet. 

Here ED = 9, DF = 16 ; therefore the square of 
y4Z> = 9x 16 = 144: thus ^i> = 12 feet. Therefore ^5 = 24 
feet. 

(2) The height of an arc is 2 feet, and the diameter of 
the circle is 10 feet. 

Here ED = 2, DF = 8 ; therefore the square of 
^Z> = 2x8 = 16: thus AD^^ feet. Therefore AB^8 
feet. 

91. Having given the chord of an arc and the dia- 
meter of the circle, to find the height of the arc. 

Here we know AC, which is half the given diameter, 
and AD which is half the given chord. We first obtain 
CD by Art. 60; and subtracting this from CE we have 
DE. 



92. Examples. 

(1) The chord of an arc is 24 feet, and the diameter of 
the circle is 25 feet. 

Here ^0=12^ feet, and AD = \2 feet. 

1 1 49 

12^+12 = 24|j 12^-12 = -; 24|x-= -; the square 

49 . 7 
root of — is -, that is 3^; and 12| — 3^ = 9: therefore 

DE = d feet. 



CHORDS OF A CIRCLE. 4-1 

(2) The chord of an arc is 8 feet, and the diameter of 
the circle is 10 feet. 

Here AC =6 feet, and AD = 4: feet. 

5 + 4 = 9, 5-4=1; the square root of 9 is 3 ; and 
5-3 = 2: therefore Z>^=2 feet. 

93. Given the chord of an arc and the diameter of 
the circle, to find the chord of half the arc. 

Here we know AC and AD. We first obtain DE, as in 
Art. 91 ; and then we find AE either by Art. 55 or by 
Art. 84. 

94. Examples. 

(1) The chord of an arc is 14 inches, and the diameter 
of the circle is 50 inches. 

Here /i(7=25, AD = 7; thus by Art. 60 we shall ob- 
tain CD = 24. Therefore DE=l. Then by Art. 55, or by 
Art. 84, we find that AE is the square root of 50. Pro- 
ceeding to four decimal places we obtain ^^=7*0710. 
Thus the chord of half the arc is about 7 071 inches. If we 
proceed to seven decimal places we obtain 7' 07 10678. 

(2) The chor*! of an arc is 58 inches, and the diameter 
of the circle is 200 inches. 

Here AC= 100, AD = 29 : thus by Art. 60 we shall find 
that CD is the square root of 9159 : proceeding to four 
decimal places we obtain CD = 95-1027 nearly. Therefore 
we take DE= 4 2973. We may now calculate AE by the 
i*ule of Art. 55 or by that of Art. 84; if AC, AD, and 
DE were all known exactly, the two rules would give 
precisely the same result ; but in the present case DE is 
not known exactly, and we shall find that the two rules 
give results which differ slightly. The rule of Art. 84 is 
the simplest, and by this we find that ^^ is the square 
root of 85946; hence we obtain ^^=2932 very nearly. 
Thus the chord of half the arc is about 29"32 inches. 

95. Given the chord of half an arc and the diameter 
of the circle, to find the chord of the arc. 

Here we know AE and EF. We first obtain ED by 
Art. 82, and then AD by Art. 60. 



4-2 CHORDS OF A CIRCLE. 

96. Example;! 

(1) The chord of hiU;i!i arc is 12 inches, and the dia- 
meter of the circle is 36 inches. 

As in Art. 83 we find that ED = 4:. Then, by Art. 60, 
we find that AD is the square root of 128: lience we ob- 
tain AD = \V2,\4r nearly, and therefore A B = 22' 628 nearly. 
Thus the chord of the arc is nearly 22*628 inches. 

(2) The chord of half an arc is 4 feet, and tlie diameter 
of the circle is 12 feet. 

As in Art. 83 we find that ED ^U. Then bv Art. 60 

16 
we see that AD is the square root of 16-—, that is 

128 1 

of — ; therefore AD is - of the square root of 128 : 

hence proceeding to three decimal places we obtain 
AD = 377l, and therefore ^5 = 7542. Thus the chord 
of the arc is about 7542 feet. 

97. Given the chord of an arc and the chord of half 
the arc^ to find the diameter of the circle. 

Here we know AD and AE We first obtain ED hy 
Art. 60, and then EF by Art. 80. 

98. Examples. 

(1) The chord of an arc is 48 inches, and the chord of 
half the arc is 26 inches. 

Here AD = 24, AE=2G; thus, by Art. 60, we obtain 

ED = 10. Then, by Art. 80, we have j^/'-"^^- --67-6. 

Thus the diameter of the circle is 67"6 inches. 

(2) The chord of an arc is 20 inches, and the chord of 
half the arc is lii'o inches. 

Here AD= 10, AE=10-5; thus, by Art. 60, we find 
that ^Z) is the square root of 1()"2.5: proceeding to four 
decimal places we obtain ED = S'20\5. Then, by Art. 80, 

we have Lt =^ ————— , which we find = 34-437 very 
,-{"20 Id •' 

nearly. Thus the diameter of the circle is verv nearly 
34-437 inches. 



CHORDS OF A CIRCLE. 



48 



99. As an exercise we will calculate the length of the 
side of an equilateral triangle inscribed in a circle, and also 
the length of the side of a regular polygon of twelve sides. 

Describe a circle. If we 
draw in succession chords 
equal to the radius AS, 
BC, CD, ... we shall find 
that exactly six of them can 
be placed in the entire cir- 
cumference. In other words, 
if a regular hexagon be in- 
scribed in a circle, a side of 
the hexagon is exactly equal 
to the radius of the circle. 

Draw the straight lines FB, BD, DF ; thus we form 
an equilateral triangle. 

Suppose the radius of the circle 1 inch : required FB. 
This is an example of Art. 95. Let be the centre of the 
circle ; draw OA cutting BF Sit K. 




We find that A K= - ; thus BK is the square root of 



that is - of the square root of 3, Therefore BF is the 

square root of 3 : proceeding to seven places of decimal-! 
we obtain BF = r7 320508. 

Thus the side of the equilateral triangle inscribed in 
the circle is 17320508 inches. 

Again, let OL be perpendicular to AF, and produce it 
to meet the circumferen ^e again at M. Join A M. Then 
AM is one of the sides of a regular polygo!i of twelve 
sides inscribed in the circle. We can calculate AM by 
Art. 93. 

AL = -., OA = 1 ; thus OL — - of the square rout of 3 

= •8660254: therefore Z3/= -1339746. Then AM is the 
square root of '2079492, which we shall find to bo -51764 
very nea^rly. 

Thus the side of a regular polygon of twelve sides 
inscribed in the circle is "51764 inches very nearly. 



44 EXAMPLES. VII. 



EXAMPLES. VII. 

1. Tlio height of an arc is 15 inches, and the chord of 
half the arc is 4 feet 6 inches : find the diameter of the 
circle. 

2. The height of an arc is 2*28 feet, and the chord of 
half the arc is 7'15 feet : find the diameter of the circle. 

3. The chord of half an arc is 3 feet 4 inches, and the 
diameter of the circle is 25 feet : find the height of the arc. 

4. The chord of half an arc is 6*43 feet, and the diameter 
of the circle is 23*65 feet : find the height of the arc. 

5. The height of an arc is 1 foot 3 inches, and the 
diameter of the circle is 1 1 feet 3 inches : find the chord of 
half the arc. 

6. The height of an arc is 3*24 feet, and the diameter 
of the circle is 2876 feet : find the chord of half the arc. 

7. The chord of an arc is 20 feet, and the height of the 
arc is 4 feet : find the diameter of the circle. 

8. The chord of an arc is 15' 78 feet, and the height of 
the arc is 2*8 feet : find the diameter of the circle. 

9. The chord of an arc is 15 inches, and the diameter 
of the circle is 20 inches : find the chord of half the arc. 

10. The chord of an arc is 80 inches, and the diameter 
of the circle is 100 inches : find the chord of half the arc. 

11. The chord of half an arc is 2 feet 6 inches, and the 
diameter of the circle is 4 feet 2 inches : find the chord of 
the arc. 

12. The chord of half an arc is 2*4 feet, and the diame- 
ter of the circle is 16 feet : find the chord of the arc. 

13. The chord of an arc is 12 yards, and the chord of 
half the arc is 19 feet 6 inches : find the diameter of the 
circle. 

14. The chord of an arc is 49 feet, and the chord of 
half the arc is 25 feet : find the diameter of the circle. 



GIROUMFERENCE OF A CIRCLE. 45 



VIII. CIRCUMFERENCE OF A CIRCLE. 

100. We often require to know the proportion which 
the length of the circumference of a circle bears to the 
length of the diameter : the proportion cannot indeed be 
stated exactly, but it can be stated with sufficient accuracy 
for any practical purpose. 

101. The diametei" of a circle being given, to find the 
circmnference. 

22 . 
Rule. Multijdy the diameter ty 3^, that is by — ; i?i 

other words, midtij^ly the diameter by 22, and divide the 
product by 7. 

102. Examples. 

(1) The diameter of a circle is 4 feet 8 mches. 

4 feet 8 inches = 56 inches, 

66x^ = 8x22 = 176. 

7 

Thus the circumference is about 176 inches, that is, 
about 34 feet 8 inches. 

(2) The diameter of a circle is 4*256 feet. 

4-2 5 6 

22 



85 12 
8512 



7 9 3-6 3 2 



13-3 7 6 
Thus the circumference is about 13'376 feet. 



46 CIRCUMFERENCE OF A CIRCLE. 

103. The Rule of Art. 101 makes the circumference a 
little greater than it ought to be. The circumference of a 
circle is in fact less than 31 § times the diameter, but 
greater than 3|? times. The rule of multiplying the dia- 
meter by 3^ is generally found sufficiently accurate in 
practice. 

104. We may if we please put the Rule of Art. 101 in 
the form of a proportion, and say, as 7 is to 22 so is the 
diameter of ar^y circle to the circumference. 

105. The following proportion is still more accurate : 
as 113 is to 355 so is the diameter of any circle to the 
ciicumferencG. This rule also makes the circumference a 
little greater than it ought to be ; but the error is exces- 
sively small, being at the rate of rather less than a foot in 
nineteen hundred miles. 

106. We may also put the proportion in the following 
form : the diameter of any circle is to the circumference as 
1 is to 3"141592653589793... ; the calculation of this pro- 
portion has been carried to more than 600 places of deci- 
mals. We may use as many as we please of the figures 
which have been obtained : it is very common to take 3*1416 
as a sufficient approximation. 

In modern mathematical books the Greek letter tt is 
generally used to denote the number 314159265.. . 

107. Accordingly the Rule for finding the circumfer- 
ence of a circle when the diameter is given, may be stated 
thus : midtiply the d/tanneter by 3^ ; or^ if greater accu- 
racy is required, multiply the dicnfueter hy 3*1416. 

The latter form of the Rule also makes the circumfer- 
ence a little greater than it ought to be; but the error 

will not be so much as ,--^^^ part of the circumference : 

400000 

so that the error will be at the rate of less than a foot in 
seventy-five miles. 

108. When we are told to multiply the diameter by 
3*1416, we may, if we please, multiply 3*1416 by the diame- 
ter. A similar remark applies to all rules relating to the 
multiplication of numbers. 



CIRCUMFERENCE OF A CIRCLE. 4^7 

109. Exiiinples. 

(1) The diameter of a circle is 427 inches. 

31416 

42-7 

21 9 9 12 
6 2 8 3 2 
125664 

13 414 6 3 2 
Thus the circumference is nearly 134' 14632 inches. 

(2) The diameter of a circle is 8000 miles. 

3-14 16 

8 00 

2 5 13 2-8000 
Thus the circumference is nearly 25132*8 miles. 

110. The beginner should exercise himself in actualh 
measuring the diameter and circumference of some circle, 
as for example, a wheel. Although he may not be able to 
obtain very accurate results, yet he may convince himself 
that the circumference is about 3^ times the diameter. 

111. The clrcurnference of a circle he ing g i cen, to fin d 
the diameter. 

22 
E,ULE. Divide the circumference hy 'i\, that is hy -- ; 

in other words, m,idtiply the circumference by 7, and 
divide the product by 22. Or, if greater accuracy i^ 
required, divide the circumference by 3'1416. 

112. Examples. 

(1) The circumference of a circle is 50 feet. 

50 . 

7 



2 350 

11 I 1 7 5 
15-9 
Thus tlie diameter is about 15'9 feet 



48 CIRCUMFERENCE OF A CIRCLE. 

(2) The circumference of a circle is 360 feet. 

3-1 4 1 6 ; 3 6 0-0 U 1 4-5 9 
81416 

4 5 8 4 
3 1416 
14 4 2 4 
125664 



185760 
157080 

286800 
282744 



4056 
Thus the diameter is about 114"59 feet. 

113. We will now solve some exercises which depend 
on the Rules already given. 

(1) Find the diameter of a carriage wheel which is 
turned round 1000 times in travelling a mile. 

Here 1000 times the circumference of the wheel is 

equal to 1760 yards ; thus the circumference is 1*76 yards. 

7 
Then, by Art. Ill, the diameter is — x 1'76 yards, that is, 

7 X -08 yards, that is, "56 of a yard. 

(2) Suppose that the distance of the earth from the 
sun is about 95000000 miles, and that the earth describes 
a circle round the sun in 365j days : find the number of 
miles described by the earth in one minute. 

The circumference of the circle described by the earth 
is about 2 x 95000000 x 3*1416 miles, that is, about 596904000 
miles. In 365^ days there are 525960 minutes. Divide 
the number of miles by the number of minutes; thus we 
obtain very nearly 1135 miles. 



EXAMPLES. VIII. 49 

EXAMPLES. YIII. 

Assuming- that tlie circumference of a circle is 3f times 
the diameter, find the circumferences of the circles with 
the following diameters : 

1. 14 feet. 2. 86 yards 1 foot. 

3. 213 yards 2 feet 8 inches. 

4. 1 furlong 60 yards. 

Assuming that the circumference of a circle is 3*1416 
times the diameter, find the circumferences of the circles 
with the following diameters : 

5. 27 feet. 6. 61 yards 2 feet. 

7. 55,5 yards 1 foot 6 inches. 

8. 1 furlong 80 yards. 

Assuming that the circumference of a circle is 3^ times 
the diameter, find the diameters of the circles with the fol- 
lowing circumferences : 

9. 66 yards. 10. 10 chains. 
11. 3 furlongs 4 chains. 12. 1 mile. 

Assuming that the circumference of a circle is 3' 1416 
times the diameter, find the diameters of the circles with 
the following circumferences : 

13. 1 foot. 14. 25 feet. 

15. 108 yards 1 foot. 16. 1 furlong. 

17. Suppose that the planet Mercury describes in 
88 days a circle round the Sun, of which the radius is 
37000000 miles, find the number of miles described by the 
planet in one second. 

18. The diameter of a carriage wheel is 28 inches : 
find how many turns the wheel makes in travelling half 
a mile. 

19. A road runs round a circular shrubbery ; the outer 
circumference is 600 feet, and the inner circumference is 
480 feet : find the breadth of the road. 

20. The difference between the diameter and the cir- 
cumference of a circle is 10 feet : find the diameter. 

T. M. 4 



50 ARC OF A CIRCLE. 



IX. ARC OF A CIRCLE. 

114. Let C be the centre of a circle, AB any arc of the 
circle, AD a quarter of the 

circumference. The length of 
AB h to the length of AD in 
the same proportion as the 
angle ACB is to the angle 
ACD, that is, as the angle 
ACB is to a right angle. 
Therefore the length of AB 
is to the circumference of the 
circle in the same proportion 
as the angle ACB is to four 
right angles. 

115. Angles are usually expressed in degrees, 90 of 
which make a right angle ; and consequently in four right 
angles there are 360 degrees. A degree is subdivided into 
60 minutes, and a minute into 60 seconds. 




116. Symbols are used as abbreviations of the words 
degrees, mirrutes, and seconds. Thus 6^ 23' 47" is used to 
denote 6 degrees, 23 minutes, 47 seconds. 

117. The number of degrees in the angle subtended 
by cin arc of a circle at the centre beiiig given, to find the 
length of the arc. 

Rule. As 360 is to the number of degrees in the 
angle, so is the circumference of the circle to the length of 
the arc. 

118. Examples. 

(1) The circumference of a circle is 48 inches, and the 
angle subtended by the arc at the centre is 54 degrees. 

360 : 54 :: 48 : the required length, 

54 X 48 54 X 4 18x4 



360 30 10 

Thus the length of the arc is 7 "2 inches 



= 7-2. 



ARC OF A CIRCLE. 51 

(2) The circumference of a circle is 25000 miles, and 
the angle subtended by the arc at the centre is one degree. 

360 : 1 :: 25000 : the required length, 

3 6^)2 50 (6 9-4 

216 

340 
324 



160 
144 

16 

Thus the length of the arc is about 69 "4 miles. 

119. The length of an arc of a chicle being given, to 
find the number of degrees m the angle subtended by the 
arc at the centre of the circle. 

Rule. As the circumference of the circle is to the 
length of the arc, so is 360 to the number of degrees in the 
angle. 

120. Examples. 

(1) The circumference of a circle is 50 feet, and the arc 
is 8 feet. 

50 : 8 :: 360 : the required number of degrees, 

8 X 360 288 ,„„ 

= = 57#. 

50 5 ^ 

Thus the angle is 57 f degrees. 

(2) The circumference of a circle is 25000 miles, and 
the arc is 750 miles. 

25000 : 750 :: 360 : the required number of degrees, 

750x360 75x36 3x36 



25000 250 10 

Thus the angl« is 10"8 degrees. 



= 10-8. 



4—3 



52 ARC OF A CIRCLE. 

121. The chord of mi arc heing knoicn, and also the 
chord of half the arc, to find tJie length of the arc. 

Rule. From eight times the chord of half the arc suh- 
tract the chord of the whole arc, and divide th,e remainder 
by three. 

This Rule is not exact ; it gives the length of the arc 
smaller than it ought to be. If the arc subtend at the 
centre of the circle an angle of 45 degrees, the error is 

about — — — of the length of the arc : the error increases 

20000 

rapidly as the angle increases, and diminishes rapidly as 
the angle diminishes. 

122. Examples. 

(1) The chord of an arc is 14 inches, and the radius 
of the circle is 25 inches. 

Bv Art. 94 the chord of half the arc is about 7"0710678 



inches. 



7-0710678 

8 

5 65 68 5424 
14 



3 4 2-5 6 8 542 4 



14-189 5 141 
Thus we obtain for the length of the arc 14-1895141 inches. 

(2) The chord of an arc is 58 inches, and the radius of 
the circle is 100 inches. 

By Art. 94 the chord of half the arc is about 29 32 
inches. 

2 9-3 2 
8 

2 3 4-5 6 

58 



3 17 6-5 6 



5 8-8 5 
Thus we obtain for the length of the arc 58'85 inches. 



ARC OF A CIRCLE. 53 

123. The error which arises from the use of the Rule 
in Art. 121 is, as we have said, much less for a small arc 
than for a large arc. It may therefore be expedioit in 
some cases to calculate by the Rule the length of half ihe 
arc, and to double this result instead of calculating the 
length of the whole arc immediately. We should proceed 
thus : from eight times the chord of one fourth of the arc 
subtract the chord of half the arc ; midtiply the remain- 
der hy two, and divide the product by three. 

124. The following Rule for finding the length of an 
arc is much more accurate than that in Art. 121, and may 
be used when a very close approximation is required : to 
256 times the chord of one fourth of the arc add the chord 
of the. arc ; subtract 40 times the chord of half the arc, 
and divide the remainder by 45. 

This Rule gives the length of the arc a little larger 
than it ought to be. If the arc subtend at the centre of 
the circle an angle of 45 dtgrees the error is less than 

of the arc : the error increases rapidly as the 

angle increases, and diminishes rapidly as the angle di- 
minishes. 

125. We will now solve some exercises. 

(1) The radius of a circle is 1 foot : find the perimeter, 
that is the length of the whole boundary, of a sector of 60 
degrees. 

Since the radius is 1 the circumference of the circle is 
2 X 3-1416, that is, 6 2S32 ; then 

360 : 60 :: 6-2832 : the length of the arc. 

Thus the length of the arc is 1-0472. Add the length of 
the two radii, that is 2 ; thus the whole perimeter is 3-0472 
feet. 

(2) The perimeter of a sector of 60 degrees is 20 feet : 
find the radius. 

Use the result of the j>receding exercise. Thus we have 

the proportion 

3-0472 : 20 :: 1 : the required radius. 

20 
Hence the required radius = .j-—— feet = 65634 feet 

very nearly. 



54. EXAMPLES. IX. 



EXAMPLES. IX. 

1. The radius of a circle is 10 inches, and the angle 
subtended by an arc at the centre is 72" : find the length of 
the arc. 

2. The radius of a circle is 19 feet 7 inches, and the 
angle subtended by an arc at the centre is 10° 24' : find the 
length of the arc. 

3. The radius of a circle is 2 feet, and the length of an 
arc is 15 inches : find the angle subtended at the centre by 
the arc. 

4. The radius of a circle is 1 foot, and the length of an 
arc is equal to the radius : find the angle subtended at the 
centre by the arc. 

5. The chord of an arc is 36 inches, and the chord of 
half the arc is 19 inches : find the arc. 

6. The chord of an arc is 56 inches, and the radius of 
the circle is 197 inches ; find the arc. 

7. The chord of an arc is 6 inches, and the radius of 
the circle is 9 inches : find the arc. 

8. The radius of a circle is 5 inches : find the perime- 
ter of a sector, the angle of which is 90". 

9. The radius of a circle is 16 inches : find the perime- 
ter of a segment, the arc of which subtends an angle of 90" 
at the centre of the circle. 

10. The radius of a circle is 1 inch : find the perimeter 
of a semicircle. 

11. The perimeter of a semicircle is 100 feet : find the 
radius. 

12. The radius of a circle is 25 inches, and the angle 
subtended by an arc at the centre is 32° 31' 12"*4 : find the 
length of the arc. 



55 



THIRD SECTION, AREAS, 



X. TABLE OF SQUARE MEASURE. 



126. It will be convenient to place here the Table of 
Measures of Area, which is usually called the Table of 
Square Measure. 

144 square inches make 1 square foot. 
9 square feet make 1 square yard. 
36 square feet make 1 square fathom. 
272 J square feet or 30 j square yards make 1 square 

rod or pole. 
ItOO square poles make 1 square furlong. 
64 square fm-longs make 1 square mile. 

Hence we obtain the following results : 



Square 


Square 


Square 


Square 


Square Square 


Inches. 


Feet. 


Yards. 


Kods. 


Furlongs. Mile. 


144 


1 








1296 


9 


1 






39204 


272i 


30^ 


1 




62726400 


435600 


48400 


1600 


1 



4014489600 27878400 3097600 102400 



64 



127. The following terms are also used in expressing 
areas : a square link, a square chain, a rood, and an acre. 

A square chain contains 22 x 22, that is, 484 square 
yards. A rood is 40 poles, that is, 1210 square yards. An 
acre is 4 roods, that is, 4840 square yards : thus an acre is 
equal to 10 square chains. 

A square chain contains 100 x 100, that is, 10000 square 
links ; so that an acre is equal to 100000 square links. 



56 



RECTANGLE. 



XL llECTANGLE. 



128. Suppose we have a rectaugle which is 4 mches 
lon<^ and 3 inches broad. 
Draw straight hues, an inch 
apart, parallel to the sides. 
The rectangle is thus divided 
into 12 equal figures, each of 
which is a square being an 
inch long and an inch broad : 
such a square is called a 
square inch. The rectangle 
then contains 12 square 

inches ; this fact is also expressed thus : the area of the 
rectangle is 12 square inches. 

The number 12 is the product of the numbers 4 and 3, 
which denote respectively the length and the breadth of 
the rectangle. 






129. If a rectangle be 8 inclies long and 5 inches broad, 
we can shew in the same manner that its area is 8 times 
5 square inches, that is 40 square inches. Similarly, ii" a 
rectangle be 9 inches long and 7 inches broad, its area 
is 9 times 7 square inches, that is, 63 square inches. And 
so on. 



130. In the same manner, if a rectangle be 4 feet loJig 
and 3 feet broad, its area is 12 square feet ; that is, the 
rectangle might be divided into 12 equal figures, each 
being a foot long and a foot broad. If a rectangle be 
4 yards long and 3 yards broad, its area is 12 square yards. 
And so on. 



131. The beginner should obscrvo very carefully the 
way in which areas are measured; it is a case of the 
general principle which applies to all measurable things. 
For example, when we meiisure lengtlis v/e fix on some 
length for a standard, as an inch or a foot, and we compare 
other lengths with the standard ; thus when we say that a 



RECTANGLE. 57 

certain line is 17 inches lon^, we mean that the line is 
17 times as long as our standard, which is one inch. In 
like manner when we measure areas we fix on some area 
for a standard, and we compare other areas witli the stand- 
ard. The most convenient standard is found to be the area 
of a square ; it may be a square inch, or a square foot, or 
any other square. 

132. In order then to find the area of a rectangle we 
must express the length and the breadth in terms of the 
same denomination; and then the product of the nu.mbers 
which denote the length and the breadth will denote the 
area. If the length and the breadth are both expressed in 
inches, the area will be expressed in square inches ; if the 
length and the breadth are both expressed in feet, the 
area will be expressed in square feet ; and so on. 

1.S3. The student will now be able to understand the 
way in which we estimate the areas of figures, and to use 
currectly the rules which will be given ; the rules will be 
stated with brevity, but this will present no difficulty to 
those who have read the foregoing explanations. 

134. Tujiiicl the area of a rectangle. 

lliJLE. Multiply the length hy the hreadth, and the 
■product will he the area. 

Sometimes the words hase and height are used respect- 
ively for the length and breadth of a rectangle. 

135. Examples. 

(1) The length of a rectangle is 3 feet 4 inches, and its 
breadth is 2 feet 6 inches. 

3 feet 4 inches = 40 inches, 2 feet 6 inches = 30 inches; 
40x30=1200. 

Thus the area is 1200 square inches. 

Or thus : 3 feet 4 inches = 3^ feet, 2 feet 6 inches = 2^ ft. : 
, ^i 10 5 25 ^, 

Tt.iis the area is S}^ square feet 



58 RECTANGLE, 

(2) The length of a rectangle is half a mile, and its 
breadth is 220 yards. 

Half a mile = 880 yards ; 880 x 220 = 193600. 

Thus the area is 193600 square yards. 

Or thus : 220 yards = - of a mile, ^ x - = — . 

Thus the area is q-x of a square mile. 
Id 

136. If we know the area of a rectangle, and also its 
length, we can find the breadth by dividing the number 
which expresses the area by the number which expresses 
the length : and similarly if we know the area and the 
breadth we can find the length. Of course we must take 
care to use corresponding denominations for the area and 
the known length or breadth : see Art. 132. 

137. Examples. 

(1) The area of a rectangle is 96 square inches, and its 

length is 1 foot 4 inches. 

96 
1 foot 4 inches = 16 inches j 77; = 6. 

lo 

Thus the breadth is 6 inches. 

(2) The area of a rectangle is 10 square feet, and its 
breadth is 1 yard. 

1 yard = 3 feet; ^ = H. 

Thus the length is 3^^ feet, that is 3 feet 4 inches. 

138. A square is a rectangle having its length and 
breadth equal ; hence to find the area of a square we mul- 
tiply the number which denotes the length of a side of the 
square by itself For example, if the length of the side of 
a square be 7 inches, the area of the square is 7 times 
7 square inches, that is, 49 square inches. Thus we see 
thereaso.u for using the term the square of a number to 
denote the product of the number into itself; and we un- 
derstand the connexion of the Rules in Chapter V. with 
the theorem of Art. 30. 



RECTANGLE. 69 

139. The statements made in the Table given in Chap- 
ter X. will be easily understood and remembered by the 
aid of the explanations in the present Chapter. Take, for 
example, the first statement that 144 square inches make 
1 square foot : a square foot is a rectangle 12 inches long 
and 12 inches broad; and therefore by the method of 
Art. 128, we see that a square foot contains 12x12 square 
mches, that is 144 square inches. 

140. If we know the area of a square we can find the 
length of a side of the square by extracting the square 
root of the number wlikh denotes the area. For example, 
suppose the area of a square to be 121 square inches; the 
square root of 121 is 11 : thus the length of a side of the 
square is 1 1 inches. Again, suppose the area of a square 
to be 150 square inches. Here the square root cannot be 
exactly found, and so the length of the side cannot be 
determined accurately: if we proceed to three decimal 
places we obtain 12*247 inches for the required length. 

141. The student must distinguish carefully between 
square feet and feet square. For example; by three 
square feet we mean an area which can be divided into 
three others each of which is a square foot ; by three, feet 
square is meant a square the side of which is three feet 
long, so that the figure contains nine square feet. Simi- 
larly by four feet square is meant a square the side of 
which is four feet long, so that the figure contains sixteen 
square feet. 

142. We will now solve some exercises which depend 
on the Rules of the present Chapter. 

(1) A room is 18 feet 6 inches long and 11 feet 3 inches 
broad : find the expense of carpeting the room, supposing 
the carpet to be 30 inches wide, and to cost 6 shillings per 
yard. 

We first find the length of carpet required. The 

length of the room is 18^ feet, and the breadth is llj feet ; 

87 45 
hence the area of the room is -- x -^ square feet, that is 

2 4 

square feet. The breadth of the carpet is 2j feet. 

8 



60 RECTANGLE. 

Hence, by Art. 186, the required length of carpet is ob- 

1665 1665 2 

tained by dividing — ^ — by 2^ ; so that it is —^ x - feet, 

333 

that is — - feet. We have now a simple example in 

Arithmetic : if one yard costs 6 shillings, iind the cost 

o o »") 1 o •^ o 

of — ^ feet. The result is - x ^ x 6 shillings, that is 
4 3 4 

333 

— shillings, that is £d>. Qs. 6d. 

It is scarcely necessary to remark that in actual ex- 
pedience rather more carpet would be required than our 
solution indicates, in order to allow for the waste which 
arises from arranging the pieces mtli due regard to the 
correspondence of the pattern. 

(2) A room, is 18 feet 6 inches long, 11 feet 3 inches 
broad, and 10 feet high : find the entire area of the four 
walls, 

37 
There are Urn Vv'alls each containing -^xlO square 

45 
feet, and two others each containing -7- x 10 square feet : 

therefore the entire area is equivalent to that of a rect- 

45 
angle the height of which is 10 feet, and the base 37 + — 

liy 

feet; thus the entire area is -~ x 10 square feet, that is 
595 square feet. 

(3) A rectangular gnibs plot is 160 feet long and 100 
feet broad; a gravel walk 4 feet v.ide surrounds the grass 
plot : firid the area of the walk. 

The extreme length of the rectangle including the walk 
is 168 feet, aud the extreme breadth is 108 feet; therefore 
the area of this rectangle is 168 x 108 square feet, that is 
l!5i44 square feet. The area of the grass plot is 160 x 100 
square feet, that is 16000 square feet. Tlie area of the 



RECTANGLE. 



61 



F 



K 



G 



D 



H 



walk is therefore 18144 — 16000 square feet, that is 2144 
square feet. 

(4) A rectangle is divided into four rectangles by two 
straight lines drawn parallel to the sides at given distances 
from them : find the areas of the four rectangles. 

Let ABDG be the 

rectangle ; suppose, for ^ E B 

example, that AB is 
16 inches, and AC \9> 
9 inches ; and that AE 
is 10 inches and ^i^ is 
7 inches. Through-E'iet 
EKG be drawn parallel 
to A C; and through F 
let FKIIhe drawn pa- 
rallel to AB. 

Then EB=Q inches, and FC= 2 inches. 
The area of AEKF= 10x7, that is 70, square inches. 
The area of EBHK^Q x 7, that is 42, square inches. 
The area of FKGC- 10 x 2, that is 20, square inches. 
The area of KHGD = Q x 2, that is 12, square inches. 

The sum of these four areas is 144: square inches, and 
is equal, as it should he, to 16x9 square inches. 

This exercise is very simple, but very instructive ; it 
affords a visible representation of an important arithmeti- 
cal proposition, which in the present case stands thus: the 
product of the sum of 10 and 6 into the sum of 7 and 2 is 
equal to the sum of 10 x 7, 6 x 7, 10 x 2, and 6x2. 

5 3 

(5) A rectangle is '- of an inch long, and - of an inch 

b t 

wide : find the area. 

5 3 
By the Rule of Art. 134 the area is - x ^ of a square 

8 / 



15 
inch, that is ^-: of a square inch. 
o6 



But in the demonstration 



of the Rule in Art. 128, we supposed that the length and 
the breadth were expressed in ichole numbers, so that it is 
expedient to shew explicitly that the Rule will also hold 
when fractions occur ; this we shall now do. 



62 EXAMPLES. XI. 

Reduce the fractions to a common denominator, thus 

they become '— and — of an inch. Now let us take —z of 
Do of) 5d 

an inch as the unit of lengtli ; then tlie length of the rect- 
angle is 35, and the breadth is 24 of these units: and there- 
fore the area of the rectangle is 35 x 24 square units. And 
a square inch contains 56 x 56 of these square units. Thus 

the area of the rectangle is '-^ — — of a square inch ; that 

56 X 56 

5 X .3 15 

is - — 1_ of a square inch : that is — of a square inch. 



EXAMPLES. XI. 

Find the area in square yards of squares having the 
following lengths of sides : 

1. 14 yards. 2. 24 yards. 3. 27| yards. 

4. SOj yards. 

Find tlie area in square yards and square feet of squares 
having the following lengths of sides : 

5. 10 yards 2 feet. 6. 12 yards 1 foot. 

7. 18 yards 2 feet. 8. 20 yards 1 foot. 

Find the area in square yards, feet, and inches of 
squares having the following lengths of sides : 

9. 3 yards 2 feet 4 inches. 10. 5 yards 2 feet 8 inches. 

11.8 yards 1 foot 9 inches. 12. 14 yards 1 foot 10 inches. 

Find the area in acres, roods, and poles of squares 
having the following lengths of sides : 

13. 4 chains 50 links. 14, 7 chains 25 links. 

15. 12 chains 45 links. 16. 26 chains 56 links. 

Find the areas of squares having the following dia- 
gonals: 

17. 255 feet. 18. 88 yards 2 feet 3 inches. 

19. 12 chains 25 links. 20. 18 chains 36 links. 



EXAMPLES. XL 63 

Find the sides of squares having the following areas : 
21. 1764 square yards. 22. 7225 square yards, 

23. 74529 square yards. 24. yV square mile. 

25. 160 acres. 26. 2^ acres. 

27. 6406401 6 square feet. 

28. 3 acres 1 rood 13 poles 5| sq. yards. 

Find in feet to three decimal places the sides of squares 
having the following areas : 

2-9. 120 square feet. 30. 287 square feet. 

31. 478 square yards 1 square foot. 

32. 526 square yards 2 square feet 90 square inches. 

33. 150 acres. 34. 2| acres. 

35. Find the diagonal of a square whose area is 7 square 
inches. 

36. The area of a chess board having 8 squares along 
each side is 100 square inches : find the length of the side 
of one of its squares. 

Find the area in square feet of rectangles having the 
following dimensions in feet : 

37. 14 by 20. 38. 24 by 18. 39. 15^ by 18. 

40. 18i by 20i 

Find the area in square yards and feet of rectangles 
having the following dimensions : 

41. 5 yards 2 feet by 6 yards. 

42. 7 yards 1 foot by 8 yards 2 feet. 

43. 10 yards 1 foot by 12 yards 1 foot. 

44. 9 yards 2 feet by 18 yards 2 feet. 

Find the area in square yards, feet, and inches, of rect- 
angles having the following dimensions : 

45. 2 yards 1 foot by 3 yards 1 foot 3 inches. 

46. 3 yards 1 foot 4 inches by 4 yards 2 feet. 

47. 4 yards 2 feet 8 inches by 4 yards 2 feet 10 inches. 

48. 6 yards 1 foot 9 inches by 8 yards 2 feet 1 1 inches. 



64 EXAMPLES. XL 

Find the area in acres, roods, and poles of rectangles 
having the following dimensions; 

49. 5 chains 14 links by 6 chains 25 links. 

50. 7 chains 4 links by 8 chains 12 links. 

51. 9 chains 24 links by 10 chains 36 links. 

52. 10 chains 80 links by 12 chains 40 links. 

Find the breadth in the following rectangles, having 
given the area and the length : 

53. Area 1056 square feet, length 11 yards. 

54. Area an acre, length 110 yards. 

55. Area a square mile, length 5 miles. 

56. Area 1000 acres, length 2^ miles. 

57. Area 2| acres, length 115| yards. 

58. Area 5j acres, length 32 chains. 

59. Area 7 acres 1 rood 15 poles, length 453 yards 
2 feet 3 inches. 

60. A plank is IS inches broad: fir.d-what length must 
be cut off that the area may be a square yard. 

61. A rectangle is 9 inches by 18 inches: find what 
decimal it is of a square Nard. 

62. Express as a fraction of an acre the rectangle 
which is 121 yards long and 25 yards broad. 

63. A street is a quarter of a mile long: find the num- 
ber of square yards in a pavement 4| feet wide down one 
hide of the street. 

64. A rectangular garden is to be cut from a rectan- 
gidar field, so as to contain three quarters of an acre; one 
side of the field is taken for one side of the plot, and mea- 
sm-es 2^ chains : find the length of the other side. 

65. The diagonal of a rectangle is 458 feet, and one 
side is 442 feet: find the area. 

66. The sides of four squares being respectively 1, 2, 4, 
and 10 feet ; find the side of the square which is equal to 
the sum of the four. 



EXAMPLES. XL 65 

67. The sides of three squares bemg 6, 6, and 7 feet : 
find the side of the square which is equal to the sum of the 
three. 

68. The window of a house is 8 feet 2 inches by 5 feet 
3 inches : find the number of panes of glass in it, each mea- 
suring 14 inches by 9. 

69. A lawn measures 150 feet by 120 feet: find how 
many pieces of turf are required to cover it, each piece 
being 3 feet 4 inches by 1 foot 3 inches. 

70. Find how many slates measuring 16 inches by 12 
inches will be required to cover a roof which measures 
24 feet by 18 feet. 

71. Find how many bricks measuring 9 inches by 4^ 
inches will be required to cover a space of 18 feet by 
12 feet 9 inches. 

72. Find how many planks 12 feet long by 10 inches 
wide will be required to floor a room wliich is 24 feet by 
20 feet. 

73. Find how many planks 12 feet 6 inches long by 
9^ inches wide will be required to floor a room which 
is 50 feet by 16 feet. 

74. Find how many persons can stand in a room mea- 
suring 15 feet by 9 feet ; supposing each person to require 
a space of 27 inches by 18 inches. 

75. A procession is formed of 504 ranks of men, 14 in 
a rank : if the men were arranged in a solid square, find 
how many there would be in a side. 

76. If one stalk of wheat will grow on nine square 
inches of ground, find how many stalks will grow on an 
acre. 

77. Find how many trees there are in a wood half a 
mile long and a quarter of a mile wide, supposing on an 
average four trees grow on each square chain. 

78. A country in the form of a rectangle 600 miles 
long by 200 miles broad supports a population of 20,000,000 : 
find the average number of acres required to support on© 
person. 

T. M. 5 



66 EXAMPLES. XI. 

79. A room is 25 feet by 18 feet ; in the central part is 
a Turkey carpet which measures 21 feet by 15 feet: find 
how many yards of oilcloth 27 inches wide will be required 
to cover the rest of the floor. 

80. The side of a square is 85 yards, and a path 10 
yards wide goes round the square outside it : find how 
many stones 1 foot 4 inches long by 10 inches wide will be 
required to pave the path. 

81. A rectangular court measures 63 feet by 36 feet; 
a path 4 feet 6 inches wide goes round the court outside it : 
find how many bricks measuring 9 inches by 4^ inches will 
be required to pave the path. 

82. It is found that 1296 bricks, each measuring 9 
inches by 4^ inches, have been employed in paving a certain 
court yard: find how many tiles 6 inches square will be 
required for a pavement one-ninth of the size. 

83. If the adjacent sides of one rectangle be 9 and 16, 
and of another 36 and 25, compare the sides of the squares 
respectively equal to these rectangles. 

84. Find what length of wall paper 27 inches wide will 
be required for a room 18 feet long, 12 feet broad, and 
10 feet 6 inches high. 

85. Find how many square feet of paper will cover the 
walls of a room which is 24 feet 10 inches long, 16 feet 
broad, and 18 feet 6 inches high. 

86. A rectangle measures 48 feet by 28 feet : find the 
area of a square which has the same perimeter as the 
rectangle. 

87. A rectangle contains 1323 square feet; and it is 
three times as long as it is broad : find its sides. 

88. Seven slieets of note paper together weigh one 
ounce; each sheet measures 9 inches by Q'^ inches: find 
the v/eight of a sheet of the same kind of paper which mea- 
sures 18^ inches by 11. 

89. Shew by examples that if a square and a rectangle 
have equal perimeters the area of the square is greater 
than that of the rectangle. 



EXAMPLES. XI. 67 

90. Shew by examples that if a square and a rectangle 
have equal perimeters, the area of the square exceeds the 
area of the rectangle by the area of a square the side of 
which is half the difference of the sides of the rectangle. 

91. Find the rent at £\. 135. per acre of a rectangular 
field of which the length is 1 furlong 20 poles, and the 
breadth 10 poles 1 yard. 

92. Find the rent at ^4. 10.9. an acre of a piece of land 
4235 yards long and 280 yards wide. 

93. A rectangular court measures 18 feet 6 inches by 
12 feet 3 inches : find the expense of paving it at 4 pence 
the square foot. 

94. The diagonal of a square court yard is 30 yards : 
find the cost of gravelling the court at the cost of a shilling 
for nine square yards. 

95. Find the expense of paving an area which mea- 
sures 32 feet 3 inches by 16 feet 6 inches at 65. 4c?. per 
square yard. 

96. The length of a street is 1 furlong 92 yards 1 foot 
6 inches, and its breadth is 22 yards 8 inches : find the cost 
of paving it at ^^d. per .square yard. 

97. In a rectangular court which measures 96 feet by 
84 feet there are four rectangular grass plots measuring 
each 22^ feet by 18 feet : find the cost of paving the 
remaining part of the court at %^d. per square yard. 

98. Find the expense of paving a road of a uniform 
breadth of 4 yards round the inside of a rectangular piece 
of ground the length of which is 85 yards and breadth 
56 yards, the cost of paving a square yard being Is. 2d. 

99. Find the side of a square court yard the ex- 
pense of paving which is J38. IO5. 6d. at 3^. dd. per square 
yard. 

100. Determine the side of a square garden that cost 
£67. 165. 2|c?. for trenching at 2|d per square yard. 

5—2 



68 EXAMPLES, XL 

101. The rent of a square field at £2. \As. 6d. per 
acre amounts to £27. 5s. : find the cost of putting a paling 
round the field at 9d. per yard. 

Find how many yards of carpet will be required for 
rooms of the following dimensions : 

102. 18 feet by 16 feet ; the carpet being 1 yara wide. 

103. 24 feet by 16 feet 6 inches; the carpet being 
1 yard wide. 

104. 21 feet by 15 feet; the carpet being 27 inches 
wide. 

105. 17 feet 3 inches by 9 feet 9 inches; the carpet 
being 27 inches wide. 

106. 28 feet by 23 feet 9 inches ; the carpet being 30 
inches wide. 

107. 27 feet 3 inches by 22 feet 6 inches ; the carpet 
being 30 inches wide. 

Find the expense of carpeting rooms, the dimensions 
and the cost of the carpet being the following : 

1U8. 12 feet 4 inches by 16 feet 3 inches; Is. 6d. per 

square foot. 

109. 24 feet 8 inches by 16 feet 3 inches; 13*. 6d. per 
square yard. 

110. 23 feet 9 inches by 16 feet 3 inches; 25. 9d. per 
square yard. 

Find the expense of carpeting rooms, the dimensions 
of the room and the width and the cost of the carpet being 
the following : 

111. 34 feet by 18 feet 6 inches ; carpet 2 feet wide at 
4s. 6d. per yard. 

112. 18 feet 9 inches by 17 feet 6 inches ; carpet 2 feet 
wide at 45. 9d. per yard. 

113. 15 feet 9 inches by 12 feet 5 inches; carpet 1 yard 
18 inches wide at 6 shillings per yard. 

114. 18 feet 6 inches by 12 feet 6 inches; carpet 27 
inches wide at 3 shillings per yard. 



EXAMPLES. XL 69 

115. 15 feet 9 inches by 12 feet 5 inches ; carpet 27 
inches wide at 4 shillings per yard. 

116. 21 feet 8 inches by 16 feet 6 inches; carpet 27 
inches wide at 3^. 4:\d. per yani. 

117. 17 feet 6 inches by 17 feet 6 inches; carpet 
2 feet 4 inches wide at Zs. 9d. per yard. 

118. Supposing the cost of a carpet in a room 25 feet 
long at 5s. per square yard to be ^6. 5^. : find the breadth 
of the room. 

119. Find the quantity of carpeting required for the 
central portion of a room, this portion being 13 feet 6 
inches wide and 18 feet 9 inches long. Find also tlie 
expense, the carpet being 27 inches wide, and 4*. Gd. per 
yard. If between the edge of the carpet and the walls 
there is a distance all round of 2| feet, find how much of 
the area of tlie iloor will remain uncovered. 

120. Find how many yards of paper will be required 
for the walls of a room which is 23 feet long, IS feet wide, 
and 12 feet high ; the paper being a yard wide. 

121. Find how many yards of paper will be required 
for the walls of a room wliich is 24 feet long, 19 feet 
6 inches wide, and 14 feet high; the paper being three 
quarters of a yard wide. 

122. A room is 34 feet long, 18| feet wide, and 12 feet 
high : find the expense of papering the walls at Is. 6d. per 
square yard. 

123. Find the expense of papering a room 6 yards 
1 foot 1 inch long, 6 yards feet 4 inches broad, 12 feet 
high, with paper a foot wide at 9d. per yard. 

124. A room is 24 feet long, 15 feet broad, and 11 feet 
high : find the expense of painting the walls at 3d. per 
square foot ; allowing for a fire place which is 4 feet 6 
inches by 3 feet, a door which is 7 feet by 4 feet, and two 
windows each 6 feet 6 inches by 5 feet. 



70 PARALLELOGRAM. 



XII. PARALLELOGRAM. 



143. We have shewn in Art. 28 that a parallelogram is 
equivalent to a rectangle having the same base and height: 
this is the reason of the rule now to be given. 

144. To find the area of a parallelogram. 

Rule. Multiply the base hy the height and the pro- 
duct will le tJie area. 



145. Examples : 

(1) The base of a parallelogram is 5 feet, and its height 
is 3 feet. 

5x3 = 15. Thus the area is 15 square feet. 

(2) The base of a parallelogram is 3 feet 9 inches, and 
its height is 2 feet 3 inches. 

3 feet 9 inches =45 inches, 2 feet 3 inches = 27 inches. 
45 X 27 = 1215. Thus the area is 1215 square inches. 

Or thus : 3 feet 9 inches = 3| feet, 2 feet 3 inches = 2^ feet. 

15 9 135 
3| X 2| = — X - = ~ = S/e . Thus the area is 8/g^ square 

feet. 



146. If we know the area of a parallelogram, and also 
one of the two dimensions, the base or the height, we can 
tind tlie other : see Art. 136. 



PARALLELOGRAM. 71 

147. We will now solve some exercises. 

(1) Tlie area of a rhombus is 180 square feet, and each 
side is 15 feet long : find the height. 

— = 12. Thus the height is 1 2 feet. 
15 

(2) Two adjacent sides of a parallelogram are 8 feet 
and 16 feet respectively ; the area is two-thirds tliat of a 
square which has the same perimeter : find the height of 
the parallelogram. 

The perimeter of the parallelogram is 16 + 32 feet, that 
is 48 feet; hence the side of a square having the same 
perimeter is 12 feet, and therefore the area of the square 
is 144 square feet. Thus the area of the paralielogiam is 
2 
- of 144 square feet, that is 96 square feet. If we take 

the side which is 8 feet hmg for base, the height is 

96 

— -, that is 12 feet. If we take the side which is 16 feet 
o 

96 
long for base, the height is — , that is 6 feet. 

(3) Each side of a rhombus is 18 feet, and one of the 
diagonals also is 18 feet : find the area. 



*o^ 



By drawing this diagonal the rhombus is divided into 
two equilateral triangles ; and the height of each triangle, 
by Art. 68, is 18 x "866... feet. Also this height is equal 
to the height of the rhombus. Thus the area of the rhom- 
bus in square feet is 18 x 18 x 'See..., that is about 280*6 
square feet. 



72 EXAMPLES. XII. 



EXAMPLES. XII. 

Find the areas of the parallelograms having the fdlow- 
ing bases and heights : 

1. Base 14 yards, height 5 yards, 

2. Base 15 yards 2 feet, heiglit 11 yards 1 foot, 

3. Base 16 yards 2 feet 3 inches, height 14 yards 
2 feet 8 inches. 

4. Base 14 chains 16 links, height 9 chains 48 links. 

Find the heights of the parallelograms having the fol- 
lowing areas and bases : 

5. Area 1125 square feet, base 15 yards. 

6. Area 3i acres, base 242 yards. 

7. Area 93 square feet 140 square inches, base 5 yards 
1 foot 7 inches. 

8. Area 160 square yards 3 square feet 33 square 
inches, base 13 yards 1 foot 9 inclies. 

9. The base of a parallelogram is 4 feet 6 iuclies, and 
its height is 2 feet 8 inches ; the side adjacent to the base 
is 3 feet : find the length of the perpendicular on this side 
from any point in the opposite side. 

10. The adjacent sides of a parallelogram are 8 feet 
and 16 feet, and its area is half that of a square having the 
same perimeter : find the perpendicular distance between 
each pair of opposite sides. 

11. Each side of a rhombus is 24 feet, and one of the 
diagonals also is 24 feet : find the area. 

12. Each side of a rhombus is 32 feet, and each of the 
larger angles is equal to twice each of the smaller angles : 
find the area. 



TRIANGLE. 73 



XIII. TRIANGLE. 

148. We have shewn in Art. 29 that a triangle is equi- 
valent to half a rectangle having the same base and height: 
this is the reason of the rule now to be given. 

149. To find the area of a triangle. 

Rule. Half the product of the base i7ito the height 
will he the area. 

It is obvious that we may multiply together the base 
and half the height, or multiply together the height and 
half the base, or multiply together the base and the height 
and take half the product. 

150. Examples : 

(1) The base of a triangle is 3 yards, and its height is 
4 feet 6 inches. 

3 yards = 9 feet, 4 feet 6 inches = 4| feet. 

« .1 « 9 81 1 ^81 81 „^, 
9x4i = 9x- = -; -of- = - = 20i. 

Thus the area is 20 j square feet. 

(2) The base of a triangle is 45 feet, and its height is 
36 feet. 

Half of 3G is 18; 45x18 = 810. Tlius the area is 810 
square feet. 

151. If we know the area of a triangle, and also one 
of the two dimensions, the base or the height, we can find 
the other. For if twice the number expressing the area be 
divided by the number expressing the height, the quotient 
is the base ; and if twice the number expressing the area 
be divided by the number expressing the base, the quotient 
is tlie Jceiffht. 



74 



TRIANGLE. 



152. The three sides of a triangle being given, tofMd 
the area. 

Rule. From half the sum of the three sides suhtract 
each side separately ; multiply the half sum and the three 
remainders together : the square root of the product will 
he the area, 

153. Examples: 

(1) The sides of a triangle are 2 feet 2 iiiclies, 2 feet 
4 inches, and 2 feet 6 inches respectively. 

2 feet 2 inches = 26 inches, 2 feet 4 inches = 28 inches, 
2 feet 6 inches = 30 inches. 

26 + 28 4- .30 = 84, ^ of 84 = 42; 

42-26 = 16, 42-28 = 14, 42-30 = 12. 

42 X 16 X 14 X 12= 112896. The square root of 112896 
is 336. Thus the area is 336 square inches. 

(2) The sides of a triangle are 24, 25, and 26 feet 
respectiyely. 

24 + 25 + 26 = 75, ^ of 75 = 37*5. 

37-5-24 = 13-5, 37-5-25 = 12-5, 375 -26 = 11-5. 

37-5 X 13-5 X 12-5x1 1-5 = 72773-4375. The square root 
of 72773*4375 cannot b« found exactly ; if we proceed to 
three decimal places we obtain 269-766 : so that the area is 
about 269 766 square feet. 

154. We will now solve some exercises. 

(1) Find the area of the gable end of a house, the 
breadth being 24 feet, the dis- 
tance of the eaves from the 
ground 30 feet, and the per- 
pendicular height of the roof 
10 feet. 

The figure is composed of a 
rectangle and a triangle. AB 
or EG is the breadth ; the 
eaves are the junctions of the 
walls and the roof, as at E and 
(7, so that AE or BG is the 




TRIANGLE. 76 

height of the eaves from the ground. The perpendicular 
height of the roof is the perpendicular from I) on EG. 

The highest part of the roof is called the ridge, so that 
Z> is on the ridge. The triangle CDE is called the gable 
top. 

Here the area of the rectangle ABGE is 24 x 30 square 
feet, that is 720 square feet ; and the area of the triangle 
is 24 X 5 square feet, that is 120 square feet. Thus the 
whole area is 840 square feet. 

(2) The side of an equilateral triangle is one foot ; 
find the area. 

3 3 1 

Half the sum of the sides = - ; - — 1 = - . 

-x^x-x- = — . The number of square feet in the 

3 

area is therefore equal to the square root of t-; , that is 

lo 

to - of the square root of 3. 

We may also obtain this result thus. It is shewn in 
Art. 65 that the height of the triangle is half of the square 
root of 3, and therefore by Art. 149, the area is one-fourth 
of the square root of 3. Thus the area is approximately 
•433 of a square foot; or to seven decimal places •4330127. 

(3) The sides of a right-angled triangle are 8 feet and 
15 feet respectively : find the perpendicular from the right 
angle on the hypotenuse, and the two parts into which it 
divides the base. 

By Art. 149 the area of the triangle is 60 square feet. 

By Art 55 the length of the hypotenuse is 17 feet. 

120 
By Art. 151 the perpendicular is — - feet, that is 

7tV feet 

Then, by Art. 60, the shorter of the two parts into 

which the perpendicular divides the base, in feet, is the 

120 120 
square root of 8x8- -jy x -— - , that is the square root of 



76 



TRIANGLE. 



^, 14400 ^, , . ., , f 4096 ,, , . 64 

64— , that IS the square root of ^^> that is — . 

Therefore the other part, in feet, is 17 — — , that is -:r=-. 

(4) Having given the sides of a triangle to find the 
diameter of the circle described round the triangle. 

The investigation which we shall now give is valuable 
not only for the result which will be obtained, but also for 
the illustration which it affords of the method by which 
geometrical tniths are demonstrated. 

Let ABC be the triangle, AE 
a diameter of the circle described 
round the triangle, AD the per- 
pendicular from A on the base BC. 
Join CE. 

By Art. 33, the angle ACE is 
a right angle ; so that this angle is 
equal to the angle ADB. 

By Art. 32, the angle A EG is 
equal to the angle ABD. There- 
fore, by Art. 23, the angle BAD 
must be equal to the angle EAC. 

Therefore, by Art. 34, the triangles ABD and AEG 
are similar ; so that AB \^ to AD 2i^ A E is, to AC ', and 
therefore AB xAC=AD x AE. 




Thus AE= 



ABxAC 



AD 



ABxAGxBG 
ADxBG ' 



Hence we have the following result: the diameter of 
the chxie described round a triangle is equal to the pro- 
duct of the sides of the triangle divided by twice the area 
of the triangle. 

Suppose, for example, that the sides of the triangle are 
26 inches, 28 inches, and 30 inches resjjectively. By 
Art. 153 the area is 336 square inches. Thus the diameter 
of the circle described round the triangle in inches 

_ 28x 28x30 65 

~ 2x336 ~ 2 ~^^^' 



EXAMPLES. XIIL 11 



EXAMPLES. XIII. 

Find the areas of the triangles having the following 
dimensions : 

1. Base 18 feet, height 8 feet. 

2. Base 8 yards 1 foot, height 5 yards 2 feet. 

3. Base 10 yards 2 feet 6 inches, height 7 yards 1 foot 
3 inches. 

4. Base 14 chains 15 links, height 12 chains 24 links. 

Find the areas of the right-angled triangles having the 
following dimensions : 

6. Hypotenuse 421, side 29. 

6. Hypotenuse 730, side 152. 

Find approximately the areas of the right-angled tri- 
angles having tlie following dimensions : 

7. Hypotenuse 10, side 7. 

8. Hypotenuse 13, side 9. 

Find the areas of the triangles having the following 
sides : 

9. 5, 5, 6. 10. 65, 65, 112. 

11. 85, 85, 154. 12. 373, 373, 504. 

13. 68, 75, 77. 14. 20, 493, 507. 

15. 105, 116, 143. 16. Ill, 175, 176. 

17. 43, 875, 888. 18. 319, 444, 455. 

19. 633, 875, 888. 20. 3501, 3604, 3605. 



78 EXAMPLES. XIII, 

Calculate to three decimal places the areas of the tri- 
angles having the following sides : 

21. 2, 3, 4. 22. 6, 7, 9. 

23. 7, 8, 13. 24. 15, 16, 17. 

25. 23, 33, 40. 26. 17, 63, 73. 

27. The sides of a triangle are 11, 24, and 31: shew 
that the area is 66\/3. 

28. The sides of a triangle are 61, 62, and 63 : shew 
that the area is 744\/5. 

29. The sides of a triangle are 68, 75, and 77 ; a straight 
line is drawn across the triangle parallel to the longest 
side, and dividing each of the other sides into two equal 
parts : find the area of the two parts into which the tri- 
angle is divided. 

30. The sides of a triangle are 111, 175, and 176; two 
straight lines are drawn across the triangle parallel to the 
longest side, and dividing each of the other sides into 
three equal parts: find the areas of the three parts into 
which the triangle is divided. 

31. The sides of a triangle are 13, 14, and 15 feet: 
find the perpendicular from the opposite angle on the side 
of 14 feet. 

32. The sides of a triangle are 51, 52, and 53 feet: 
find the perpendicular from the opposite angle on the side 
of 52 feet, and find the area of the two triangles into which 
the original triangle is thus divided. 

33. The side of a square is 100 feet; a point is taken 
inside the square which is distant 60 feet and 80 feet re- 
spectively from the two ends of a side : find the areas of 
the four triangles formed by joining the point to the four 
corners of the square. 

34. ^5C is a triangle, and AD is the perpendicular 
from A on BG. If ^^ = 13 feet, and the lengths of the 
perpendiculars from D on AB and AG hQ b feet and 
10*4 feet respectively, find the sides and the area of the 
triangle. 



EXAMPLES. XIIL 79 

35. The base of a triangular field is 1166 links, and 
the height is 738 links; the field is let for ^24 a year: 
find at what j)i*ice per acre the field is let. 

36. The sides of a triangular field are 350, 440, and 
750 yar«ls; the field is let for X26. 5s. a year: find at what 
price per acre the field is let. 

37. Find to the nearest square inch the area of a tri- 
angle whose sides are 5, 6, and 7 feet. 

38. A field is in the form of a right-angled triangle, 
the t\vo sides which contain the right angle being lOv) yards 
and 200 yards : find its area. If the triangle be divided 
into two parts by a straight line drawn from the right 
angle perpendicular to the opposite side, find the area of 
each part. 

39. The sides of a triangle are in the proportion of the 
numbers 5, 12, and 13; and the perimeter is 50 yards: 
find the area. 

40. The sides of a triangle are in the proportion of 
the numbers 13, 14, 15; and the perimeter is 70 yards: 
find the area. 

41. Find the cost of painting the gable end of a house 
at \s. 9d. per square yard; the breadth being 27 feet, the 
distance of the eaves from the ground 33 feet, and the 
perpendicular height of the roof 12 feet 

Find the diameters of the circles described round the 
triangles having the following sides : 

42. 293, 285, 68. 

43. 136, 125, 99. 

44. 123, 122, 49. 

45. 267, 244, 161. 



80 



Q UA DEILA TERALS. 



XIV. QUADRILATERALS. 



155. A quadrilateral cau be divided into two triangles 
oy drawing a diagonal ; tlien the area of each triangle can 
be found, and the sum of the area» of the triangles will be 
the area of the quadrilateral. 

156. Examples : 

(1) The diagonal 
AG oi 'A quadrilate- 
ral A BCD is 12 
feet; the perpendi -^ 
cular BE is 3 feet, 
and the perpendicu- 
lar DF is 4 feet. 

The area of the triangle ^Ci? = -xl2x3 = 18; 

the area of the triangle ACD = - x 12 x 4 = 24 ; 

18 + 24 = 42. 
Thus the area of the quadrilateral is 42 square feet. 

(2) A diagonal of a quadrilateral is 88 yards, and the 
perpendiculars on it from the opposite angles are 31) yards 
and 25 yards respectively. 




-X 88x30 = 1320; 



-X 88x25 = 1100. 



1320 + 1 100 = 2420. Thus the area of the quadrilateral 
is 2420 square yards, that is half an acre. 



Q UADRTLA TERALS. 



81 



157. It is obvious that in Examples like those of the 
preceding Article, instead of calculating separately the 
areas of the two triangles, we may find the area of the 
quadrilateral by using the following rule: multiply the 
871071 of the perpendiculars by the diagonal^ and take half 
the product. 

Thus in the first example of Art. 156 the sum of the 
perpendiculars is 7 feet; and therefore the area in square 

feet = - X 12 X 7 = 42 ; in the second example the sum of 

the perpendiculars is 55 yards, and tlioefore the area in 

square yards = — x 88 x 55 = 2420. 

158. In the particular case in 
which the diagonals of a quadri- 
lateral intersect at right angles 
the rule just given amounts to 
this : take half the product of the 
two diagonals. 

Bvthe aid of a figure the truth 
of this rule becomes self-evident. 
Let A BCD be a quadrilateral 
such that its diagonals AC and 
.5Z) intersect at right angles; let 
E be the point of intersection. 
Through A and C draw straight 
lines parallel to BD ; through 

B and D draw straight lines parallel to A C. Thus a rect- 
ai'gle KLMN is formed. Now it is easy to see thnt, the 
triangle AEB is equal to the triangle BKA, the tri- 
angle BEC is equal to the triangle CLB, tlie triangle 
CED is equal to the triangle DMC, and the triangle 
DEA is equal to the triangle AND. Thus the quadri- 
lateral ABCD is half of the rectangle KLMN-, and there- 
fore the area of the quadrilateral is equni to half the pro- 
duct of A C and BD. 

159. The diasfonals of a rhombus intersect at right 
angles ; and therefore the rule of the preceding Article 
may always be applied to a rhombus. 




T. M. 



82 Q UA DRILA TERA LS. 

160. It is usual to give a special rule for finding the 
area of a trapezoid. 




Let A BCD be a quadrilateral having the sides AB 
and CD parallel. From C draw CE perpendicular to AB ; 
and from A draw AF perpendicular to CD. Then 



the area of the triangle ABG = - AB^ CE; 



1 

2 



the area of the triangle ADC =^ CD y. AF. 

Now we may admit that^i^=C^; and therefore the 
area of the quadrilateral is equal to the product of CE 
into half the sum of AB and CD. Thus we obtain the 
Rule which will now be given. 

161. To find the area of a trapezoid. 

Rule Midti2?ly the sum of the tico parallel sides hy 
the perpendicular distance between them, and half the 
product will he the area. 

1 62. Examples : 

^ (1) The two parallel sides of a trapezoid are 2 feet 
6 inches, and 3 foot 4 inclies respectively; and the perpen- 
dicular distance between them is 1 foot 8 inches. 

2 feet 6 inches = 2^ feet, 3 feet 4 inches = 3J feet. 

1 foot 8 inches = If feet ; 2| + 3f^ = 5f , 

1 .6 12 1 3.5 .5 175 ,„! 
2X5|xl|=-x-x-= - = 4e. 

Thus the area of the trapezoid is 4|^ square feet. 



Q UADRILA TERALS. 83 

(2) The two parallel sides of a trapezoid are 4*32 feet 
and 5*48 feet respectively ; aiid the jjerpeudicular distance 
between them is 2*18 feet. 

4-32 4- 5-48 -9-8, ^ of 9*8 = 4-9, . 

At 

2-18 X 4-9 = 10-682. 
Thus the area of the trapezoid is 10*682 square feet. 

163. We have established the rule for finding the area 
of a trapezoid in a very simple manner in Art. 160; there 
is also another process which we will give as it is interest- 
ing and instructive. 

Let A BCD be a quadrilateral having the sides AB and 
CD parallel. Through G the middle point of BG draw 
the straight line ^G^ A" parallel to AD, meeting the paral- 
lel sides of the trapezoid at H and K respectively. 




Then the triangles B GR und CGK are equal; and 
thus the trapezoid A BCD is equivalent to the parallelo- 
gram AHKD. And since HB is equal to CAT, it follows 
that AH is equal to half the sum ot AB and CD. Thus 
the trapezoid is equivalent to a parallelogram having its 
base equal to half the sum of the parallel sides of the 
trapezoid, and its height equal to the perpendicular dis- 
tance between those sides. Hence we have the rule given 
in Art. 161. 

Through G draw a straight line parallel to AB meet- 
ing AD at Z. Then L is the middle point of AD, and 
LG = AH'y so that half the sum of the parallel sides is 
equal to the straight line which joins the middle points of 
the other sides. 

6—2 



S4 QUADRILATERALS. 

164. We will now solve some exercises 

(1) -^^Ci) is a quadrilateral ; jj 

^^ = 3 feet, BC==A feet, 
(7Z> = 6feet, DA^I feet; 

and the angle ABC is a right 
angle : find the area of the quad- 
rilateral. 

By Art. 55 we have A C equal 
to the square root of 9 + 16, ihat 
is to the square root of 25 : so 
that AC=p). 

The area of the triangle ^J5C=-x4x3 = 6. 

The area of the triangle ACD can now be found by 
Art. 152. 




5 + 6 + 7 = 18, 2 of 18 = 9, 9 



5-4, 9-6 = 3, 9-7 = 2, 
9 x4x3x2 = 2i6. 



The square root of 216 cannot be found exactly; if we 
proceed to tliree decimal places we obtain 14-697. Thus 
the area of the quadrilateral is about 20-697 square feet. 

(2) The diagonals of a rhombus are 80 and 60 feet re- 
spectively : find the area ; find also the length of a side, 
and the height of the rhombus. 

2 X 80 X 60 = 2400. Thus the area is 2400 square feet. 

The diagonals of a rhombus intersect at the middle 
point of each; thus to find the side of the rhombus we 
must determine the hypotenuse of a right-angled triangle 
the sides of which are 40 and 30 feet respectively. By 
Art. 55 the hypotenuse is the square root of 2500 ; so that 
the side of the rhombus is 50 feet. 



2400 
50 



= 48. Thus the height of the rhombus is 48 feet. 



EXAMPLES. XIV. 85 



EXAMPLES. XIY. 

Find the areas of the quadrilaterals having the follow- 
ing dimensions : 

1. Diagonal 50'08 feet ; perpendiculars 1012 and 8'4 
feet. 

2. Diagonal 54 feet ; perpendiculars 23 feet 9 inches 
and 18 feet 3 inches. 

3. Diagonal 10 chains 14 links ; perpendiculars 6 chains 
27 links and 8 chains 6 links. 

4. Diagonal 3 chains 27 links ; perpendiculars 2 chains 

15 links and 1 chain 75 links. 

5. Diagonal 18 yards 2 feet, sum of the perpendicu- 
lars 16 yards 1 foot. 

6. The area of a quadrilateral is 37 acres 1 rood 

16 poles ; one diagonal is 25 chains : find the sum of the 
perpendiculars on this diagonal from the two opposite 
angles. 

Find the areas of the trapezoids which haye the follow- 
ing dimensions : 

7. Parallel sides 3 feet and 5 feet ; perpendicular dis- 
tance 10 feet. 

8. Parallel sides 10 feet and 12 feet; perpendicular 
distance 4 feet. 

9. Parallel sides 14 yards and 20 yards ; perpendicular 
distance 12 yards. 

10. Sum of the parallel sides 625 links; perpendicular 
distance 160 links. 

11. Sum of the parallel sides 1225 links; perpendicular 
distance 240 links. 

12. Parallel sides 750 links and 1225 links ; perpendi- 
cular distance 1540 links. 



86 EXAMPLES. XIV. 

13. The area of a trapezoid is 3| acres ; the sum of 
the two parallel sides is 242 yards : find the perpendicular 
distance between them. 

14. The area of a trapezoid is 8 acres 2 roods 17 poles; 
the sum of the parallel sides is 297 yards : find the perpen- 
dicular distance between them. 

15. In Example 7 a straight line is drawn across the 
figure parallel to the parallel sides and midway between 
them : find the area of the two parts into which the trape- 
zoid is divided. 

16. In Example 9 two straight lines are drawn across 
the figure parallel to the parallel sides and dividing each 
of the other sides into three equal parts : find the areas of 
the three parts into which the trapezoid is divided. 

17. The diagonals of a quadrilateral are 26 feet and 
24 feet respectively, and they are at right angles : find the 
area. 

18. The diagonals of a rhombus are 88 yards and 110 
yards respectively: find the area. 

19. The diagonals of a rhombus are 64 yards and 
36 yards respectively : find its area and the cost of turfing 
it at 4 pence per square yard. 

20. The area of a rhombus is 52204 square feet, and 
one diagonal is 248 feet: find the other. 

21. ABCD is a quadrilateral; AB = 28 feet, BC=45 
feet, CD = 51 feet, DA = 52 feet; the diagonal AC=5S 
feet : find the area. 

22. ABCD is a quadrilateral; ^4^ = 48 chains, 
BC=20 chains, the diagonal AC =52 chains, and the per- 
pendicular from D on AC =30 chains : find the area. 

23. The sides of a quadrilateral taken in order are 
27, 36, 30, and 25 feet respectively ; and the angle con- 
tained by the first two sides is a right angle : find the 
area, 

24. The sides of a quadrilateral taken in order are 
6, 5, 4, and 3 feet respectively ; and the angle contained by 
the first two sides is 60^ : find the area. 



EXAMPLES, XIV. 87 

25. A railway platform has two of its opposite sides 
parallel and its other two sides equal ; the parallel sides 
are 80 feet and 92 feet respectively ; the equal sides are 
10 feet each : find the area. 

26. ABCD is a quadrilateral ; AB = 8+5 feet, 
^C=613 feet, (7Z> = 810 feet; AB is parallel to CD, and 
the angle at ^ is a right angle : find the area. 

27. ABCD is a quadrilateral ; the sides AB and DC 
are parallel. ^5=165 feet, CD=\2Z feet; the perpen- 
dicular distance oi AB and DC is 100 feet. ^ is a point 
in ^^.sucli that AE is equal to half the difference of 
AB and CD : find the area of the triangle EBC, and of the 
quadrilateral AECD. 

28. The diagonals of a rhombus are 88 and 234 feet 
respectively: find the area; fiud also the length of a side, 
and the height of the rhombus. 



'&' 



29. The area of a rhombus is 35414-t square feet, and 
one diagonal is 672 feet : find the other diagonal ; find also 
the length of a side, and the height of the rhombus. 

30. Two adjacent sides of a quadrilateral are 228 feet 
and 704 feet respectively, and the angle contained by them 
is 90*^ ; the other two sides of the quadrilateral are equal, 
and the angle contained by them is 60^ : shew that the 
area of the quadrilateral in square feet is 

80256 + 130900^3. 



88 



RECTILINEAL FIGURE. 



XV. RECTILINEAL FIGURE. 

165. To find (he area of any i-ectilineal figure. 

Rule. Divide the figure into convenient parts^ find 
the area of every part, and the sum will he the area of 
the figure. 

In general the parts into wliich the rectilineal figure 
can be most conveniently divided will all be triangles : but 
in some cases we may have a square, a parallelogram, or a 
trapezoid, as one of the parts. 

166. Examples : 

(1) ^5(7Z>^is a five-sided 
figure : BH and DK are per- 
pendiculars on AG, and EL 
is a perpendicular on AD. 
The following lengths are in 

ic=10-4, AD = 8'7, 
Bff^48, DK=e5, 
EL = S2. 

The area of the triangle ABC=\x 10-4 x 4'8 = 24-96 ; 
the area of the triangle A CD = ^ x 10-4 x 6-5 = 33-8 ; 
the area of the triangle AED = \ x 87 x 3'2 = 13-92. 

Thus the area of the rectilineal figure in square feet 
= 24-96 + 33-8 + 13-92 = 72-68. 




(2) ABCDEF is a six-sided figure; 



BK, CL, EM, 




RECTILINEAL FIGURE. 89 

FN are perpendiculars on AD. The following lengths 
are in feet : 

BK=^, CL = 4:, EM =4:1, FN= 5-1; 

also AK=Z4, KL = 3"2, LD = 4'l, AN =3% N3f= 5'3. 

It follows from these lengths that AD = 10'7, and that 
A3f=S-f5; hence iV/i)= 10-7 -86 ^2-1. 

The area of the triangle AKB = i x 3'4 x 3 = 5*1, 
the area of the trapezoid BKLC =^ x 7 x 3'2 = 11'2, 
the area of the triangle DLG =^ x 4*1 x 4 = 8*2, 
the area of the triangle ANF =^ x 3*3 x 51 = 8'415, 
the area of the trapezoid FN ME =-| x 9-8 x 5*3 = 25-97, 
the area of the triangle EMD = ^ x 2-1 x 47 = 4-935. 

Thus the area of the rectilineal figure in square feet 
= 5-1 + 11-2 + 8-2 + 8-415 4- 2597 + 4-935 = 63-8-2. 

167. We will now solve some exercises. 

(1) The side of a regular hexagon is one foot : find the 
area. 

By the aid of the figure in Art. 99, we see that a regu- 
lar hexagon can be divided into six equilateral triangles; 
this can be done by drawing straight lines from to ^, ^, 
C, Z>, E, and F. Now, by Art. 154, the area of each equi- 
lateral triangle in square feet is \ of the square root of 3 ; 
therefore the area of the hexagon in square feet is f of the 
square root of 3, that is, f of the square root of 3. 

(2) A regular polygon of twelve sides is inscribed in a 
circle of which the radius is one foot : find the area of the 
polygon. 

In Art. 99 it is shewn that AM is the side of a regular 

polygon of twelve sides inscribed in the circle ; so that the 

area of the polygon is twelve times the area of the triangle 

0AM. The area of the triangle OAM^\-^OM^AL', 

now 0M= 1 foot, AL — \ of AF-\ a foot. Thus the area 

of the triangle 0AM— ^oi a square foot. Therefore the 

12 
area of the polygon = — square feet = 3 square feet. 



90 EXAMPLES. XV, 



EXAMPLES. XV. 

1. ABODE is a five-sided figure ; the following lengths 
are in feet: AG=\^, AD-\% the perpendiculars from 
B and D on AC are 8*4 and 4*6 respectively, and the per- 
pendicular from E on AD is 5 feet : find the area. 

2. ABODE is a five-sided figure; BX, OL, EM sere 
perpendiculars on ^Z> ; the following lengths are in feet : 
AD = 15% BX=r6, CL=5% EM =4% AX =^27, 
DL = 3'9 : find the area. 

3. ABODEF is a six-sided figure; BK, OL, EM, 
FN are perpendiculars on ^Z> ; the following lengths are 
in feet: ^i> = 18-4, BX=5, 0L = 1, EM =6, FN =4, 
AX =4-1, AN=41, DL = 5S, DM=4.'9 : find the area. 

4. ABODEF is a figure having six equal sides ; 
AB = 57'S feet, BF=U-4 feet, and the portion BOEF 
forms a rectangle : find the area. 



'O' 



5. ABODE is a five-sided figure, having the angle at 
E a right angle ; the following lengths are in feet : 
AB^\4., BO^I, OD = 10, DE=12, EA = 5, A0=l1. 
Find the area. 

6. Find the area of a regular hexagon each side of 
which is 20 feet. 

7. Find the area of a regular hexagon which is in- 
scribed in a circle, the diameter of which is 100 feet. 

8. The length of the side of a field, which is in the 
form of a regular hexagon, is 10 chains : find the area. 

9. The radius of a circle is one foot : find the area of a 
regular polygon of eight sides inscribed in the circle. 

10. Find the area of a regular polygon of 24 sides in- 
scribed in a circle, the radius of which is one foot. 



CIRCLE. 91 



XVI. CIRCLE. 

168. To find tJie area of a circle. 

22 
Rule. Multiply the square of the radius hy — ; or, 

if greater accuracy is required, multiply the square of 
the radius by 3'1416. 

169. Examples : 

(1) The radius of a circle is 5 feet. 

The square of 5 is 25; and 25 x ^ = -— - = 78^. Thus 

7 7' 

the area of the circle is about 78f square feet. 

(2) The radius of a circle is 3 miles. 

The square of 3 is 9; and 9 x 3*1416 = 28*2744. Thus 
the area of the circle is about 2S'2744 square miles. 

170. Both the rules in Art. 168 make the area of the 
circle a little greater than it ought to be ; but the second 
rule is sufficiently accurate for most practical purposes. If 
a more accurate result is required we must take as many 
decimal places of the number 31415926... as may be ne- 
cessary. 

171. The area of a circle being given, to find the 

radius, 

22 
Rule. Divide the area by — , and extract the square 

root of the quotient ; or, if greater accuracy is required^ 
divide the area hy 3*1416, and extract the square root of 
the quotient. 



92 CIRCLE. 

172. Examples: 

(1) The area of a circle is 100 square feet. 

92 7 700 

100-^^=100x^= -—= 31-8181...; the square root 

of this is 5'64... Thus the radius of the circle is 5'64 feet. 

(2) The area of a circle is an acre. 

An acre is 4840 square yards ; dividing 4840 by 3*1416 
we have the quotient 1540 '61...; the square root of this is 
39*25... Thus the radius of the circle is about 3925 yards. 

173. To find the area of a circular ring, that is of 
the space between the circumferences of two concentric 
cii'cles. 

Rule. Find the area of each circle, and subtract the 
area of the inner circle from the area of the outer circle. 

Or, Multiply the sum of the radii hy their difference 

22 

and the product hy — , or if greater accuracy is required 

hy 3-1416. 

Thus the area is half the product of the sum of the 
circumferences into the difference of the radii ; or half the 
product of the difference of the circumferences into the 
sum of the radii. 

174. Examples: 

(1) The radii of the two circles are 10 feet and 12 feet 
respectively. 

The area of the inner circle in square feet 
= 10x10x31416 = 314*16; 
the area of the outer circle in square feet 

= 12 X 12 X 3-1416 =452-3904, 
452-3904 - 314*16 = 138-2304. 

Thus the area of the ring is 138-2304 square feet. 



CIRCLE. 93 

Or thus, 12+10 = 22, 12-10-2, 

22x2x3-141 6 = 138-2304. 

(2) The radii of the two circles are 3 yards and 5 feet 
respectively. 

3 yards = 9 feet ; 9 + 5 = 14, 9-5=4, 

14x4x3-1416 = 175-9296. 

Thus the area of the ring is 1759296 square feet. 

175. If one circle fall entirely within the other, it is 
obvious that the rule of Art. 173 will give the area of the 
space between the circumferences of the two circles, even 
when the circles are not concentric. 

176. The rule given in Art. 168 for finding tlie area of 
a circle is that to which the beginner's attention should 
be chiefly, if not entirely directed. Other rules may be 
given, which are of course equivalent to that : they are of 
small practical importance, but tliree such rules will be 
placed here for use if required. 

Multiply the yxidlus hy the circumference^ and take 
half the product. 

Multijily the aqiiare of the diameter hy -7854. 

Divide the square of the circunnference &?/ 4 x 3-1416; 
or m,uUiply tJie square of the circumference hy '07958. 

177. The first of the three Rules given in the preced- 
ing Article is of interest in connexion with the theory of 
our subject. The Rule amounts to the statement that the 
area of a circle is equal to the area of a triangle which has 
the circumference of the circle for its base, and the radius 
of the circle for its height. A strict demonstration of this 
statement would be unsuitable for beginners ; but it is 
easy to give a notion of the grounds on which the state- 
ment rests. 

Suppose we inscribe in a circle a regular polygon with 
a large number of sides. Then the three following facts 
are suflBciently obvious : the area of this polygon will not 
differ much from the area of the circle ; the perimeter of 



94 CIRCLE. 

the polygon will not differ much from the circumference of 
the circle; and the perpendicular drawn from the centre 
of the circle on a side of the polygon will not differ much 
from the radius of the circle. By joining the angular 
points of the polygon with the centre of the circle, the 
polygon is divided into a set of equal triangles ; and this 
set of triangles is equivalent to a single triangle having the 
perimeter of the polygon for its base, and the perpendi- 
cular from the centre of the circle on a side of the polygon 
for its height. Hence the truth of the statement becomes 
apparent. 

As an illustration of the three facts which we have 
noticed with respect to a regular polygon of a large num- 
ber of sides inscribed in a circle, we may refer to the 
results which have been obtained for a regular polygon of 
timlve sides: see Arts. 99 and 167. Suppose the radius of 
the circle to be one foot : then the perpendicular from the 
centre of the circle on a side of the polygon is about "866 
of a foot ; the perimeter of the polygon is 12 x -.51764 feet, 
that is, 6"2117 feet very nearly, while the perimeter of the 
circle is 6"2S32 feet ; the area of the polygon is 3 square 
feet, while the area of the circle is 3'1416 square feet. 

Again, from Example 10 of Chapter xv. it appears that 
the area of a regular polygon of twenty-four sides in- 
scribed in a circle of one foot radius is 3*1058 square feet. 

From these results we may readily allow that by taking 
a regular polygon of a very large number of sides, the 
differences between the quantities which are compared 
will become almost imperceptible. 

178. We will now solve some exercises. 

(1) The diameter of a circular courtyard is 80 feet; 
a gravel walk a yard wide runs round it on the inside: 
find the area of the walk. 

The outer boundary of the walk is the circumference 
of a circle of 40 feet radius ; the inner boundary is the 
circumference of a circle of 37 feet radius. Hence by 
Art. 173 the area of the gravel walk in square feet 

= 77 X 3 X 31416 = 7257096. 



CIRCLE. 95 

(2) The radius of a circle is 15 inches: find the radius 
of a circle the area of which is three quarters of the area 
of this circle. 

By the rule for finding the area of a circle we see that 
the areas of two circles are in the same proportion as the 
squares of their radii. Hence we have the proportion 

1 : - :: the square of 15 : the square of the required radius. 

Therefore the square of the required radius 

= 7x15x15 = 16875. 
4 

Hence the required radius is the square root of this num- 
ber; proceeding to two decimal places we obtain 12"99. 
Thus the required radius is very nearly 13 inches. 

(3) The radius of a circle is 20 inches : it is required 
to draw three concentric circles in such a manner that the 
whole area may be divided into four equal parts. 

This amounts to three exercises like that just solved. 
The area of the inner circle is to be a quarter of that of 
the given circle. Hence, proceeding as before, we shall 
find that the radius of the inner circle is equal to the 
square root of 100. 

The area between the inner circle and the second circle 
is also to be a quarter of that of the given circle. There- 
fore the whole area of the second circle is to be half that 
of the given circle. Hence, proceeding as before, we shall 
find that tiie radius of the second circle is equal to the 
square root of 200 

Similarly, we shall find that the radius of the third cir- 
cle is equal to the square root of 300. 

Thus the radii of the three circles in inches will be 
found to be respectively 10, 14*14, and 17*32, by proceed- 
ing to two decimal places. 



96 EXAMPLES. XVI. 



EXAMPLES. XVI. 

Assuming that the circumference of a circle is 3^ times 
the diameter, find in square feet the areas of the circles 
with the following radii : 

1. 21 feet. 2. 16 yards 2 feet. 

3. One furlong. 

Assuming that the circumference of a circle is 3-1416 
times the diameter, find in square feet the areas of the 
circles wiih the following radii : 

4. 25 feet. 5. 992 feet. 

6. A quarter of a mile. 

Assuming that the circumference of a circle is 3| times 
the diameter, find in feet the radii of the circles with the 
following areas: 

7. 100 square feet. S. One rood. 

9. 5 acres 3 roods 8 poles. 

Assuming that the circumference of a circle is 3-1416 
times the diameter, find in feet the radii of the circles with 
the following areas : 

10. 500 square feet. 

11. 6 acres 2 roods 11 poles. 12. A square mile. 

[In all future Examples unless anything is stated to the 
coTifrary, we shall assume that the circumference of a circle 
is 3-1416 times the diameter.] 

13. The radius of the inner circle of a ring is 14 feet, 
and the radius of tlie outer circle is 16 feet : find the area! 

14. The radius of the inner circle of a ring is 14 yards 
2 feet, and the radius of the outer circle is 18 yards 2 feet : 
find the area. 



EXAMPLES. XVI. 97 

15. A circle of radius 10" 15 feet falls entirely within 
another circle of radius 13*35 feet : find the area between 
the circles. 

16. The radius of the inner boundary of a ring is 14 
inches ; the area of the ring is 100 square inches : find the 
radius of the outer boundary. 

17. The radius of the outer boundary of a ring is 18 
feet, the area of the ring is 300 square feet : find the 
radius of the inner boundary. 

18. The area of a quarter of a circle is 7 square yards : 
find the radius of the circle. 

19. The circumference of a circle is 700 feet : find the 
area. 

20. The circumference of a circle is half a mile : find 
the area. 

21. The area of a circle is half an acre : find the cir- 
cumference. 

22. The area of a circle is equal to that of a rectangle 
which is 400 feet by 256 : find the circumference of the 
circle. 

23. The radius of a circle is 8 feet : find the radius of 
another circle of half the area. 

24. The radius of a circle is 18 inches: find the radius 
of another circle of one-fifth the area. 

25. A circle of 10 inches radius is divided into three 
parts by t.vo concentric circles : find the radii of these cir- 
cles, so that the three parts may be of equal area. 

2:5. A room 25 feet 3 inclies long, and 14 feet 6 hichos 
wide, has a semicircular bow 21 feet in diameter thrown 
out on one side: find the area of the whole room. 

27. If a pressure of 15 lbs. on every square inch be 
applied to a circular plate 3 feet in d anieter, find the total 
pressure to the nearest hundredweight. 

28. Find the expense of paving a circular court 40 
foet in diameter at 2s. 3d. per square foot. 

T.M. 7 



98 EXAMPLES. XVI, 

29. The inner diameter of a circular building is 68 feet 
10 inches, and the thickness of the wall is 22 inches: find 
how many square feet of ground the base of the wall 
occupies. 

30. In a circular riding-school of 100 feet in diameter 
a circular ride, within the outer edge, is to be made of a 
uniform width of 10 feet: find the cost of doing this at 4df. 
per square foot. 

31. A circular gi*ass-plot whose diameter is 40 yards 
contains a gravel walk, one yard wide, running round it 
one yard from the edge : find what it will cost to turf the 
grass-plot at 4d per square yard. 

32. A road runs round a circular shrubbery ; the outer 
circumference is .500 feet and the inner circumference is 
420 feet: find the area of the road. 

33. Find the side of a square which is equivalent in 
area to a circle of 80 feet radius. 

34. Find the radius of a circle which is equivalent in 
area to a square the side of which is 80 feet. 

35. The side of a square is 16 feet; a circle is in- 
scribed in the square so as to touch all its sides : find the 
area between the circle and the square. 

36. The side of a square is 18 feet; a circle is de- 
scribed round the square : find the area I'etween the circle 
and the square. 

37. The sides of a right-angled triangle are 27 feet 
and 43 feet respectively : find the area of the circle de- 
scribed on the hypotenuse as diameter. 

38. The area of a semicircle is 645 squ ire feet : find 
the length of the perimeter of the semicircle. 

39. The radius of a circle is 1 foot; an equilateral tri- 
angle is inscribed in the circle : find the area between the 
circle and the triangle. (See Art. 99.) 

40. The sides of a right-angled triangle are 370 feet 
and 168 feet respectively: find the area of the circle which 
has the hypotenuse of this triangle for diameter. 



EXAMPLES. XVI. 99 

41. A rectangle is 8 feet long and 7 feet broad : find 
the area of the circle which has the same perimeter. 

42. The sides of a triangle are 13, 14, and 15 feet : 
find the area of the circle which has the same perimeter. 

If a circle has the same j)erimeter as a rectangle the 
circle has the greater area; verify this statement in the 
following cases : 

43. Rectangle 18 feet by 10. 

44. Rectangle 27 feet by 13. 

If a circle has the same perimeter as a triangle the 
circle has the greater area; verify this statement in the 
following ases : 

45. Sides of a triangle 9, 10, 17 feet. 

46. Sides of a triangle 11, 16, 19 feet. 

If a circle has the same area as a rectangle, the circle 
has the less perimeter ; verify this statement in the follow- 
ing cases : 

47. Rectangle 15 feet by 12. 

48. Rectangle 24 feet by 21. 

If a circle has the same area as a triangle the circle 
has the less perimeter ; verify this statement in the follow- 
ing cases : 

49. Sides of a triangle 5, 6, 7 feet. 

50. Sides of a triangle 12, 15, 17 feet. 

51. A circle is 4 feet in circumference : find the area 
of a square inscribed in it. 

52. A circle is 7 feet in circumference : find the area 
of a square inscribed in it. 



7-2 



100 SECTOR OF A CIRCLE 



XVll. SECTOR OF A CIRCLE AND SEGMENT 

OF A CIRCLE. 

179. To find the area of a sector of a 
circle. 

Rule. As 360 is to the nuniber of de- 
grees in the angle of the sector so is the 
area of the circle to the area of the sector. 

180. Examples : 

(1) The radius of a circle is 25 feet, and the angle of 
the sector is 80 degrees. 

The area of the circle = 25 x 25 x 3 1416 = 1963'5. 

360 : 80 :: 1963*5 : required area, 

80x1963-5 8x1963-5 2x1963-5 ,^^ ^ 

= = = 43d'3... 

360 36 9 

Thus the area of the sector is about 436-3 square feet. 




•** 



(2) The radius of a circle is 12 feet, and the angle of 
the sector is 75 degrees. 

The area of the circle^ 12 x 12 x 31416. 
360 : 75 :: 12x12x3 1416 : the required area, 

75X12X12XS-U16 ^ 75X12X31416 ^3^ ^ 31416 = !)4-248. 
ooO oO 

Thus the area of the sector is 94 248 square feet. 

181. The following is another rule for finding the area 
of a sector of a circle : multiply the arc hy tfie radius and 
take liolf the product. 

The truth of this rule will be obvious from the remarks 
made in Art. 177. 



AND SEGMENT OF A CIRCLE, 101 

182. Examples: 

(1) The radius of a circle is 4 feet, and the arc of the 
sector is equal to the radius. 

- X 4 X 4 = 8. Thus the area of the sector is 8 square feet. 

(2) The radius of a circle is 2 feet 6 inches, and the 
arc of the sector is 1 foot 5 inches. 

-X 30x17 = 255. Thus the area of the sector is 255 
square inches. 

183. Suppose we require the area 
of a figure which is the difierence of 
two sectors having a common angle. 
Let OAB be one sector, and OCD 
the other sector ; so that ABDC is 
the figure of which the area is re- 
quired. We may proceed thus : 

We may calculate separately the 
area of each sector and subtract the 
less from the greater. 

Or, we may calculate the area of the entire ring be- 
tween the two circles, of which AB and CD are arcs ; and 
then use the proportion, as 360 is to the number of de- 
grees in the angle at O so is the area of the ring to the 
required area. 

Or we may use this Rule : multiply the sum of the 
arcs hy the difference of the radii and take half the 
product. Or this : midtiply the difference of the arcs hy 
the sum, of the radii and take half the product. 

184. Examples : 

(1) The radii are 15 feet and 10 feet, and the arcs 
6 feet and 4 feet respectively. 

By Art. 181, the area of the larger sector in square feet 
= -xl5x6 = 45, and the area of the smaller sector in 




102 



SECTOR OF A CIRCLE 



square feet =-x 10 x 4 = 20 : thus the required area in 
square feet = 45 — 20 = 25. 

Or, using the third Rule of the preceding Article, we 
have the sum of the arcs = 10 feet, and the difference of 
the radii = 5 feet ; thus the required area in square feet 

= ^x 10x5 = 25. 

(2) The radii are 7 feet and 5 feet respectively, and 
the angle at is 45 degrees. 

Here we use the second Rule of the preceding Article. 

By Art. 173 the area of the entire ring in square feet is 
12 X 2 X 31416, that is, 75-3984. Then 

360 : 45 :: 75'3984 : the required area. 

Thus the required area in square feet 

75-3984 



8 



9-4248. 




185. Any chord AB of a cir- 
cle, which is not a diameter, di- 
vides the circle into two segments, 
one greater than a semicircle, and 
the otlier less. 

When we know the area of the 
lesser segment, we can, by subtract- 
ing this from the area of the circle, 
determine the area of the greater 
segment ; so that it is sufficient to 

give a Rule for finding the area of a segment less than a 
semicircle. 

In the Examples, unless the contrary is expressly stated, 
we shall always refer to the lesser segment. 

Let be the centre of the circle ; then it is obvious 
that the segment ABC is equal to the difference of the 
sector OACB and the triangle OAB. Hence we have the 
Rule which we shall now give. 



AND SEGMENT OF A CIRCLE. 103 

186. To find the area of a segment which is less than 
a semicii'de. 

Rule. Find the area of the sector ivhich has the some 
arc, and subtract the area of the triangle formed by the 
radii and the chord. 

187. Examples : 

(1) The radius of a circle is 10 inches, and the angle of 
the sector is 60 degrees : find the area of the segment. 

The area of the circle in square inches = 10 x 10 x 3*1416 
= 314'16. By Art. 179, the area of the sector in square 

inches = — ~ — = 52"36. The triangle in this case is equi- 
6 

lateral, and by Art. 152 its area in square inches is the 

square root of 15 x 5 x 5 x 5, that is, about 4330. 

Thus the area of the segment in square inches 
= 52-36 -43-30 = 9-06. 

(2) The radius of a circle is 4 feet, and the angle of the 
sector is a right angle : find the area of the segment. 

The area of the circle in square feet = 4 x 4 x 3-1416; 
therefore the area of the sector in square feet = 4 x3'1416 
= 12-5G64. The area of the triangle in square feet 

= ^x4x4 = 8. 

Thus the area of the segment in square feet = 4-5664. 

188. In the examples of the preceding Article we were 
able to find the areas of the triangles, and thus to deduce 
the areas of the segments. But in general, if we know 
only the radius of the circle and the angle, we cannot 
find the area of the triangle by methods given in the 
present book, though we could find it vvith the aid of Tri- 
gonometry. 



104 SECTOR OF A CIRCLE 

189. To find the area of a segment of a circle, the 
chord of the arc and the height being known. 

Rule. Add together one- fourth of the square of the 
chord, and tico-fiflhs of the square of the height^ and 
multiply the square root of the sum by four-thirds of 
the height. 

This Rule is not exact; it gives the area of the segment 
greater than it ought to be, but the error is very small, 
provided the angle of the corresponding sector be small; 

when this angle is 60 degrees, the error is less than 

part of the area, and when this angle is 90 degrees, the 

error is less than --— - part of the area. 
4000 

190. Examples: 

(1) The chord is 12 inches, and the height is 1 inch. 

1 2 2 

-xl2x r2 = .'i6, -xlxl==-; 

4 5 5 

2 

36 + - = 36"4 : the square root of this is 6-0332, 
o 

^xl X 6-0332 =-8-0443. 

Thus the area of the segment is about 8-0443 square 
inches. 

(2) The chord is 20 inches, and the height is r4 inches. 

1 2 

-X 20x20= 100, - X 1-4 X 1-4 = -784; 

4 5 

the square root of 100-784= 10*0391, 

-X 1-4x10-0391 = 18-73965. 

Thus the area of the segment is about 18-74 square 
incnes. 



AND SEGMENT OF A CIRCLE. 105 

191. If the angle of the corresponding sector be so 
large that the Rule is not suflficiently accurate, we may 
divide the segment into a triangle and two equal smaller 
segments ; we can then calculate the area of the triangle 
by an exact Rule, and calculate the area of the smaller seg- 
ments by the Rule in Art. 189. See the figure in Art. 78, 
where the segment ADBE is made up of the triangle 
ABE, and the segments having the chords AE and EB. 

192. "We will now solve some exercises. 

(1) The radius of a circle is 25 inches, and the chord of 
the sector is 14 inches : find the area of the sector. 

This exercise cannot be solved exactly, but only ap- 
proximately. By Art. 122 the length of the arc is about 
14"l595141 inches; and thus the area of the sector in 

square inches = - x 25 x 14-1895141 = 177-36S92. 

(2) The radius of a circle is 25 inches, and the chord of 
a segment is 14 inches : find the area of the segment. 

We have just found for the area of the sector 177*36892. 
The area of the triangle can be obtained by Art. 152, the 
sides being 25, 25, and 14 respectively : the area will be 
found to be 168 square inches. Thus the area of the seg- 
ment is about 9*36892 square inches. 

Or we mav calculate the area of the segment by the 
Rule in Art. 189. 

We must first determine the height. With the figure 
of Art. 78 we have AC^1r>, AB=\4. Thus AI) = 7 ; and 
then, by Art. 60, we get CD = 2i. Therefore DE= 1. 

1 2 2 

-xl4xl4 = 49, ,xlxl=-. 
4 5 5 

2 

49+ =49-4: the square root of this is 7-02851... 

^x 1x7-02851 = 9-37135. 

Thus, by this Rule, we obtain for the area of the seg- 
ment 937135.,. square inches; this differs but slightly 
from the former result. 



106 EXAMPLES. XVIL 



EXAMPLES. XYII. 

Find in square feet the areas of the sectors of circles 
having the following dimensions : 

1. Radius 24 feet, angle 25*^. 

2. Radius 12 feet, angle l^O''. 

3. Radius 48 feet, angle 28'^. 

4. Two concentric circles have radii of 10 feet and 

15 feet respectively : find the area of the figure bounded by 
these circles, and by radii inclined at au angle of 40^ to 
e.ach other. 

5. Two concentric circles have radii of 10 feet and 
18 feet respectively: find the area of the figure bounded by 
these circles, and by radii iuchnetl at an angle of 50" to 
each other. 

6. The area of a sector is 150 square feet; the angle 
of the sector is 50" : find the radius. 

7. The area of a sector is 230 square feet ; the angle 
of the sector is 40° : find the whole perimeter of the sector. 

8. The area of a sector is 45 square feet ; the radius is 
8 feet : find the angle. 

9. The area of a sector is 94 square feet ; the radius is 

16 feet : find the arc. 

10. The area of a sector is 357 square feet ; the arc is 
^Q feet : find the radius. 

11. The area of a sector is 125 square feet; the area 
of the circle is 400 square feet : find the angle. 

12. The area of a sector is 115 square feet; the area 
of the circle is 700 square feet : find the arc. 

13. The chord of a sector is 58 inches ; the radius is 
100 inches : find the area of the sector. 



EXAMPLES, XVII. 107 

14. The chord of a sector is 6 iuches ; the radius is 
9 inches : find the area of the sector. 

15. The radius is 10 feet ; the angle of the sector is 
30° ; find the area of the segment. 

16. The radius is 10 feet ; the angle of the sector is 
120° : find the area of the segment. 

17. The radius of a circle is 10 feet ; two parallel 
chords are dra^^^l each equal to the radius : find the area 
of the zone between the cliords. 

18. The radius of a circle is 12 feet ; two parallel 
chords are drawn on the same side of the centre, one sub- 
tending an angle of 60^ at the centre, anil the other an angle 
of 90** : find the area of the zone between the chords. 

19. The radius of a circle is 12 feet ; two parallel 
chords are drawn on opposite sides of the centre, one sub- 
tending an angle of 60" at the centre, and the other an 
angle of 90°: find the area of the zone between the chords. 

20. The radius of a circle is 15 feet : find the areas of 
the two parts into which it is divided by a chord equal to 
the radius. 

Find, by Art. 189, in square feet to three decimal places 
the areas of the segments of circles having the following 
dimensions : 

21. Chord 17-3205 feet ; height 5 feet. 

22. Chord 14-1421 feet ; height 2-9289 feet. 

23. Chord 10 feet; height 1*3397 feet. 

24. Chord 5-1764 feet; height '3407 feet. 

25. Chord 2*6105 feet ; height '0855 feet. 



108 



SIMPSON'S RULE. 



XYIII. SIMPSON'S RULE. 




193. We shall now give a very important Rule by 
which the areas of certain figures may be approximately 
found. 

Let there be an area 
bounded by the straight line 
AG, the straight lines Aa and 
Gg at right angles to AG^ 
and the curve ag. 

Divide AG into any even 
number of equal parts AB, 
£G, CD, ... ; at the points 
of division draw straight lines 
Bb, Cc, Dd,... at right an- 
gles to AG, to meet the ciirve. 
The straight lines A a, Bb, Cc, 
... Gg are called ordinates. 
Then the area may be approximately found by the follow- 
ing Rule : Add together the first ordinate, the last ordi- 
nate, twice the sum of all the other odd ordinates, and 
four times the sum of all the even ordinates ; miUtiply 
the residt by one-third of the common distance between 
two adjacerit ordinates. 

194. In the figure there are seven ordinates ; the even 
ordinates are Bb, Dd, and Ff ; the other odd ordinates 
besides the first and last are Cc and Ee. 

195. The Rule given in Art. 193 is sometimes called 
the method of equidistant ordinates ; but it is more usually 
called Simpson's Rule, although it was not invented by 
Simpson. 



SIMPSON'S RULE. 



109 



196. Examples : 

(1) Suppose there are seven ordinates, tlie common 
distance being 1 foot ; and that these ordinates are respec- 
tively 4-12, 4-24, 4-36, 4-47, 4'58, 469, and 4-80 feet. 



4-12 
4-80 
8-92 


4-36 

4-58 

8-94 
2 

17-88 


424 
4-47 
4-69 

13 40 

4 


8-92 
17-88 
53-60 




3 80-40 
26-80 



53-60 
Thus the area is about 26-S square feet. 

(2) Suppose there are five ordinates, the common dis- 
tance being 2 feet; and tliat these ordinates are respec- 
tively 1-26, 1-44, 1-59, 1-71, and 1-82 feet. 



1-26 


1-59 


1-44 


3 08 


1-82 


2 


1-71 


318 


306 


3-18 


3 15 


12-60 






4 


18-86 






12-60 


2 

3|37-72 



12-57 



Thus the area is about 12-57 square feet. 



197. In the figure of Art. 193, the curve is concave 
towards the straight \\\\& AG ; moreover the ordinates con- 
tinually increase from one end of this straight line to the 
other. But Simpson's Rule is applicable to areas in which 
the curve has other shapes, as in the following figures : 




The result will in general be more accurate the more 
ordinates are used ; and the Rule ought not to be trusted 
if the curve be very irregular. 



110 



SIMPSON'S RULE, 





198. If the area be bound- 
ed bv the straio-lit line AG 
and the curve AdG, the same 
Rule applies ; only here the 
first and last ordinates are no- ^\ 
thing, and so do not occur in 
the calculation. 

If the area be bounded by 
the closed curve AdG8A, tlie same Rule may be applied : 
we have now to take as the ordinates the breadths 5/3, 
cy, dd,.., 

199. The beginner would not be able to follow a strict 
investigation of Simpson's Rule ; 
but he may see without difficulty 
that there are grounds for con- 
fidence in the approximate accu- 
racy of the Rule. 

We will call tlie portion of the 
area between two consecutive or- 
dinates a 2^ic''^ ,' find we will con- 
sider the first two pieces of the 
figure in Art. 193. 

Suppose the straight lino ab drawn ; we thus obtain a 
trapezoid, and it is obvious from the diagram that the first 
piece is greater than this trapezoid ; so that the area of 
the first piece is greater than the product oi AB into 
half the sum of Aa and JBb. 

Therefore twice the area of the first piece is greater 
than the product of AB into the sum of Aa and Bb. 

Similarly, twice the area of the second piece is greater 
than the product of BG into the sum of Bb and Cc. 

Thus twice the area of the first two pieces is greater 
than the product of AB into tlie sum of Aa and Gc and 
twice Bb. 

Again : suppose the straight line bt to touch the curve 
at b, and let Aa and Bb be produced to meet this straight 
line. Thus another tra})ezoid is formed; and by Art. 163, 
the area of this trapazoid is equal to the product of AG 



A B 



SIMPSON'S RULE. Ill 

into Bh, that is, to the product oi AB into twice Bh. An 1 
it is obvious from the diagram that the sum of the first two 
pieces is less than this trapezoid. 

Thus the area of the first two pieces is less than the 
product of AB into twice Bh. 

Hence we may expect that when we combine these tv/o 
results, the errors will to some extent balance each other ; 
and that three times the area of the first two pieces will be 
very nearly equal to the product of AB into the sum of 
Aa and Cc and four times Bh. 

By proceeding in this way with tlie other pairs of pieces 
in the figure of Art 1S3, we may obtain a sufiicicnt confi- 
dence in Simpson's Rule. With respect to the figures in 
Art. 197, the process would be similar, though not abso- 
hitely the same ; the main fact is tliat we combine two 
results, one of which is too large, and the other too small, 
and trust that the errors will to some extent balance each 
other. 

200. If ag were a straight line, instead of a curve, in 
the figure of Art. 19:5, Simpson's Rule would give the exact 
value of the area ; but in such a case the whole figure 
would form a trapezoid, and the area is most easily found 
by taking the product oi AG into half the sum of ^« and 
Gg. Also if ag is a curve of a certain form the Rule will 
give an exact result ; but we cannot explain in an elemen- 
tary manner what this form is. 

201. It may happen that the boundary of a figure is a 
curve which is too irregular to allow of the immediate ap- 
plication of Simpson's Rule ; in such a case we may adopt 
the following method : Draw a rectilineal polygon, difier- 
ing as little as possible from the figure, and determine the 
area of this polygon exactly; then by Simpson's Rule calcu- 
late separately the areas of the portions which lie between 
the polygon and the figure ; add these to the area of the 
polygon or subtract them from the area of the polygon, 
according as they fall without or within the polygon, and 
the final result will be approximately the area of the 
figure. 

202. In Land Surveying it is often necessary to deter- 
mine the area of a figure which has its boundary composed 
in an irregular manner of curves and numerous short 



112 SIMPSON'S RULE. 

straight lines ; it is found in practice that the area can be 
obtained with ease and sufficient accuracy by a method of 
adjustment of the boundary to a more commodious form. 

Thus, suppose a field 
to be represented on a plan x-rr^ 

by the figure ^J56^Z>. Draw . /^■~r><^- '^^^^^^ 

a straigTit line from A to ""/■- ^ — '\ ) 

B; the small portions lost W X 

and gained will obviously V U 

balance each other, exactly A r\ 

or very nearly, and the area !) 

will thus remain almost or yf Q 

quite unchanged. Similar- v' ^-^-7:7:--^ ..^— -~J> 

ly, straight lines may be ^^^^^ ^^.,^-^-^^^^-^^^177^^ 

drawn from B to 6', from 

C to D, and from D io A, with a like balance of loss and 
gain. Thus we have a four-sided rectilineal figure equi- 
valent to the original figure ; and the area can be easily 
obtained. 

'Jlie skill and judgment of the surveyor will be exer- 
cised in drawing the straight lines, so that the greatest pos- 
sible accuracy may be secured. A piece of transparent horn 
with straight edges is very useful in drawing the straight 
lines ; the horn is placed over the irregular boundary and 
is shifted about until there appears to be an equal portion 
on each side of the edge between the edge and the boundary. 

203. An experimental method of determining the area 
enclosed by an irregular boundary may be noticed. 

Suppose a field to be represented on a plan by the 
figure ABCD of Art. 202. Cut the figure out of stout 
paper or cardboard of uniform thickness, and weigh it in 
a very delicate balance. Also cut out a square inch from 
the sanie paper or cardboard, and weigh it. Then by 
pro[)ortion we can find how many square inches the figure 
ABCD contains. And from observing the scale on which 
the plan is drawn, we shall know the area on the ground 
wliicl) corresponds to a square inch on the plan ; and thus 
finally we can determine the area of the field. 

An interesting application has been made of this pro- 
cess to determine the proportion of the water to the laud 
ou the surface of the earth. Camh. Phil. Trans. Vol. vi. 



EXAMPLES, KVIIL 113 



EXAMPLES. XVIII. 

Apply Simpson's Rule to find in square feet the areas of 
figures having the following dimensions : 

1. Ordinates 3, 8, 15, 24, 35, 48, 63 feet ; common dis- 
tance 1 foot. 

2. Ordinates 4, 14, 36, 76, 140 feet ; common distance 
1 foot. 

3. Ordinates 0, 20, 32, 36, 32, 20, feet ; common dis- 
tance 2 feet. 

4. Ordinates 0, 1*25, 4, 6'75, 8, 6'25, feet; common 
distance 1 foot. 

5. Ordinates 6-082, 6-164, 6-245, 6*325, 6*403, 6'481, 
6-557 feet ; common distance 1 foot 8 inches. 

6. Ordinates 2*7 14, 2*759, 2*80*2, 2-844, 2*884 feet ; com- 
mon distance 9 inches. 

7. Ordinates 14*2, 14*9, 15*2, 15-1, 14*5, 14'1, 137 feet; 
common distance 3 feet. 

8. Ordhiates 0, 1*11, 2*48, 4*17, 6*24, 8*75, 11*76, 15-33, 
19*52 feet; common distance 1 foot. 

9. Ordinates 10-204, 9*804, 9*434, 9*090, 8*771, S-475, 
8*197, 7-937, 7692 feet ; common distance 1 foot. 

10. Ordinates 2-4849, 2-5849, 2*6391, 2*7081, 2-77*26, 
2-833*2, 28904, 2-9444, 2-9957 feet ; common distance 1 foot. 

11. Ordinates 0, "4359, -6, *7141, "8, -8660, 9165, -9539,' 
•9793, 9950, 1 foot ; common distance *1 of a foot. 

10 n ^- f 1<^ 1^ 10 10 iO 10 10 iO 

12. Urdmates ^^, ^^, ^.^, j^, ^^, ^„, j^, ^^ , 

10 10 10 . . ,. , - ,, . . 

— -, -— , — - feet : common distance l ol a root. 
18 19 :^0 

X.M. 8 



114 SIMILAR FIGURES, 



XIX. SIMILAR FIGURES. 

204. In Chapter vi. we have draAvn attention to the 
natnre of Similar Figures ; and we have now to point out 
the relation which holds between the areas of Similar 
Figures, We shall state a most important proposition, and 
then proceed to apply it to various problems. 

205. TTie areas of similar figures are as the squares 
of corresponding lengths. 

For example, suppose we have two similar triangles, 
and that tiie side of one trianofle is three times the cor- 
responding side of the otlier ; then the area of the larger 
triangle is nine times the area of the smaller triangle, the 
number 9 being the square of the number 3. And it is 
easy to see the reason for this fact : the larger triangle has 
its base three times the base of the smaller, and, because 
the triangles are similar, the height of the larger triangle 
is also three times the height of the smaller ; but the area 
is half the product of the base into the height ; and there- 
fore the area of the larger triangle is 9 times the area of 
the smaller triangle. 

In like manner, if two triangles are similar, and the 
side of one triangle is five times the corresponding side of 
the other, the area of the larger triangle is twenty-five 
times the area of the smaller triangle. 

206. We have found in Art. 154, tliat the area of an 
equilateral triangle, of which the side is 1 foot, is •4330127 
square feet : suppose we require the area of an equilateral 
triangle, of which the side is 7 feet. 

The square of 1 is 1, and the square of 7 is 49; there- 
fore we have the proportion 

1 : 49 :: '4330127 : required area. 

Thus the required area in square feet 

= 49 X -4330127 = 21-2176223. 



SIMILAR FIGURES. 115 

In the same way we may proceed with other examples, 
and it is obvious that we shall have the following Rule for 
finding the area of any equilateral triangle : Multiply the 
square of the length of a side by '4330127. 

207. Circles are similar figures ; and the areas of two 
circles are in the same proportion as the squares of their 
radii, see Art. 178. So also sectors of circles having the 
same angle are similar figures ; and the corresponding 
segments are similar figures ; the areas of two similar 
sectors are in the same proportion as the squares of the 
radii, and so also are the areas of two similar segments. 

208. Suppose we require the radius of a circle, such 
that the area of the segment corresponding to an angle of 
60*^ shall be 20 square inches. 

In Art. 187 we have found that if the radius is 10 
inches, the area of the segment corresponding to this 
angle is 9*06 square inches; thus we have the propor- 
tion 

9*06 : 20 :: 100 : the square of the required radius. 

Therefore the square of the required radius 

20x100 „^^^, 

the square root of this number = 14*857. 

Thus the required radius is 14*857 inches. 

Again ; suppose we require the radius of a circle such 
that the area of the segment corresponding to an angle of 
90*^ shall be 10 square feet. 

By Art. 187 we have the proportion 

4*5664 : 10 :: 16 : the square of the required radius. 

Therefore the square of the required radius 

the square root of this number = 5'92. 
Thus the required radius is 6*92 feet. 

8—2 



116 



SIMILAR FIGURES. 




209. ABC'is a triangle; 
the side AB is 10 feet: it is 
required to divide the tri- 
angle into four equal parts 
by straight lines parallel to 
BG. 

This problem resembles 
one solved in Art. 178. Sup- 
pose DE the straight line 
nearest to A. Then the area 
of ADE will be one-fourth of the area of ABC. 

Thus we have the proportion 

I : J :: the square of ^^1^ : the square of ^Z). 

Therefore the square of AD = ^ of the square of AB 
= ^ X 100 = 25 ; the square root of this number = 5, Thus 
A£> = 5 feet. 

In like manner, if FG be the next straight line 

1 : f :: the square of AB : the square of ^i^. 

Therefore the square of AF=l of the square of AB 
= f X 100 = 50 ; thus the number of feet in ^i^=the square 
root of 50 = 7-0710678. 

In like manner, if HK be the next straight line, we 
find that the number of feet in AII=^ the square root of 
75 = 8-G602540. 

210. ABGD is a trape- 
zoid ; the perpendicular dis- 
tance of the parallel sides ^1^ 
and 1)0 is 3 feet; AB=- 10 
feet, DG-d feet: it is re- 
quired to divide the trape- 
zoid into two equal parts by 
a straight line parallel to 
AB. Produce AD and BG 
to meet at O. Let EF de- 
note tlie required straight 

line. Draw OM perpendicular to AB, meeting EF at L, 
nndBGsitK. 




SIMILAR FIGURES. 117 

Since EF divides the trapezoid into two equal parts, 
the triangle OEF will be equal to half the sura of the 
triangles OAB and ODC. The three triangles ODC, OEF, 
and OAB are similar; and their areas are therefore as the 
squares of the corresponding lengths OK, OL, and OM. 

Hence the square of OL must be equal to half the sum 
of the squares of OiTand OM, 

Now by Art. 77, we have 0^=4'5; therefore 0M = 15. 
The square of 4*5 = 20'25; the square of 7'5 = 56"25. Tims 
the square of OZ = half of 76*5 = 38*25; and therefore the 
number of feet in OZ = the square root of 38"25 = G*184G. 

The number of feet in irZ = 6-1846-4'5 = l-684G. Thus 
the position of EF is determined. 

211. We will now solve some exercises. 

(1) A pliin of an estate is drawn on the scale of 1 inch 
to 20 feet : find what space on the plan will correspond to 
8000 square yards of the estate. 

The scale is that of 1 inch to 240 inches. The required 
space will be obtained by dividing 8000 square yards by 
the square of 240. 

8000 SO 10 5 



240x240 24x24 24x3 36 

5 
Thus the required sjDace is ~ of a square yard, that i.s. 

36 

5 

7— of 9 square feet, that is, Ix square feet. 
36 

(2) If a square inch on a plan corresponds to 4 square 
yards of the original, find the scale. 

4 square yards = 4 x 9 x 144 square inches. The square 
root of 4 X 9 X 144 = 72. Thus the scale is that of 1 inch' to 
72 inches. 

(3) The sides of a rectangle are in the proportion of 
4 to 5, and the area is 180 square feet : find the sides. 



118 SIMILAR FIGURES. 

If the sides of a rectangle are 4 and 5 feet respectively, 
the ai'ea is 20 square feet. Thus we have the proportion 

20 : 180 : : the square of 4 : the square of the required 
corresponding side. 

Therefore the square of this side = — — — = 16 x 9 = 144; 

thus this side = 12 feet; and therefore the other required 
side = 15 feet. 

(4) An equilateral triangle and a circle have the same 
perimeter : compare their areas. 

Suppose each side of the triangle to be 1 foot ; then the 
area is '43301 square feet. The perimeter of the triangle 
is 3 feet. If the perimeter of a circle be 3 feet, the area 
of the circle will be found by Chap. xvi. to be "7162 square 
feet. Divide '7162 by '43301 ; the quotient is 1*6.5... 
Thus the area of the circle is 1'65... times the area of the 
equilateral triangle. 

We shall obtain the same final result whatever be the 
length of the side of the equilateral triangle. If, for ex- 
ample, we suppose each side to be 7 feet, we shall obtain 
for the areas of the triangle and of the circle respectively 
49 times the former values, but the proportion of the areas 
will remain unchanged. 



'o^ 



EXAMPLES. XIX. 119 



EXAMPLES. XIX. 

1. A field containing 3600 square yards is laid down 
on a plan to a scale of 1 inch to 10 feet : find the number 
of square inches of the plan it will occupy. 

2. A field containing 6 acres is laid down on a plan to 
a scale of 1 inch to 20 feet : find how much paper it will 
cover. 

3. Determine the scale used in the construction of a 
plan upon which every square inch of surface represents a 
square yard, 

4. Determine the scale used in the construction of a 
plan upon which a square foot of surface represents an 
area of ten acres. 

5. A field is ten thousand times as large as the plan 
which has been made of it : find what length on the plan 
will represent a length of 20 yards in the field. 

6. An estate, which has been surveyed, is one hundred 
miUiou times as large as the plan wliich has been made of 
it : express the scale of the plan in terms of inches to 
a mile. 

7. The sides of a rectangle are in the proportion of 
2 to 3; and the area is 210 square feet : find the sides. 

8. The sides of a triangle are in the proportion of the 
numbers 13, 14, and 15 ; and the area is 24276 square feet : 
find the sides in feet. 

9. The sides of a triangle are in the proportion of the 
numbers 7, 15, and 20; and the area is 2226 square feet : 
find the sides in feet. 

10. An equilateral triangle and a square have the 
same perimeter : compare the areas. 

11. A square arid a regular hexagon have the same 
perimeter : compare their areas. 



120 EXAMPLES. XIX. 

12. A circle and a square have the same perimeter : 
compare their areas. 

13. A circle and a regular hexagon have the same 
perimeter : compare their areas. 

14. Find the side of an equilateral triangle^ so that 
the area may be 100 square feet. 

15. Find the side of a regular hexagon, which shall be 
equal in area to an equilateral triangle, each side of which 
is 150 feet. 

16. Find the radius of a circle, such that the area of a 
segment corresponding to an angle of 90*^ may be 50 square 
feet. 

17. One side of a triangle is 15 feet ; it is required to 
divide the triangle into five equal parts by straight lines 
parallel to one of the otiier sides : hnd the distances from 
the vertex of the points of division of the given side. 

IS. An equilateral triangle and a square have the 
same area : compare their perimeters. 

19. The parallel sides of a trapezoid are respectively 
16 and 20 feet, and the perpendicular distiince between 
them is 5 feet; it is required to divide the trapezoid into 
two equal trapezoids : find the distance of the dividing 
straight line from the shorter of the parallel sides. 

20. The side of a square is 12 feet ; the square is di- 
vided into three equal parts by two straight lines parallel 
to a diagonal : find the perpendicular distance between the 
parallel straight lines. 

21. A circle and a regular polygon of twelve sides 

have the same perimeter: sliew, by Arts. 99 and 167, that 

3"2154 
the area of the circle is ^ , ^ , - times the area of the 

o"141d 

polygon. 



121 



FOURTH SECTION. VOLUMES. 



XX. DEFINITIONS. 

212. We shall commence this part of the subject with 
definitions of some terms which we shall have to employ. 

Although it is convenient to collect the definitions in 
one Chapter, it is not necessary for the beginner to study 
them closely all at once ; it will be sufficient to read them 
with attention, and then to recur to them hereafter as oi>- 
casion may require. 



213. Parallel planes are such as do not meet one an- 
other, although produced. 

Thus the floor and the ceiling of a room are parallel 
planes. 

214. A straight Une is said to be at right angles to a 
plane, or perpendicular to a plane, when it makes right 
angles with every straight line which it meets in that 
plane. 

This is the strict geometrical definition ; but without 
regarding it, the student will gain an adequate idea of what 
is meant by a perpendicular to a plane from considering 
the illustration aflbrded by a straight rod when fixed in the 
ground so as to be upright. 

So also the student will readily understand when one 
plane is perpendicular or at right angles to another, with- 
out a strict geometrical definition. Thus the walls of a 
room are perpendicular to the floor and to the ceiling ; 
and a door while moving on its hinges remains perpen- 
dicular to the floor and to the ceiling. 



122 



DEFINITIONS. 




215. A parallelepiped is a 
solid bounded by six parallelo- 
grams, of which every opposite 
two are equal and in parallel 
planes. 

The diagram represents a 
pa-allelepiped. A BCD and 
EFGH'A.Y(iQ(\\\2i\ parallelograms in parallel planes; ABFE 
and DGGH are equal parallelograms in parallel planes ; 
and AD HE and BCGF are equal parallelograms in paral- 
lel planes. 

A ' parallelepiped is called rectangular when the six 
bounding [»araileiogi"auis are rectangles, and oblique when 
they are not. A couiinon brick furnishes an example of a 
rectangular parallelepiped. A rectangular parallelepiped 
which has its six bounding rectangles all equal is called a 
cube. This will be found equivalent to saying that a cube 
is a solid bounded by six equal squares, of which every 
ojjposite two are in parallel planes. 

216. Plane figures which form the boundaries of solids 
are called /aces of the solid; the straight lines which form 
the boundaries of the plane figures are called edges of the 
solid. Thus a parallelepiped has six faces and twelve 
edges. 

217. A prism is a solid bounded by plane rectilineal 
figures, of which two are equal and in parallel planes, and 
the rest are parallelograms. Tiie two bounding figures 
which are equal and in parallel planes are called the ends 
of the prism. 

The diagram represents a 
prism having for its ends the 
equal pentagons ABODE and 
FGHIK \ these figures are 
in parallel planes. The other 
bounding figures of the solid are parallelograms, as ABGF, 
BGHG, and so on. Such a prism is called a pentagonal 
prism ; if the ends are hexagons, the prism is called a hex- 
agonal prism ; and so on. 

A prism is called right when the parallelograms be- 
tween the ends are rectangles, and oblique when they are 
not. 




DEFINITIONS, 123 

In a right prism the ends are at right angles to the 
other faces. 

Thus we see that a parallelepiped is included among 
prisms, and that a rectangular parallelepiped and a cube 
are included among right prisms. 

In a rectangular parallelepiped every face is at right 
angles to the four faces which it meets. 

218. A pyramid is a solid bounded by three or more 
triangles which meet at a point, and by another rectihneal 
figure. 

The point is called the vertex of the pyramid, and the 
rectilineal figure opposite to the vertex is called the hase 
of the pyramid. 

When three triangles meet at the vertex the base of 
the pyramid is a triangle. When four triangles meet at 
the vertex the base of the pyramid is a quadrilateral ; the 
famous pyramids of Egypt, of which the student has pro- 
bably seen pictures, are of this kind, the bases being squares. 
AVhen five triangles meet at the vertex the base is a pen- 
tagon. And so on. 

219. A frustum of a sulid is a slice of it, contained 
between the base and any plane parallel to the base; the 
base and the opposite face are called the ends of the frus- 
tum. Thus, if a pyramid be cut into two pieces by any 
plane parallel to the base, one of the two pieces is a frus- 
tum of a pyramid, and the other is a pyramid. 

220. A wedge is a solid bounded 

by five planes; the base is a rect- 
angle, the two ends are triangles, and 
the other two faces are trapezoids. 

The fine of intersection of the two 
trapezoids is called the edge; that side of the base which 
is parallel to the edge is called the length of the base. 

If the length of the base is equal to the edge, the tra- 
pezoids are parallelograms, and the wedge is an oblique 
triangular prism. If the parallelograms are rectangles, the 
wedge is a right triangular prism. 



124 



DEFINITIONS. 




221. A prismoid is a solid having for its ends any two 
parallel plane rectilineal figures of the same number of 
sides, and having all its faces trai:>ezoid8. 

If the ends are similar figures, 
simihxrly situated, the prismoid is 
a frustum of a pyramid. 

^ If the ends are rectangles, the 
prismoid is a frustum of a wedge: 
the term prismoid is by some wri- 
ters restricted to tliis solid. 

222. A sphere is a solid having every point of its sur- 
face equally distant from a certain point, called the centre 
of the sphere. 

A radius of a sphere is a straight line drawn from the 
centre to the surface. 

A diameter of a sphere is a straight line drawn through 
the centre, and terminated both ways by the surface. 

The intersection of a sphere with any plane is a circle; 
if the plane passes through the centre of the sphere the in- 
tersection is called a gi*eat circle of the sphere. 

A sphere is sometimes called a glohe, and sometimes a 
round hotly : marbles and billiard balls are familiar exam- 
ples of spheres. 

If a sphere be cut into two parts by a plane, each part 
IS called a segment of a sphere ; the base of the segment is 
the circle which is formed by tiie intersection of the sphere 
and the plane: if a diameter of the sphere be drawn per- 
pendicular to the base of the segment, the portion inter- 
cepted by the segment is called the height of the segment. 

A zone of a sphere is the part of the sphere contained 
between two parallel planes; the height of the zone is the 
perpendicular distance between the two parallel planes. 

223. Another method of defining a sphere 
may be given. Let ABC be a semicircle, 
AC being the diameter. Suppose the semi- 
circle to be cut out in paper or cardboard, 
and let AChQ kept fixed, and the semicircle 
be turned roimd A C, The figure ABC, as it 
turns round, sweeps out a solid, and this solid 
is a sphere. 





DEFINITIONS. 125 

It is not necessary to introduce this method, because 
we can give a good definition, and form a clear conception 
of a sphere \nthout it; but there are otlier solids for which 
such a method must be used, and so we have applied it to 
the sphere in order to render it more readily intelligible. 

224. A cylinder is a solid produced by turning a rect- 
angle round one of its sides, which remains fixed. 

Thus let A BCD be a rectangle; let 
^^ be kept fixed, and let the rectangle 
be turned round AB. 

Then the figure ABGD, as it turns 
round, sweeps out a solid, which is called 
a cylinder. 

AB m called the axis of the cylinder. 
The circles described by AD and BC are called the ends 
of the cylinder; either end may be called the base of the 
cylinder. 

An uncut lead pencil is an example of a cylinder. 

If a cylinder be cut by any plane parallel to the ends, 
the intersection of the plane and the cylinder is a circle; 
and each of tlie two parts into which tlie cylinder is divided 
is a cylinder. 

225. The solid which we have called a cylinder is more 
strictly called a right circular cylinder; the word right 
refers to the fact that the axis AB is, perpendicular to the 
base, and the word circular refers to the f;ict that the 
base is a circle. Other cylinders besides right circular 
cylinders occur in mathematical investigations. 

A cylinder bears some resemblance to a prism which 
has for its base a rectilineal figure wuth a very large num- 
ber of sides, each side being very small. A right circular 
cylinder resembles a right prism, the ends of which are 
regular polygons. And an oblique prism will give an 
idea of cyhnders which are not right circular cylinders. 

In future, when we use the single word cylinder, we 
shall mean that what we say is applicable to any cylinder; 
but it will be sufficient for the beginner to think of a right 
ch'cular cylinder. 




126 DEFINITIONS. 

226. A cone is a solid produced by turning a right- 
angled triangle round one of the sides which contain the 
right angle, this side remaining fixed. 

Thus let ABC be a triangle 
having a right angle at B\ let 
AB be kept fixed, and let the 
triangle be turned round AB. 
Then the figure ABC, as it turns 
round, sweeps out a solid, which is 
called a cone. 

The point A is called the verier, of the cone; AB is 
called the axis of the cone. The circle described by BG 
as it turns round is called the base of the cone. A straioht 
line drawn from the vertex to the circumference of the 
base, as AG, is called the slant side of the cone, and some- 
times the slant height of the cone. 

If a cone be cut by any plane parallel to the base, the 
intersection of the plane and the cone is a circle; for 
example, GH in the diagram represents such a circle. 
When a cone is cut into two parts by a plane parallel to 
the base, the part between the vertex and the plane is a 
cone; the other part is called a frustum of a cone, see 
Art. 219. The slant side or slajit height of the frustum of 
a cone is that portion of the slant side of the cone which is 
cut off by the frustum; for example, in the diagram GG m 
the slant side of the frustum cut off by the plane GH. 

227. The solid which we have called a cone is more 
strictly called a right circular cone ; the word right refers 
to the fact that the axis AB is perjiendicular to the base, 
and the word circular refers to the fact that the base is a 
circle. Other cones besides right circular cones occur in 
mathematical investigations. 

A cone bears some resemblance to a pyramid which has 
for its base a rectilineal figure, with a veTy large number of 
sides, each side being very small. A right circular cone 
resembles a pyramid in which the base is a regular polygon, 
and the triangular faces are all equal. And a pyramid, the 
base of which is not restricted to be a regular polygon, and 
the triangular faces of which are not all equal, wiU give an 
idea of cones which are not right circular cones. 



DEFINITIONS. 



127 



In future, wheu we use the single word cone, we shall 
mean that what we say is applicable to any cone; but it 
will be sufficient for the beginner to think of a right circu- 
lar cone. 

228. A solid ring is a solid produced by turning a 
circle round any straight line in the plane of the circle 
which does not cut the circle. 



E 



D 




Thus, let ABC be a circle, and 
DE any straight line in the plane 
of the circle, which does not cut 
the circle^ let DE be kept fixed, 
and let the circle be turned round 
DE. Then the figure ABC, as it 
turns round, sweeps out a solid, 
which is called a solid ring, or briefly 
a ring 

229. Any face of a parallelepiped may be called the 
base; then the height of the parallelepiped is the perpen- 
dicular drawn to the base from any point of the opposite 
face. 

The height of a pyramid or a cone is the perpendiculai 
drawn to the base from the vertex. 

The height of a prism, or a cylinder, or a prjsmoid, or a 
frustum of a solid, is the perpendicular drawn to one end 
from any point of the other end; either end may be called 
a base. 

The height of a wedge is the perpendicular drawn from 
any point of the edge to the base. 



128 SOLID MEASURE. 



XXI. SOLID MEASURE. 



230. A Tiil)le of Soiid Measure might be given as ex- 
tensive as the Table of Square Measure of Art. 126; but it 
will be sufficient to observe that 

1728 cubic inches make I cubic foot, 
27 cubic feet make 1 cubic yard. 

231. The connection which subsists between the sys- 
tems of measures and of weights must be noticed. 

The grain is thus determined: a cubic inch of pure 
water weighs 252'458 grains. 

A pound Avoirdupois contains 7000 grains. 

A cubic foot of pure water weighs 1728 x 252-458 grains, 

,, ,. 16 X 1728 X 252458 , ., . .. -n , 

that IS, -— Avoirdupois ounces: it will be 

found that this number to three decimal places is 997*137. 
Thus it is usually sufficient in practice to take 1000 Avoir- 
dupois ounces as the weight of a cubic foot of pure water. 

A gallon is a measure which will hold 10 Avoirdupois 
pounds of pure water, that is 70000 grains. Hence the num- 
ber of cubic inches in a gallon is ^r^rvrvT^ : it v/ill be found 

252*458 

that this number to three decimal places is 277'274. Thus 
it is usually sufficient in practice to take 27 7^^ as the num- 
ber of cubic inches in a gallon. 



RECTANGULAR PARALLELEPIPED. 129 




XXII. RECTANGULAR PARALLELEPIPED. 

232. Suppose we have 
a rectangular parallelepi- 
ped, which is 4 inches long, 
3 inches broad, and 2 inches 
high. Let the rectangular 
parallelepiped be cut by 
planes, an inch apart, pa- 
rallel to the faces; it is 
thus divided into 24 equal 
solids, each of whicli is a 
cube, being an inch long, 
an inch broad, and an inch 
high : such a cube is called a caUc inch. The rectangular 
parallelepiped then contains 24 cubic inches; this fact is 
also expressed thus : the volume of the rectangular paral- 
lelex^iped is 24 cubic inches. 

The word content, or the word solidity, may be used 
instead of the word volume. 

The number 24 is the product of the numbers 4, 3, 
and 2, which denote respectively the length, the breadth, 
and tlie height of the rectangular parallelepiped. 

233. If a rectangular parallelepiped be 8 inches long, 
7 inches broad, and 5 inches high, we can shew in the same 
manner that its volume is 8 times 7 times 5 cubic inches, 
that is, 280 cubic inches. Similarly, if a rectangular paral- 
lelepiped be 15 inches long, 12 inches broad, and 10 inches 
high, its volume is 15 times ]2 times 10 cubic inches, that 
is, 1800 cubic inches. And so on. 



234. In the same manner, if a rectangular parallelepi- 
ped be 4 feet long, 3 feet broad, and 2 feet high, its volume 
is 24 cubic feet; that is, the rectangular parallelepiped 
might be divided into 24 equal solids, each being a foot 
long, a foot broad, and a foot high. If a rectangular 



T. M. 



9 



130 RECTANGULAR PARALLELEPIPED, 

parallelepiped be 4 yards long, 3 yards broad, and 2 yards 
liigii, its volume is 24 cubic yards. And so ou. 

235. The beginner will observe, that the way in which 
volumes are measured is another case of the general prin- 
ciple explained in Art. 131. We fix on some volume as a 
standard, and we compare other volumes with this standard. 
The^most convenient standard is found to be the volume 
of a cube; it may be a cubic inch, or a cubic foot, or any 
other cube. 

236. In order then to find the volume of a rectangu- 
lar parallelepiped we must express the length, the breadth, 
luid the height in terms of the same denomination; and 
the product of the numbers which denote the length, the 
breadth, and the height, will denote the volume. If the 
length, the breadth, and the height are all expressed in 
inches, the volume will be expressed in cubic inches; if the 
length, the breadth, and the lieight, are all expressed in feet, 
the volume will be expressed in cubic feet; and so on. 

237. In the example given in Art. 232, we find that 
the .volume is equal to 4 x 3 x 2 cubic inches. Now suppose 
we take for the base of the rectangular parallelepiped the 
rectangle which is 4 inches by 3; then the height is 2 inches, 
and the area of the base is 12 square inches. Thus the 
number denoting the volume is equal to the product of the 
numbers denoting the area of the base and the height. If 
we take for the base the rectangle vvhich is 4 inches by 2; 
then the height is 3 inches : and, as before, the number 
denoting the volume is equal to the product of the num- 
bers denoting the area of the base and the height. Or 
we may take for the base the rectangle which is 3 inches by 
2; then the height is 4 inches: and, as before, the number 
denoting the volume is equal to the product of the numbers 
denoting the area of the base and the height. 

238. The student will now bo able to understand the 
way in which we estimate the volumes of solids, and to 
use correctly the rules which will be given : the rules will 
be stated with brevity, but this will present no difficulty to 
those who have read the foregoing explanations. 



RECTANGULAR PARALLELEPIPED. 131 

239. To find the volume of a rectangular parallel- 
epiped. 

Rule. Multiply together the length, the breadth, and 
the height, and the product will he the volume. 

Or, Midtiply the area of the lase ly the height, and 
the product will he the volume. 

240. Examples. 

(1) The length of a rectangular parallelepiped is 
2 feet 6 inches, the breadth is 1 foot 8 inches, and the 
height is 9 inches. 

2 feet 6 inches = 30 inches, 1 foot 8 inches = 20 inches, 
30x20x9 = 5400. 

Thus the volume is 5400 cubic inches. 

(2) The area of the base of a rectangular paral- 
lelepiped is 15 square feet, and the height is 3 feet 
9 inches. 

3 feet 9 inches = 375 feet. 
15x3'75 = 56-25. 

Thus the volume is 56*25 cubic feet. 

241. If we know th.e volume of a rectangular paral- 
lelepiped, and also the area of its base, we can find the 
height by dividing the number which expresses the volume 
by the number whicli expresses the area of the base ; and 
similarly if we know the volume and the height we can 
find the area of the base. Of course we must be careful to 
use corresponding denominations for the volumes and the 
known area or heisrht : see Art. 132. 



o 



242. Examples. 

(1) The volume of a rectangular parallelepiped is 
676 cubic inches, and the area of the base is half a square 
foot : find the height. 

576 
Half a square foot = 72 square inches; -7rr- = 8. 



Thus the height is 8 inches. 



9—2 



182 RECTANGULAR PARALLELEPIPED. 

(2) The volume of a rectaugular parallelepiped is 
8 cubic feet, and the height is 1 foot 4 inches : find the 
area of the base. 

Q O O 

1 foot 4 inches = \\ feet : rr = t ^ V = 6. 
Thus the area of the base is 6 square feet. 

243. A cube is a rectangular parallelepiped having its 
length, breadth, and height equal; hence to find the 
volume of a cube we multiply the number which de- 
notes the length by itself, and multiply the product by 
the number again. Thus we see the reason for using the 
term ciibe of a nimiber to denote the result obtained by 
multiplying a number by itself, and the product by the 
number again. 

244. The statements made in Art. 230 as to the con- 
nection between cubic inches, cubic feet, and a cubic yard, 
will be easily understood by the aid of the explanations of 
the present Chapter. Take, for example, the first state- 
ment that 1728 cubic inches make 1 cubic foot : a cubic 
foot is a cube 12 inches long, 12 inches broad, and 12 inches 
high ; and therefore by the method of Art. 232 we see that 
a cubic foot contains 12x12x12 cubic inches, that is, 1728 
cubic inches. 

245. We will now solve some exercises. 

(1) Find how many bricks will be required to build a 
wall 25 yards long, 15 feet high, and 1 foot lO^^ inches 
thick ; a brick being 9 inches long, 4^ wide, and 3 deep, 

The number of cubic inches in the wall is 

45 
25x3xl2xl5xl2x— : 

2 

the number of cubic inches in a brick is 9 x-x 3: divide 

2 

the former number by the latter, and the quotient is 30000, 

which is therefore the number of bricks required. 



EXAMPLES. XXIL 133 

(2) A reservoir is 15 feet 4 inches long by 8 feet 

3 inches wide : find how many cubic feet of water must be 
drawn off to make the surface sink 1 foot. 

' The vohime of the water which must be drawn off is 

that of a rectangular parallelepiped 15j feet long, 8J feet 

wide, and 1 foot deep ; therefore the number of cubic feet 

.. X. ^ -46 33 ,,,,. 23x11 

m the volume is — x -r- x 1, that is — - — , that is 126^. 

3 4: ^ 

(3) Find the length of a cubical vessel which will hold 
100 gallons. 

The vessel is to hold 27727'4 cubic inches ; so that the 
number of inches in the length will be found by extracting 
the cube root of this number: it will be found that the 
cube root is 30-267. 

(4) A vessel is in the shape of a rectangular paral- 
lelepiped; it is without a lid; externally the length is 

4 feet, the breadth 3 feet, and the depth 2 feet; the thick- 
ness of the material is half an inch : find the number of 
cubic inches of the material. 

Externally the dimensions in inches are 48, 36, and 24 ; 
tlms the volume is 41472 cubic inches. 

Internally the dimensions in inches are 47, 35, and 23| ; 
thus the volume is 38657|' cubic inches. 

The difference is 28141 cubic inches. This is the re- 
quired result. 



EXAMPLES. XXII. 

Find the number of cubic feet and inches in cubes hav- 
ing the following lengths ; 



1. 2 feet 8 inches. 


2. 1 fathom. 


3. 1 yard 1 foot 9 inches. 


4. 1 pole. 



134 EXAMPLES. XXIL 

Find the number of cubic feet and inches in rectangu- 
lar parallelepipeds which have the following dimensions : 

5. 4 feet 8 inches, 3 feet 6 inches, 2 feet 4 inches. 

6. 7 feet 9 inches, 4 feet 6 inches, 2 feet 3 inches. 

7. 6 yards 2 feet 7 inches, 3 feet 4 inches, 2 feet 
11 inches. 

8. 10 yards 9 inches, 5 yards 1 foot 7 inches, 2 yards 
8 inches. 

Find the number of cubic feet and inches in rectangular 
parallelepipeds which have the following dimensions : 

9. Area of base 16 square feet, height 4 feet 3 inches. 

10. Area of base 1000 square inches, height 1 yard. 

11. Area of base 12 square feet 80 square inches, 
height 2 feet 7 inches. 

12. Area of base 1 square yard 5 square feet 120 square 
inches, height 1 yard 1 foot 6 inches. 

Find the heights of the rectangular parallelepipeds 
which have the following volumes and bases : 

13. Volume 6 cubic feet, base 8 square feet. 

14. Volume 3 cubic feet, base 3 feet 4 inches by 2 feet 
6 inches. 

15. Volume 124 cubic feet 1668 cubic inches, base 
24 square feet 143 square inches. 

16. Volume 198 cubic feet 856 cubic inches, base 
34 square feet 4 square inches. 

Find the areas of the bases of the rectangular paral- 
lelepipeds which have the following volumes and heights : 

17. Volume 15 cubic feet, height 9 inches. 

18. Volume 9 cubic feet 43 cubic inches, height 2 feet 
1 inch. 

19. Volume 99 cubic feet 428 cubic inches, height 
5 feet 10 inches. 

20. Volume 296 cubic feet 144 cubic inches, height 
8 feet 6 inches. 



EKAMPLES. XXII. 135 

Find to tlie nearest gallon tlio volumes of rectangular 
parallelepipeds having the following dimensions : 

21. 6 feet, 6 feet, 6 feet. 

22. 6 feet 8 inches, 6 feet 8 inches, 6 feet 5 inches. 

23. 7 feet 4 inches, 7 feet 6 inches, 6 feet 9 inches. 

24. 8 feet 6 inches, 8 feet 4 inches, 8 feet 2 inches. 

Find to the nearest hundredweight the weights of the 
water which can be contained in rectangular parallelepipeds 
having the following dimensions : 

25. 5 feet, 5 feet, 5 feet. 

26. 5 feet 6 inches, 5 feet 6 inches, 5 feet 3 inches. 

27. 6 feet 9 inches, 6 feet 5 inches, 5 feet 10 inches. 

28. 9 feet 4 inches, 8 feet 7 inches, 8 feet 2 inches. 

29. Shew that a cube 6 inches long is equivalent to 
the sum of three cubes which are respectively 3 inches, 
4 inches, and 5 inches long. 

30. Find the number of cubic chains in a rectan- 
gular parallelepiped whose edges are 94 chains 50 links, 
1 chain 5 Unks, and 31|^ links. 

31. The beams of wood used in building a house are 
3 inches thick, and 10 inches wide; 200 of them are used 
which together amount to 1000 cubic feet : find the length 
of each beam. 

32. Find how many bricks will be required to build a 
wall 90 feet long, 18 inches thick, and 8 feet high; a brick 
bchig 9 inches long, 4-| wide, and 3 deep. 

33. A certain book is 8 inches long, 5 J inches broad, 
and 2} inches thick : find how many such books can be 
packed in a box which is 3 feet 6 inches long, 3 feet broad, 
and 2 feet deep. 

34. If a cubic foot of gold may be made to cover uni- 
formly 432000000 square inches, find the thickness of the 
coating of gold. 

35. A metre is 39"37 inches : find the number of cubic 
feet in a cube whose side is a metre. 



136 EXAMPLES. XXII. 

36. A block of stone is 4 feet long, 2^ feet broad, and 
1 J feet thick; it weighs 27 cwt. : find the weight of 100 
cubic inches of the stone. 

37. If a cubic foot of marble weigh 2716 times as 
much as a cubic foot of water, iind the weight of a block 
of marble 9 feet 6 inches long, 2 feet 3 inches broad, and 
2 feet thick. 

38. Sliew that a cubical vessel the length of which is 
14*04 inches will hold less than 10 gallons; and that a 
cubical vessel the length of which is 14*05 inches will hold 
more than 10 gallons. 

39. A reservoir is 24 feet 8 inches long by 12 feet 
9 inches wide : find how many cubic feet of water must 
be drawn off to make the surface sink 1 foot. 

40. A cistern is 13 feet 6 inches long by 9 feet 9 inches 
\\ide: find through how many inches the surface will sink 
if 260 gallons of water are drawn off. 

41. If gold be beaten out so thin that an ounce will 
form a leaf of 20 square yards, find how many of these 
leaves will make an inch thick, the weight of a cubic foot 
of gold being 10 cwt. 95 lbs. 

42. Shew that a cubic fathom of water weighs about 
6 tons. 

43. If a rectangular parallelepiped has its length, its 
breadth, and its depth respectively half as large again as 
anotlier rectangular parallelepiped, shew that the first is 
more than three times as large as the second. 

44. If a rectangular parallelepiped has its length, its 
breadth, and its depth respectively a quarter as large 
again as another rectangular parallelepiped, shew that the 
first is nearly twice as large as the second. 

45. If a rectangular parallelepiped has its length, its 
breadth, and its depth respectively a sixth, a seventh, and 
an eighth as large again as another; shew that the first is 
half as large again as the second. 

48. A vessel is in the shape of a cube ; it is without a 
lid: if the external length is 3 feet, and the thickness of 
the material one inch, find the number of cubic inches of 
the material. 



EXAMPLES. XXII. 187 

47. A vessel is in the shape of a rectangular paral- 
lelepiped ; it is without a lid ; if externally the length be 
6 feet, the breadth 5 feet, and the depth 3 feet, and the 
thickness of the material half an inch, find the number of 
cubic inches of the material. 

48. The external length, breadth, and height of a 
closed rectangular wooden box are 18 inches 10 inches and 
6 inches respectively, and the thickness of the wood is 
half an inch; when the box is empty it weighs 15 lbs., 
and when filled with sand 100 lbs.: find the weight of a 
cubic inch of wood, and of a cubic inch of sand. 

49. A box without a lid is made of wood an inch thick ; 
the external length, breadth, and height of the box are 
2 feet 10 inches, 2 feet 5 inches, and 1 foot 7 inches respec- 
tively : find what volume the box will hold. 

50. A rectangular parallelepiped having a square base 
is 3 feet 4 inches high ; the volume is 40 cubic feet 1440 
cubic inches: find the side of the square base. 

51. A rectangular parallelepiped has two faces each 
containing 2 square feet 117 square inches, two faces each 
containing 2 square feet 12 square inches, and two faces 
each containing 1 square foot 96 square inches : shew that 
the volume of the rectangular parallelepiped is 3 cubic 
feet 216 cubic inches. 

The following Examples involve the extraction of the 
cube root : 

52. Find the length of a cube which will be equivalent 
to 2 cubic feet. 

53. Find the length of a cube which will be equivalent 
to 3000 cubic feet. 

54. Find the length of a cubical vessel which wiU hold 
4000 gallons of water. 

55. Find the length of a cubical vessel which will hold 
a ton weight of water. 

56. If 100 cubic inches of a certain kind of stone 
weigh 14 lbs., find the length of a cube of this stone Avhich 
weJL'-hs half a ton. 



138 PARALLELEPIPED, PRISM, CYLINDER. 



XXIIL PARALLELEPIPED, PRISM, CYLINDER. 



246. To find the volume of a parallelepiped, a prism, 
or a cylinder. 

Rule. Multiply the area of the base by the height, 
and the product will he the volume. 

247. Examples : 

(1) The area of the base of a parallelepix^ed is 5 square 
feet, and the height is 9 inches. 

9 inches = - of a foot, 5 x 7 = — = 3|. 
4 4 4 * 

Thus the volume is 3| cubic feet. 

(2) The base of a prism is a triangle, the sides of which 
are 1 foot 1 inch, 1 foot 8 inches, and 1 foot 9 inches re- 
spectively; and the height of the prism is 1 foot 10 inches. 

We must first find the area of the base by Art. 152. 

1 foot 1 inch = 13 inches, 1 foot 8 inches = 20 inches, 
1 foot 9 inches = 21 iuches. 

13+20+21 = 54, ^ = 27, 

27-13 = 14, 27-20 = 7, 27-21 = 6. 

27 X 14 X 7 X 6 = 15876. The square root of 15876 is 126. 
Tlius the area of the base is 126 square inches. 

1 foot 10 inches =22 inches. 

126 X 22 = 2772. Thus the volume of the prism is 2772 
cubic iuches. 



PARALLELEPIPED, PRISM, CYLINDER. 189 

(3) The radius of tlie base of a cylinder is 5 inches, 
and the height is 16 inches. 

The area of the base in square inches=5x5x3'1416 
= 78-54. 

78-54 X 16 = 1256-64. 

Thus the volume is about 1256-64 cubic inches. 

248. If we know the volume of a parallelepiped, a 
prism, or a cylinder, and also the area of the base, we can 
find the height by dividing the number which expresses 
the volume by the number which expresses the area of the 
base; and similarly if we know the volume and the height, 
!we can find the area of tb*^ base. 

249. Examples : 

(1) The volume of a prism is a cubic foot, and the area 
of the base is 108 square inches : find the height. 

— — = 16. Thus the height is 16 inches. 
lOo 

(2) The volume of a cylinder is 2000 cubic inches, and 
the height is 4 feet 2 inches : find the area of the base. 

———=40. Thus the area of the base is 40 square inches. 
50 

250. It will be seen that the Rule in Art. 246 is the 
same as the second form of the Rule in Art. 239 ; and it 
will be instructive to attempt to expla-in the reason of this 
coincidence. We suppose that the beginner has convinced 
himself by the process of Art. 232, that the Rule holds for 
any rectangular parallelepiped; and we will try to shew 
that it will also hold for a right prism or a right circular 
cylinder. 

251. Refer to the diagram of Art. 29. Let there be 
two right prisms with the same height, one having the tri- 
angle ABG for base, and the other having the rectangle 
ABDE for base. By the method of Art. 29 we can shew 
that the prism on the triangular base is half the prism^'on 



140 PARALLELEPIPED, PRISM, CYLINDER. 

the rectangular base. Hence it will follow that the Rule 
of Art. 246 holds for any right prism on a triangular base. 
Therefore the Rule will also hold for a right prism which 
has any rectilineal figure for its base ; because such a 
base could be divided into triangles ; and the prism could 
be divided into corresponding prisms with triangular bases, 
for each of which the Rule holds. 

We are thus led to the notion that the volume of a 
right prism of given height depends only on the area of 
the base, and not on the shape of that base; and this may 
suggest that the Rule will also hold for a right circular 
cylinder. And we may infer that the Rule will also hold 
for other solids which are not called cylinders in ordinary 
language ; for example, the shaft of a fluted column. 

252. The Rule will also hold for oblique prisms and 
cylinders. The ground on which this rests is the following 
proposition : an oblique parallelepiped is equivalent to a 
rectangular parallelepiped tc?dch has the same base and 
an equal height. This proposition resembles that in 
Art. 28 ; and the mode of demonstration is similar ; but 
instead of one adjustment of adding an area and sub- 
tracting an equal area, we shall here in general require 
tim separate adjustments, of adding a volume and sub- 
tracting an equal volume. 

253. We will now solve some exercises. 

(1) A cubic inch of metal is to be drawn into a wire 
jj^j of an inch thick : find the length of the wire. 

The wire will be a cylinder, having the radius of its 
base ^ of an inch. Thus the area of the base in square 

inches=-^j— = •007854. As the volume is 1 cubic inch, 

we divide 1 by '007854 ; and thus we obtain for the length 
of the wire 127'3 inches. 

(2) Find the volume of a cylindrical shell, the height 
being 5 feet, the radius of the inner surface 3 inches, and 
the radius of the outer surface 4 inches. 



PARALLELEPIPED, PRISM, CYLINDER. 141 

By a cylindrical sliell is meant the solid which remains 
when from a solid cylinder another cylinder with the same 
axis or with a parallel axis is removed ; such bodies are 
usually called pipes or tubes. 

By Art. 173, the area of the base in square inches 
= 7x1x3-1416 = 21-9912; the height is 60 inches: there- 
fore the volume in cubic inches = 60 x 21-9912 = 1319-472. 

(3) The height of a cylinder is to be equal to the radius 
of the base, and the volume is to be 500 cubic inches : find 
the height. 

Since the height is equal to the radius of the base, the 
product of 3-1416 into the cube of the number of inches in 
the radius must be equal to 500: hence the cube of the 

number of inches in the radius=—--— =159-15. By ex- 

3-1416 

tracting the cube root wg obtain 5-419. Thus the radius 

is nearly 5*42 inches. 



EXAMPLES. XXIII. 

Find in cubic feet and inches the volumes of the prisms 
having the following dimensions : ; 

1. Base 6 square feet 35 square inches; height 2 feet 
6 inches. 

2. Base 15 square feet 135 square inches ; height 

3 feet 11 inches. 

3. Base 23 square feet 115 square inches; height 

4 feet 7 inches. 

4. Base 35 square feet 123 square inches; height 

5 feet 5 inches. 

Eind in cubic feet and inches the volumes of the trian- 
gular prisms having the following dimensions : 

5. Sides of the base 7, 15, and 20 inches : height 
45 inches 



142 EXAMPLES. XXIII. 

6. Sides of the base 16, 25, and 39 inches; height 
52 inches. 

7. Sides of the base 13, 40, and 51 inches; height 
58 inches. 

8. Sides of the base 25, 33, and 52 inches ; height 
62 inches. 

Find in cubic feet and decimals the volumes of the 
cylinders having the following dimensions : 

9. Radius of base 2 feet ; height 3 feet 6 inches. 

10. Radius of base 2 feet 6 inches ; licight 4 feet 
3 inches. 

11. Radius of base 3 feet 6 inches; height 5 feet 
9 inches. 

12. Radius of base 5 feet 4 inches; height 6 feet 
4| inches. 

Find the heights of the prisms which have the following 
volumes and bases : 

13. Volume 18 cubic feet 708 cubic inches ; base 

6 square feet 100 square inches. 

14. Volume 28 cubic feet 500 cubic inches; base 

7 square feet 103 square inches. 

15. Volume 3G cubic feet 349 cubic inches ; base 
9 square feet 35 square inches. 

16. Volume 65 cubic feet 782 cubic inches; base 
14 square feet 118 square inches. 

Find the radii of the bases of the cylinders which have 
the following volumes and heights : 

17. Volume 10000 cubic inches ; height 4 feet 2 inches. 

18. Volume 20 cubic feet ; height 4 feet 7^ inches. 

19. Volume 50 cubic feet; height 5 feet 4^ inches. 

20. Volume 100 cubic feet ; height 5 feet 10 inches. 



EXAMPLES. XXIII. 143 

Find to the nearest gallon the quantity of water which 
will be held by cylindrical vessels having the following 
dimensions : 

21. Kadius of base 10 inches ; height 20 inches. 

22. Radius of base 2 feet 6 inches ; height 4 feet. 

23. Radius of base 5 feet ; height 8 feet. 

24. Radius of base 7 feet 6 inches ; height 10 feet, 

25. The height of a prism is 24 feet; the base is a 
trapezoid, the parallel sides being 18 feet and 12 feet re- 
spectively, and the distance between them 5 feet : find the 
rolume. 

2G. The wall of China is 1500 miles long, 20 feet high, 

15 feet wide at the top, and 25 feet wide at the bottom : 
find how many cubic yards of material it contains. 

27. Find the number of cubic feet of earth which 
must be dug out to form a ditch 1000 feet long, 8 feet deep, 

16 feet broad at the bottom, and 20 feet broad at the top. 

28. Find to the nearest gallon the quantity of water 
which will be required to fill a ditch having the following 
dimensions : length 40 feet, depth 6 feet, breadth at the 
top 10 feet, breadth at the bottom 8 feet. 

29. A ditch is 8 feet deep, 24 feet broad at the top, 
and 16 feet broad at the bottom : find the length of the 
ditch if 250000 cubic feet of earth are dug out to make it. 

30. A ditch is 4 feet deep, 5 feet broad at the top, and 
4 feet broad at the bottom : find the length of the ditch 
to the nearest foot, if it will hold 10000 gallons of water. 

31. Find how many cubic feet of earth must be dug 
out to make a well 3 feet in diameter and 30 feet deep. 

32. Find how many cubic yards of earth must be dug 
out to make a well 4 feet in diameter and 119 feet deex). 

33. Find the number of cubic yards of earth dug out 
of a tunnel 100 yards long, whose section is a semicircle, 
with a radius of 10 feet. 

34. Find how many pieces of money | of an inch in 
diameter, and ^ of an inch thick, must be melted down iu 
order to form a cube whose edge is 3 inches long. 



144 EXAMPLES. XXIIL 

35. The diameter of a well is 4 feet, and its depth 
30 feet : find the cost of excayation at 7^. Qd. per cubic 
yard. 

36. The diameter of a well is 3 feet 6 inches, and its 
depth 40 feet : find the cost of excavation at "Is. Qd. per 
cubic yard. 

37. The diameter of a well is 3 feet 9 inches, and its 
depth 45 feet : find the cost of excavation at 7."?. ^d. i)er 
cubic yard. 

38. If 30 cubic inches of gunpowder weigh a pound, 
find what length of a gun, 6 inches bore, will be filled 
with 10 lbs. of powder. 

39. A cubic foot of brass is to be drawn into a wire 
^ of an inch in diameter : find the length of the wire. 

40. A cubic foot of brass is to be drawn into a wire 
"025 of an inch thick : find the length of the wire. 

41. Find the volume of a cylindrical shell, the radius 
of the inner surface being 5 inches, the radius of the 
outer surface 6 inches, and the height 7 feet. 

42. Find the volume of a cylindrical shell, the radius 
of the outer surface being 10 inches, the thickness 2 inches, 
and the height 9 feet. 

43. Find the volume of a cylindrical shell, the radius 
of the inner surface being 12 inches, the thickness 3 inches, 
and the height 10 feet. 

44. An iron pipe is 3 inches in bore, half an inch thick,' 
and 20 feet long : find its weight, supposing that a cubic 
inch of iron weighs 4"526 ounces. 

45. The length of a leaden pipe is 13 feet, its bore is 
1| inches, and its thickness \^ inches : find its weight, 
supposing a cubic inch of lead to weigh 6'604 ounces.^ 

46. Find the cost of a leaden pipe of 2 inches bore, 
wliich is half an inch thick and 8 yards long, at 'l^d. per lb., 
supposing a cubic foot of lead to weigh 11412 ounces. 

47. A square iron rod, an inch thick, weighs 10|lbs. : 
find the weight of a round iron rod of the same length and 
thickness. 



EXAMPLES. XXIII. 11:5 

48. Every edge of a certain triangular prism measures 
10 inclies: find the volume. 

49. The base of a certain prism is a regular hexagon 
every edge of the prism measures 1 foot : find the volume 
of the prism. 

50. The radius of the inner surface of a leaden pipe is 
\\ inches, and the radius of the outer surface is \-^ inches : 
if the pipe be melted, and formed into a solid cylinder of 
the same length as before, find the radius. 

51. The trunk of a tree is a right circular cylinder, 
.S feet in diameter and 20 feet high : find the volume of tho 
timber -wliich remains when the trunk is trimmed just 
enough to reduce it to a rectangular parallelepiped on a 
square base. 

The following examples involve the extraction of tho 
cube root : 

52. The sides of the base of a triangular prism are 52, 
51, and 25 inches respectively; and the height is 60 inches? 
find the length of a cube of equivalent volume. 

53. The height of a cylinder is 4 feet 9 inches, the 
radius of the base is 4 feet 3 inches : find the length oi 
a cube of equivalent volume. 

54. Suppose a sovereign to be | of an inch in diame- 
ter, and ^g- of an inch in thickness; if 100000 of them bo 
melted down and formed into a cubCj find the length of 
the cube. 

55. Tlio height of a cylinder is to be 10 times the 
radius of the base, and the volume is to be 25 cubic feet: 
find the radius. 

56. The height of a cylindrical vessel is to be half the 
radius of the base, and the cylinder is to hold a gallon : 
find the radius. 



T.M. 10 



146 



SEGMENTS OF A RIGHT 



XXIV. SEGMENTS OF A RIGHT CIRCULAR 
CYLINDER. RING. 



254. There are certain segments 
of a right circular cylinder, the volumes 
of which can be found by yery simple 
Rules, as wo will now shew. 

255. Suppose a right circular cy- 
linder cut into two parts by a piano 
parallel to the axis; then each part 
has a segment of a circle for base. 
The volume of each part may be found 
by the Rule of Art. 246. 

256. Suppose a solid has been 
obtained by cutting a right circular 
cylinder by a plane, inclined to the 
axis, which does not meet the base 
of the cylinder. Let the straight 
line CD, drawn from the centre of 
the base at right angles to the base 
to meet the other plane, be called 
the height of the solid. Then the 
Rule for finding the volume of this 
solid is the same as that given in 
Art. 246. 



Thus we may say that the heiglit 
of the solid is the portion of the 
axis of the cylinder which is ^'^"- -^ 
tained between the two ends. 



con- 







257. The preceding Rule may be easily justified. For 
if vre snppo.se a plane drawn through D parallel to the 
base of the cylinder, it will cut off a wedge-shaped slice, 
which may be so adjusted to the remaining solid as to form 
a complete right circular cylinder with the height CD, 



CIRCULAR CYLINDER. RING. 147 

258. In the diagram of Art. 256, it is easy to see that 
CD is half the sum of -4i^and BG; that is, the height is 
equal to half the sum of the greatest and the least straight 
lines which can be drawn on the solid parallel to the axis 
of the cylinder. See Art. 163. 

259. Suppose a solid has been obtained by cutting a 
right circular cylinder by tico planes, inclined to the axis, 
wliich do not meet each other. The volume of the solid 
will be found by inultiplying the tase of the cylinder hy 
the height of the solid; where by the height of the solid 
we must understand the portion of the axis of the cylinder 
which is contained between the two ends. The Rule follows 
from the fact that the solid may be supposed to be the 
difference of two solids of the kind considered in Art. 256. 

260. Suppose we have a solid like that represented in 
the diagram of Art. 256, and that it is bent round until A 
and i^ meet: we obtain a sohd resembling a solid ring; 
and thus a ring may be described roughly as a cylinder 
bent round until the ends meet. This is not exact, but it 
will serve to illustrate the Rule which w^e shall now give. 

261. To find the volume of a solid ring. 

Midtiply the area of a circular section of the ring hy 
the length of the ring. 

The circular section is sometimes called the cross sec- 
tion. The length of the ring is the length of the circum- 
ference of the circle which passes through the centres of all 
the cross sections; or it may be described as half the sum 
of the inner and outer boundaries of the ring. See Art. 258. 

262. Examples : 

(1) The radius of the circular section of a ring is one 
inch, and the length of the ring is ten inches. 

The area of the circular section of the ring is 3"1416 
square inches ; therefore the volume of the ring in cubic 
inches is 10 x 3-1416, that is, 31-416. 

10—2 



148 EXAMPLES. XXIV, 

(2) The inner diameter of a ring is 7 inches, and the 
outer diameter is 8 inches. 

The difference of these diameters is twice the diameter 
of the circular section; therefore the radius of the circular 
section is \ of an inch ; and the area of the circular section 
is 19635 square inches. 

The inner boundary of the ring is 7 x 3'1416 inches, and 
the outer boundary is 8 x 3-1416 inches; half the sum of 
these numbers is 23'562 inches, which is therefore the 
length of the ring. 

Therefore the volume of the ring in cubic inches 
= •19635 X 23-562 = 4-6264 very nearly. 

EXAMPLES. XXIY. 

1. Shew that the length of a ring is equal to the dif- 
ference of the outer boundary and the circumference of the 
cross section. 

2. Shew that the length of a ring is equal to the sum 
of the inner boundary and the circumference of the cross 
section. 

Find in cubic inches the volumes of the rings having 
the following dimensions : 

3. Length 20|- inches, radius of cross section | of an inch. 

4. Length 16 inches, diameter of cross section I'l inches. 

5. Outer diameter 4-8 inches, inner 4*2 inches. 

6. Inner diameter 12*3 inches, diameter of cross sec- 
tion 32 inches. 

7. Outer diameter 19 inches, diameter of cross sec- 
tion 3j inches. 

8. Outer boundary 15 inches, circumference of cross 
section 1*6 inches. 

9. The volume of a ring is 800 cubic inches, the radius 
of the cross section is 2 inches: find the length of the ring. 

10. The volume of a ring is 100 cubic inches, and the 
length is 20 inches : find the inner diameter. 



PYRAMID. CONE. 



149 



XXV. PYRAMID. CONE. 




263. To find the volume of a 
pyramid or a cone. 

Rule. Multiply the area of 
the base by the height and one- 
third of the product will be the 
volume. 



2G4. Examples : 

(1) The base of a pyramid is a square, each side of 
which is 3 feet 6 inches, and the height of the pyramid 
is 3 feet 9 inches. 

3 feet 6 inches = 3^ feet, 

3 feet 9 inches = 3| feet. 

3ix3J = -x-=-, 



1 49 15 
3"T^T 



49x5 _245 ^ 

4x4 "" 16 -^^T6- 



Thus the vojume is 15^^ cubic feet. 

(2) The radius of the base of a cone is 10 inches, and 
the height of the cone is 18 inches. 

i0xl0x3'1416 = 314-16, 
~ X 18 X 31416 = 6 X 314-16 = 1884-96. 
Tims the volume is about 1884-96 cubic inches. 

265. If wo know the volume of a pyramid or a cone, 
and also the area of the base, we can find the height by 
dividing three times the number which expresses the 
volume by the number which expresses the area ; and 
similarly, if we know the volume and the height, we can 
find the area of the base. 



150 



PYRAMID. GONE. 



266. Examples : 

(1) The Tolume of a pyramid is a cubic yard, and the 
area of the base is 18 square feet : find the height. 

A cubic yard = 27 cubic feet ; = -=4^. 

Thus the height is 4^ feet. 

(2) The volume of a cone is half a cubic foot, and its 
height is 27 inches : find the area of the base. 

3 X 864 
Haifa cubic foot = 864 cubic inches : — — — = 96. 

' 27 

Thus the area of the base is 96 square inches. 

267. We will now solve some exercises. 

(1) The base of a pyramid is a square, each side of 
which measures 10 feet ; the length of each of the four 
edges which meet at the vertex is 18 feet : find the 
volume. 

"We must determine the height of the pyramid. 

Let ABGD be the base, and 
E the vertex of the pyramid. 
Let EF be the height of the 
pyramid, that is, the perpendicu- 
lar from E on the base ; then jP 
will be the middle point of the 
diagonal AG. 

Now, by Art. 55, we shall find 
that the number of feet m AG is 
10^2; and thus the number of 
feet in AF is 5sj2. 

In the right-angled triangle 
AEF, the hypotenuse AE is 18 
feet ; and the number of feet in 
AF is 5^/2; therefore, by Art. 60, the number of feet in 
EF is the square root of 324 - 50, that is, the square root 
of 274, that is, 16 '5529454. 




EXAMPLES. XXV. 151 

Hence tlie volume of the pyramid in cubic feet 

=\ X 100 X 16-5529454 = 551-76484. 
o 

(2) The base of a pyramid is a square, each side of 
which is 10 feet; the length of the straight line drawn 
from the vertex to the middle point of any side of the base 
is 13 feet : find the volume. 

We must determine the height of the pyramid. Using 
the same figure as in the preceding Exercise, let G be the 
middle point of AD. Then EG \% \^ feet ; and GF is 
5 feet : therefore, by Art. 60, the number of feet in EF is 
the square root of 169 — 25, that is, the square root of 144. 
that is 12. 

Hence the volume of the pyramid in cubic feet 

= ^x 100x12 = 400. 
o 

"We may observe that EG is sometimes called the slant 
height of the pyramid. 

(3) A corner of a cube is cut off by a plane which 
meets the edges at distances 3, 4, and 5 inches respectively 
from their common point : find the volume of the piece 
cut off. 

The piece cut off is a triangular pyramid : we may take 
for the base the right-angled triangle the sides of which 
are 3 and 4 inches respectively, and then the height of the 
pyramid is 5 inches. Hence the volume of the pyramid in 

cubic inches = ^ ^ -ir- x 5 = 10. 



EXAMPLES. XXV. 

Find in cubic feet and inches the volumes of the pyra- 
mids having the following dimensions : ' 

1. Base 7 square feet 102 square inches ; height 2 feet 
5 inches. 



152 EXAMPLES. XXV. 

2. Base 14 square feet ^(^ square inches ; height 3 feet 
7 inches. 

3. Base 20 square feet 120 square inches ; height 
6 feet 8 inches. 

4. Base 23 square feet 21 square inches; height 
4 feet 11 inches. 

Find in cubic feet and decimals the volumes of the tri- 
angular pyramids having the following dimensions : 

5. Sides of the base 4, 5, and 7 feet ; height 6 feet. 

6. Sides of the base 7, 9, and 11 feet ; height 8 feet. 

7. Sides of the base 15, 19, and 20 feet ; height 22 feet. 

8. Sides of the base 23, 27, and 30 feet ; height 24 feet. 

Find in cubic feet and decimals the volumes of the 
cones having the following dimensions : 

9. Eadius of base 2 feet ; height 4 feet. 

10. Radius of base 3 feet 6 inches ; height 5 feet. 

11. Radius of base 4*2 feet ; height 5'3 feet 

12. Radius of base 10 feet ; height 10 feet. 

Find the heights of the pjTamids which have the follow- 
ing volumes and bases : 

13. Volume 17 cubic feet 363 cubic inches; base 2 
square feet 143 square inches. 

14. Volume 33 cubic feet 309 cubic inches; base 
4 square feet 83 square inches. 

15. Volume 91 cubic feet 792 cubic inches; base 

9 square feet 21 square inches. 

16 Volume 114 cubic feet 1152 cubic inches; base 

10 square feet 96 square inches. 

Find the radii of the bases of the cones which have the 
follov.'ing volumes and heights : 

3 7. Volume 4000 cubic inches ; height 5 feet. 
13. Volume 40 cubic feet ; height 5*3 feet. 



EXAMPLES. XXV. 153 

19. Volume 60*7 cubic feet; heiglit 5*45 feet. 

20. Volume 120 cubic feet ; height 6*24 feet. 

21. The faces of a pyramid on a square base are equi- 
lateral triangles, a side of the base being 120 feet : find the 
volume. 

22. Find the volume of a pyramid which stands on a 
square base, each side of which has 200 feet, each of the 
edges which meet at the vertex being 150 feet. 

23. A pyramid has a square base, the area of which is 
20*25 square feet ; each of the edges of the pjTamid which 
meet at the vertex is 30| feet : find the volume. 

24. The base of a pyramid is a rectangle 80 feet by 
60 feet ; each of the edges which meet at the vertex is 
130 feet : find the volume. 

25. The base of a pyramid is a square, each side of 
which is 24 feet ; the length of the straight line drawn from 
the vertex to the middle point of any side of the base is 
21*8 feet : find the volume. 

26. The base of a pyramid is a square, each side of 
which is 12 feet; the length of the straight line drawn from 
the vertex to the middle point of any side of the base is 
25 feet : find the volume. 

27. The base of a pyramid is a rectangle, which is 
21 feet by 25 feet ; the length of the straight line drawn 
from the vertex to the middJe point of either of the longer 
sides of the base is 23*3 feet : find the volume. 

28. The ibase of a pyi'amid is a rectangle, which is 
IS feet by 26 feet ; the length of the straight line drawn 
from the vertex to the middle point of either of the shorter 
sides of the base is 24 feet : find the volume. 

29. The slant side of a right circular cone is 25 feet, 
and the radius of the base is 7 feet : find the volume. 

30. The section of a right circular cone by a plane 
through its vertex perpendicular to the base is an equila- 
teral triangle, each side of which is 12 feet ; find the volume 
of the cone. 

31. The slant side of a right circular cone is 41 feet, 
and the height is 40 feet : find the volume. 



154. EXAMPLES. XXV. 

32. The slant side of a right circular cone is 55 feet, 
and the height is 42 feet: find the volume. 

33. ^ Find how many gallons are contained in a yesscl 
which is in the form of a right circular cone, the radius of 
the base being 8 feet, and the slant side 17 feet. 

34. ^ A conical wine glass is 2 inches wide at the top 
and 3 inches deep : find how many cubic inches of wine it 
will hold. 

35.^ Find the volume of a circular cone, the height of 
which is 15 feet, and the circumference of the base 16 feet. 

36. A cone, 3 feet high and 2 feet in diameter at the 
bottom, is placed on the ground, and sand is poured over 
it until a conical heap is formed 5 feet high and 30 feet in 
circumference at the bottom : find how many cubic feet of 
sand there are. 

37. Tlie volume of a cone is 22| cubic feet; the cir- 
cumference of the base is 9 feet : find the height. 

38. Find the number of cubic feet in a regular hex- 
agonal room, each side of which is 20 feet in length, and 
the walls 30 feet high, and which is finished above with a 
roof in the form of a hexagonal pyramid 15 feet high. 

39. Find the volume of the pyramid formed by cutting 
off a corner of the cube, whose side is 20 feet, by a plane 
which bisects its three conteniiinous edges. 

40. The edge of a cube is 14 inches ; one of the corners 
of the cube is cut off, so that the part cut off forms a pyra- 
mid, with each of its edges terminating in the angle of the 
cube, 6 inches in length : find the volume of the solid that 
remains. 

41. The gi-eat pyramid of Egypt was 481 feet in height, 
when complete; and its base was a square 764 feet in 
length: find the volume to the nearest number of cubic 
yards. 

42. The spire of a church is a right pyramid on a 
regular hexagonal base; each side of the base is 10 feet 
and the height is 50 feet ; there is a hollow part which is 
also a right p}Tamid on a regular hexagonal base, the height 
of the hollow part is 45 feet, and each side of the base is 
9 feet : find the number of cubic feet of stone in the spire. 



FRUSTUM OF A PYRAMID OR CONE. 155 



XXVI. FRUSTUM OF A PYRAMID OR CONE. 

268. To find the volume of a frustum of a Pyramid 
or Cone, 

Rule. To the areas of the two ends of the frustum 
add the square root of their product ; midtiply the sum, 
ty the height of the frustum^ and one-third of the product 
ic'dl de the volume. 

269. Examples : 

(1) The area of one end of a frustum of a pyramid is 
18 square inches, and the area of the other end is 98 square 
inches; and the height of the frustum is 15 inches. 

The square root of IS x 98 is 42 ; 18 + 98 + 42 = 158. 

- X 15 X 158 = 790. Thus the volume is 790 cubic inches. 

o 

(2) The radius of one end of a frustum of a cone is 
5 feet, and the j-adius of the other end is 3 feet ; and the 
height of the frustum is 8 feet. 

The area of one end in square feet = 25 x3"1416; the 
area of the other end in square feet = 9 x3'1416 ; the 
square root of the product of these numbers is 31416 mul- 
tiplied by the square root of 9 x 25, that is, 3" 141 6 x 15. 

Add these results and we obtain 3*1416 multiplied by 
the sum of 25, 9, and 15 ; that is, 31416 x 49. 

Then ^x8x49x 31416 = 410-5024. 
o 

Thus the volume is 410"5024 cubic inches. 

270. It will be seen that in the second example of the 
preceding Article we have adopted a peculiar arrangement, 
with the view of saving labour in multiplication. The same 
arrangement may always be used with advantage in finding 
the volume of the frustum of a cone, when the radii of the 
ends are known. In fact, in such a case we may, instead 
of the rule in Art. 268, adopt the following, which is sub- 




156 FRUSTUM OF A PYRAMID OR CONE. 

staniially the same, but more convenient in form : add the 
squares of the radii of the ends to the product of the 
radii : multiply the sum hy the height^ and this product 
"by 31416 ; one-third of the result will be the volume. 

271. "We will now solve some exercises. 

(1) The radii of the ends of a frustum of a right circu- 
lar cone are 7 inches and 10 inches respectively ; and the 
slant height of the frustum is 5 inches. 

"We must determine 
the height of the frus- 
tum. 

Let the diagram re- 
present a section of the 
frustum made by a plane, 
containing the axis of the 
cone. "We see that the slant height is the hypotenuse of a 
right-angled triangle, of which one side is the height of the 
frustum, and the other side is the difference of the radii of 
the ends. 

In the present case the slant height is 5 inches, and the 
difference of the radii of the ends is 3 inches ; therefore, 
by Art. 60, the height of the frustum is 4 inches. 

1^'ow 7x7 = 49, 10x10 = 100, 7x10 = 70; 
49 + 100 4- 70 = 219 ; ^ x 4 x 219 x 31416 = 917-3472. 

Thus the volume is 917*3472 cubic inches. 

(2) The ends of a frustum of a pyramid are equilateral 
triangles, the sides being 3 feet and 4 feet respectively; 
and the height is 9 feet. 

By Art. 206 the area of one en din square feet = 9 x '433...; 
and the area of the other end in square feet = 16 x '433... j 
the square root of the product of these numl3ers = 12 x '433 . . . 
Add these three results, and we obtain 37 x '433... 

Then - x 9 x 37 x -433... = 48-063. 

o 

Thus the volume is rather more than 48 cubic feet. 



EXAMPLES. XXVI. 157 



EXAMPLES. XXVI. 

Find in cubic feet the volumes of frustums of pyramids 
which have the following dimensions : 

1. Areas of ends 4*5 square feet and 12*5 square feet; 
height 1*5 feet. 

2. Areas of ends 4 square feet and 5 square feet ; 
height 2 feet 6 inches. 

3. Areas of ends 900 square inches and 6'5 E{\uare 
feet ; height 2 yards. 

4. Areas of ends 7°5 square feet and 8'25 square feet; 
height 6-125 feet. 

Find in cubic feet the volumes of frustums of cones 
which have the following dimensions : 

5. Radii of the ends 3 feet and 4 feet : height 5^ feet. 

6. Radii of the ends 4'5 feet and 5'4 feet; height 
6-5 feet. 

7. Radii of the ends 4"S feet and 6'4 feet; height 
7-2 feet. 

8. Radii of the ends 6-375 feet and 51 feet ; height 
10 feet. 

9. The slant side of the frustum of a right circular 
cone is 5 feet, and the radii of the ends are 7 feet and 
10 feet: find the volume. 

10. Find the cost of the fnistum of a right circular 
cone of marble at 24 shillings per cubic foot, the diameter 
of the greater end being 4 feet, of the smaller end 1| feet, 
and the length of the slant side 8 feet. 



158 . EXAMPLES. XX VL 

11. The ends of the frustum of a pyramid are equila- 
teral triangles, the lengths of the sides being 6 feet and 
7 feet respectively; the height of the frustum is 4 feet: 
find the volume. 

12. The ends of the frustum of a pyramid are squares, 
the lengths of the sides being 20 feet and 30 feet respect- 
ively : the length of the straight lino which joins the 
middle point of any side of one end with the middle point 
of the corresponding side of the other end is 13 feet : find 
the volume. 

13. The shaft of Pompey's pillar, which is situated 
near Alexandria in Egypt, is a single stone of granite ; the 
height is 90 feet, the diameter at one end is 9 feet, and at 
the other end 7 feet 6 inches : find the volume. 

14. The mast of a ship is 50 feet high ; the circumfer- 
ence at one end is 60 inches, and at the other 36 : find the 
number of cubic feet of Vvood. 

15. The radii of the ends of a frustum of a right circu- 
lar cone are 7 feet and 8 feet respectively ; and the height 
is 3 feet : find the volume of the cone from which the frus- 
tum was obtained. 

IQ. The radii of the ends of the frustum of a right 
circular cone are 7 feet and 8 feet respectively ; and the 
height is 3 feet: find the volumes of the two pieces ob- 
tained by cutting the frustum by a plane parallel to the 
ends and midway between them. 

17. The radii of the ends of a frustum of a right cir- 
cular cone are 7 feet and 8 feet respectively; and the 
height is 3 feet ; the frustum is cut into three, each one 
foot in height, by planes parallel to the ends: find the 
number of cubic inches in each of the pieces. 

18. The ends of the frustum of a pyramid are regular 
hexagons, the lengihs of the sides being 8 feet and 10 feet 
respectively ; the height of the frustum is 12 feet : find the 
volumes of the two pieces obtained by cutting the frustum 
by a plane parallel to the ends and midway between them. 



WEDGE. 159 



XXVII. WEDGE. 

272. To find the volume (]f a wedge, 

EuLB. Add the length of the edge 
to twice the length of the base; mul- 
tijyly the sum by the width of the 
base, and the product by the height 
of the wedge ; one-sixth of the result 
icill be the volume of the icedge. 

273. Examples : 

(1) The edge of a wedge is 12 inches j the length of 
the base is 16 inches, and the breadth is 7 inches ; the 
height of the wedge is 24 inches. 

12 + 16 + 16 = 44; ix44x 7 x24=1232. 

o 

Thus the volume of the wedge is 1232 cubic inches. 

(2) The edge of a wedge is 6\ inches ; the length of 
the base is 3 inches, and the breadth is 2 inches ; the 
height of the wedge is 4 inches. 

5i + 3 + 3 = lli = f ; ^x|x2x4==f=^15i 

Thus the volume of the wedge is 15| cubic inches. 

274. If the edge of a wedge be equal to the length 
of the base, the wedge is a triangular prism; so that 
we are thus furnished with another Rule for finding the 
volume of such a prism. This Rule is in general not 
identical with that given in Art. 246, because the dimen- 
sions which are supposed to be known are not the same in 
the two cases. If the prism be a right prism, it is easy to 
see that the two Rules are really identical. 



IGO 



WEDGE. 



275. If the edge of a wedge be shorter than the length 
of the base, the wedge can be divided into an oblique 
triangular prism and a pyramid on a rectangular base, by 
drawing a plane through one end of the edge parallel to 
the triangular face at the other end. And, in like manner, 
if the edge be longer than the length of the base, the 
wedge is equal to the excess of a certain triangular prism 
over a certain pyramid on a rectangular base. It is by this 
consideration that the Rule of Art. 272 can be shewn to 
be true. 

276. It has been proposed to extend the meaning of 
the word wedge to the case in which the base instead of 
being a rectangle is a parallelogi-am or a trapezoid. The 
Rule for finding the volume will still hold, provided we 
understand the length of the base to be half the sum of the 
parallel sides, and the breadth of the base to be the per- 
pendicular distance between the parallel sides. 

277. Suppose a solid has 
been obtained by cutting a right 
triangular prism by a plane, in- 
clined to the length of the prism, 
which does not meet the base x) 
of the prism. 



The 
may be 
Rule: 



solid 



volume of this 
found by the following 




MuUqjIy the area of the 
hase of the prism by one-third 
of the sum of the parallel edges 
of the solid. 

278. The solid considered in the preceding Article is 
a wedge in the enlarged meaning of the word noticed in 
Art. 276 : the Rule for finding the volume may be demon- 
strated by a method similar to that explained in Art. 275. 
For if through E, the upper end of the shortest of the 
three edges, we draw a plane parallel to the base ABC, 
we divide the solid into a right prism and a pyramid ; the 



WEDGE. 



161 




volumes of each of these can be obtained by known Rules, 
and it will be found that the sum of the two volumes will 
agree with that assigned by the Rule of Art. 277. 

279. Suppose a solid has 
been obtained by cutting a right 
prism having a parallelogram 
for base by a plane, inclined to 
the length of the j)rism, which 
does not meet the base of the 
prism. The volume of this 
solid may be found by the fol- 
lowing R,ule : 

Multiply the area of the 
base of the j^rlsm hy one-fourth 
of the sum of the four par- 
allel edges of the solid. 

280. It is easy to see that AE+CG = BF+DH -, for 
each of these is equal to twice the distance between the 
point of intersection oi AG and BD and the point of inter- 
section of EG and FH ', and thus the Rule of the pre- 
ceding Article might be given in this form : 

Multiply the area of the "base of the prism hy half the 
sum of tico opposite edges out of the four parallel edges. 

281. The Rule of Art. 280 follows from that of Art. 
277. For if we cut the solid into two pieces by a plane 
which passes through AE and CG^ the Rule of Art. 277 
mil determine the volume of each piece; and it will be 
found that the sum of the two volumes will agi-ce with 
that assigned by the Rule of Art. 280. 

The Rules of Arts. 277 and 279 may be extended in the 
same way as the Rule of Art. 256 is extended in Art. 259. 

282. Wo will now solve some exercises. 

(1) The edge of a wedge is 18 inches; the length of 
the base is 20 inches ; the area of a section of the wedge 
made by a plane perpendicular to the edge is 150 square 
inches : find the volume. 



T. M. 



11 



1G2 WEDGE. 

The section made by a plane perpendicular to the edge 
is a triangle ; therefore the product of the base of this 
triangle into its height is 2 x 150, that is, 300 : this product 
is the same as that of the breadth of the wedge into the 
height of the wedge. 

18 + 20 + 20 = 58: 7x58x300 = 2900. 

D 

Thus the volume is 2900 cubic inches. 

The result can be obtained more readily by the Rule of 

Art 277. 

18 + 20 + 20 = 58; ^x 58 x 150 = 2900. 

o 

(2) The edge of a wedge is 16 inches ; the length of 
the base is 24 inches, and the breadth is 6 inches ; and the 
height of the wedge is 10 inches. The wedge is divided 
into a pyramid and a prism by a plane through one end of 
the edge parallel to the triangular face at the other end. 
I'ind the volume of each part. 

The length of the base of the pyramid is 24- 16 inches, 
that is, 8 inches ; hence, by Art. 263, the volume of the 

pyramid in cubic inches = - x8x6xl0 = 160. 

The prism has three parallel edges, each 16 inches 
long ; and, by Art. 274, its volume in cubic inches 

= ^xl6x6xl0 = 480. 



EXAMPLES. XXVII. 

1. The QdigQ of a wedge is 2 feet 3 inches ; the length 
of the base 2 feet 3 inches, and the breadth is 8 inches; the 
height of the wedge is 15 inches : find the volume. 

2. The edge of a wedge is 9 feet ; the length of the 
base is 6 feet, and the breadth is 3 feet ; the height of the 
w^dge is 2 feet : find the volume. 



EXAMPLES. XXVIh 163 

3. The base of a wedge is a square, a side of which is 
15 inches ; the edge is 24 inches, and the height of the 
wedge is 24 inches : find the volume. 

4. The base of a prism is an equilateral triangle, each 
side of whidi is 4 inches: find the volume of the solid 
obtained by cutting off a piece of this prism, so that the 
sum of the three parallel edges is 15 inches. 

5. The base of a prism is a rectangle which measures 
7 inches by 8 : find the volume of the solid obtained by 
cutting off a piece of this prismj so that the sum of the 
four parallel edges is 42 inches. 

6. The edge of a wedge is 21 inches; the length of 
the base is 27 inches ; the area of a section of the wedge 
made by a plane perpendicular to the edge is 160 square 
inches : find the volume. 

7. The edge of a wedge is 25 inches ; the length of the 
base is 22 inches ; a section of the wedge made by a plane 
perpendicular to the edge is an equilateral triangle, each 
side of which is 10 inches : find the volume. 

8. The edge of a wedge is 15 inches ; the length of the 
base is 24 inches, and the breadth is 7 inches ; the height of 
the wedge is 22 inches ; the wedge is divided into a pyra- 
mid and a prism by a plane through one end of the edge 
parallel to the triangular face at the other end : find the 
volume of each part. 

9. The edge of a wedge is 2 feet 3 inches ; the length 
of the base is 2 feet 9 inches, and the breadth is 8 inches ; 
the height of the wedge is 14 inches ; the wedge is divided 
into two pieces by a plane which passes through a point in 
the edge, distant 18 inches from one end, and which ia 
parallel to the triangular face at that end : find the volume 
of each piece. 

10. The edge of a wedge is 36 inches; the length of 
the base is 27 inches, and the breadth 5 inches ; the height 
of the wedge is 12 inches. The wedge is divided by a plane, 
so that the sum of the three parallel edges in one part is , 
42 inches : find the volume of each part. 

. 11—2 




164 PRISMOID. 

XXVIII. PRISMOID. 

283. To find the volume of a Prismoid. 

Rule. Add together the areas 
of the two ends and four times 
the area of a section parallel to 
tJie two ends and midway heticeen 
them; multiply the sum hy the 
height, and one-sixth of the pro- 
duct will he the volmne. 

284. Examples. 

(1) The area of one end is 4 square feet, of the other 
end 9 square feet, and of the middle section 6 square feet; 
the heiglit is 2 feet. 

4 + 24+9 = 37, 7x37x2 = ^ = 12i 
6 o 

Thus the volume is 12J cubic feet. 

(2) The area of one end is 224 square inches, of the 
other end 216 square inches, and of the middle section 
221 square inches : the lieight is 18 inches. 

224 + 884 + 216 = 1324, - x 1324 x 18 = 3972. 

6 

Thus the volume is 3972 cubic inches. 

285. The demonstration of the Rule in Art. 283 de- 
pends on the fact that a prismoid can be divided into 
pyramids and wedg-es, some having their bases in one end 
of the prismoid, and some in the other, and all having the 
same height as the prismoid. 

286. Each side of tho middle section is equal to half 
the sum of the corresponding sides of the ends. Thus if 
the ends are rectangles of known dimensions, the area of 
the middle section can be easily found : for then four 
times the area of the middle section is equal to the area 



PBIS3I0ID. 1G5 

of a rectangle, having for eacli of its dimensions the sum 
of the corresponding dimensions of the ends. 

Each angle of the middle section is equal to the cor- 
responding angle at the ends. 

287. If the ends of a pvismoid are similar figures 
similarly situated, the pri!^moid is a frustum of a pyramid, 
and therefore the volume might be found by the llule of 
Art. 268. By comparing the two Rules, we infer that in 
this case, four times the area of the middle section is equal 
to the sum of the areas of tlic ends added to twice the 
square root of the jn'oduct of these areas. 

288. It has been proposed to exteud the meaning of 
the term prismoid so as to apply to cases in which the 
ends are not rectilineal figures. Accordingly, the follow- 
ing definition may be given : A prismoid has for its ends 
any tico parallel plane figures, and has its other houn- 
dary straight. By having the other boundary straight is 
meant that a straight line may be placed on the boundary 
at any point, so as to coincide with the surface from end 
to end. This definition will include a frustum of a cone, 
or either of the pieces obtained by cutting a frustum of a 
cone by a plane which meets both ends. 

The Rule in Art. 283 holds for this extended meaning 
of the term prismoid. 

289. Tlie Rule for finding the volume of a prismoid 
holds for many other soHds; but it would not be i^ossiblo 
to define these solids in an elementary manner: the ad- 
vanced student may consult the author's Integral Calcu- 
lus, Art. 192. 

290. We will now solve some exercises. 

(1) The ends of a prismoid are trapezoids with four 
parallel edges; the parallel sides of one end are 100 feet 
and 32 feet respectively, and tlio distance between them is 
28 feet; the corresponding dimensions of the other end 
are 80 feet, 30 feet, and 26 feet; the distance between the 
ends is 112 feet: find the volume. 



166 PRISMOID. 

The middle section is a trapezoid; one of the two 
parallel sides is half of 100 + 80 feet, that is, 90 feet; and 
the other is half of 32 + 30 feet, that is, 31 feet : the dis- 
tance between these two parallel sides is half the sum of 
the corresponding distances for the two ends, that is, half 
of 28 + 26 feet, that is, 27 feet. 

1^9 

The area of one end in square feet=--^ x 28 = 1848 ; 

the area of the other end in square feet = -— x 26 = 1430 ; 
the area of the middle section in square feet ^ 

121 

• = — x27 = 1633i; 

four times this area = 6534; 

1848 + 1430 + 6534 = 9812 ; \ x 9812 x 112 = 183157^. 

D "^ 

Thus the volume is 183157^ cubic feet. 

(2) The Q^gQ of a wedge is 21 inches ; the length of 
the base is 15 inches, and the breadth 9 inches ; the height 
of the wedge is 6 inches ; the wedge is divided into three 
parts of equal heights by planes parallel to the base : find 
the volume of each f>art. 

The parts are two prismoids and a wedge ; the height 
of each part is 2 inches. 

The first prismoid has one end a rectangle which mea- 
sures 15 inches by 9 ; it will be found that the other end 
is a rectangle which measures in the corresponding manner 
17 inches by 6. The volume, by Art. 283, is 239 cubic 
inches. 

The second prismoid has one end a rectangle which 
measures 17 inches by 6; it will be found tliat the other 
end is a rectangle ^vhich measures in the corresponding 
manner ^ 19 inches by 3. The volume, by Art. 283, is 
161 cubic inches. 

The edge of the wedge is 21 inches ; the length of the 
base is 19 inches, and the breadth 3 inches. The volume, 
by Art. 272, is 59 cubic inches. 



EXAMPLES, XXVIII. 107 

The sum of the three volumes in cubic inches is 
239 + 161 + 59, that is, 459: it will be found that this is 
equal to the volume of the original wedge, as of coui'so 
it should be. 



EXAMPLES. XXVIII. 

1. Find the number of cubic feet which must be re- 
moved to form a prismoidal cavity ; the depth is 12 fect^ 
and the top and the bottom are rectangles, the corre- 
sponding dimensions of which are 400 feet by 180, and 
350 feet by 150. 

2. Find the number of cubic feet which must be re- 
moved to form a prismoidal cavity; the depth is 12 feet, 
and the top and the bottom are rectangles, the coire- 
sponding dimensions of which are 400 feet by 180 feet, 
and 150 feet by 350 feet. 

3. Find the volume of a coal waggon the depth of 
which is 47 inches; the top and the bottom are rectangles, 
the corresponding dimensions of which are 81 inches by 
54 inches, and 42 inches by 30 inches. 

4. Find the number of gallons of water required to 
fill a canal the depth of wliich is 4| feet, and the top and 
the bottom of which are rectangles, the corresponding di* 
mensions of which are 250 feet by 16 feet, and 240 feet by 
14 feet. 

5. Find the number of cubic feet which must be re- 
moved to form a railway cutting in the form of a pris- 
moidal cavity ; the ends are trapezoids with four parallel 
edges; the parallel sides of one end are 144 feet and 
36 feet, and the distance between them is 36 feev : the 
corresponding dimensions of the other end are 108 feet, 
36 feet, and 24 feet: the distance between the ends is 
137i feet. 

6. The ends of a prismoid are rectangles, the corre- 
sponding dimensions of which are 12 feet by 10 feet and 



168 EXAMPLES. XXVIII. 

8 feet by 6 feet ; the height of the prismoicl is 4 feet : the 
prismoid is divided by a plane parallel to the ends and 
midway between them : find the volume of each part. 

7. The ends of a prismoid are rectangles, the corre- 
sponding dimensions of which are 16 feet by 11 feet, and 
10 feet by 8 feet; the height of the prismoid is 9 feet: the 
prismoid is divided into three parts, each 3 feet high, by 
planes parallel to the ends : find the volume of each of the 
parts. 

8. The ends of a prismoid are rectangles, the corre- 
sponding dimensions of which are 20 feet by 16 feet, and 
14 feet by 12 feet; the height of the prismoid is 5 feet: the 
prismoid is cut into two wedges by a plane which passes 
through one of the longer sides of one end, and the op- 
posite longer side of the other end: find the volume of 
each part. 

9. The edge of a wedge is 24 inches, the length of the 
base 8 inches, and the breadth is 7 inches; the height of the 
wedge is 16 inches; the wedge is divided into tv/o parts 
by a plane parallel to the base midway between the edge 
and the base : find the volume of each part. 

10. The edge of a wedge is 27 inches ; the length of 
the base is 18 inches, and the breadth is 15 inches; the 
height of the wedge is 12 inches; the wedge is divided into 
three parts of equal height by two planes parallel to the 
base : find the volume of each part. 

11. The ends of a prismoid are rectangles, the corre- 
sponding dimensions of which are 18 feet by 10 feet, and 
12 feet by 16 feet; the height of the prismoid is 9 feet ; a 
section is made by a plane parallel to the ends at the dis- 
tance of 3 feet from the larger end: shew that the section 
is a square. 

12. Find the volumes of the two parts in the preced- 
mg Example. 



SPHERE. 169 



XXIX. SPHERE. 

291. To find the volume of a Siihere. 

Rule. Multiply the cube of the diameter ty one-sixth 
0/3-1416, that is, hy '5236. 

292. Examples. 

(1) The diameter of a sphere is 10 inches. 
Tiie cube of 10 is 1000 j '5236 x 1000 = 523*6. 

Thus the vohime of the spliere is about 5236 cubic inches. 

(2) The diameter of a sphere is 3^ feet. 

Tiie cube of 3*5 is 42-875; 42-875 x '5236 = 22-44935. 

Thus the volume of the sphere is very nearly 22*45 
cubic feet. 

293. The volume of a spherical shell will of course bo 
obtained by subtracting the volume of a sphere having its 
diameter equal to the inner diameter of the shell from the 
volume of a sphere having its diameter equal to the outer 
diameter of the shell Thus we obtain the Rule which 
will now be given. 

294. To find the volume of a spherical shell. 

Rule. Subtract the cube of the imier diameter from 
the cube of the outer diameter, and multiply the result 
hy -5236. 

295. Examples. 

(1) The outer diameter of a spherical shell is 9 inches, 
and the thickness of the shell is 1 inch. 

Here the inner diameter will be 7 inches. 

The cube of 9 is 729 ; the cube of 7 is 343 j 729 ~ 343 = 386 ; 
•5236x386 = 202-1096. 

Thus the volume of the shell is very nearly 202*11 cubic 
inches. 



170 SPHERE. 

(2) The inner diameter of a spherical shell is 10 inches, 
and the thickness of the shell is 1^ inches. 

Here the outer diameter will be 13 inches. 

The cube of 13 is 2197; the cube of 10 is 1000; 
2197 - 1000 = 1197 ; -5236 x 1197 = 6267492. 

Thus the volume of the shell is very nearly 626'75 cubic 
inches. 

296. If one sphere fall entirely within the other, it is 
obvious that the Rule of Art. 294 will give the volume of 
the space between the surfaces of the two spheres, even 
when the spheres are not concentric. 

297. "We will now solve some exercises. 

(1) The circumference of a gi-eat circle of a sphere is 
28 inches : find the volume of the sphere. 

We first determine the diameter of the sphere; by 
Art. Ill this will be about 8'9 inches: then by Art. 291 w^e 
shall obtain for the volume of the sphere about 369'12 
cubic inches. 

(2) Find the weight of a leaden ball 5 inches in dia- 
meter, supposing a cubic inch of lead to weigh Q'Q ounces. 

The cube of 5 is 125; 125 x •5236 = 65-45. Thus the 
volume of the ball is 65'45 cubic inches; and therefore its 
weight in ounces = 65-45 x 6*6 = 431-97. 



'C5' 



(3) If a cubic inch of gold weighs 11' 194 ounces, find 
the diameter of a ball of gold which w^eighs iOOO ounces. 

The number of cubic inches in the ball will be tz—--, , 

11-194 

that is, about 89-334; this number then is equal to the 

product of the cube of the diameter into -5236. Thus to 

obtain the cube of the diameter we must divide 89-334 by 

•5236; the quotient will be found to be 170-615. The cube 

root of this number wdll be the diameter ; we shall find 

that this cube root is 5*546. Thus the diameter of the 

ball is about 5 '55 inches 



EXAMPLES, XXIX. 171 



EXxVMPLES. XXIX. 

Find the volumes of spheres having the following dia- 
meters : 

I. 11 inches. 2. 8 feet. 3. 24 feet. 4. 32-5 feet. 

Find to the nearest hundredth of a cubic foot the 
volumes of spheres having the following circumferences of 
great circles : 

5. 6 feet. 6. 8 feet. 7. 10 feet. 8. 12 feet. 

Find in cubic inches the volumes of spherical shells 
having the following dimensions : 

9. External diameter 5 inches, internal 4. 

10. External diameter 8 inches, internal 6. 

II. External diameter 10 inches, internal 7. 

12. External diameter 16 inches, internal 12. 

13. Find how many gallons a hemispherical bowl, 2 feet 
4 inches in diameter, will hold. 

14. Find how long it will take to fill a hemisplierioal 
tank of 10 feet diameter, from a cistern which supplies by 
a pipe 6 gallons of water per minute. 

15. A solid is in the form of a right circular cylinder 
•with hemispherical ends; the extreme length is 29 feet 
and the diameter is 3 feet: find the volume. 

16. A solid is in the form of a right circular cylinder 
with hemispherical ends; the extreme length is 22 feet 
and the diameter is 2 feet 6 inches : find what will bo the 
weight of water equal in bulk to this solid. 

17. A sphere, 4^ inches in diameter, is cut out of a 
cube of wood, the edge of which is 4^ inches : find the 
quantity of wood which is cut away. 

18. Find the weight of a spherical shot of iron, 6 inches 
in diameter, supposing a cubic inch of iron to weigh 4*2 
ounces. 



172 EXAMPLES. XXIX. 

19. If a sphere of lead, 4 inches in diameter, weighs 
221*3 ounces, find the weight of a sphere of lead 5 inches 
in diameter. 

20. Find the weight of gimpowder required to fill a 
hollow sphere of 7 inches diameter, supposing that 30 cubic 
inches of gunpowder weigh one lb. 

21. Find the weight of gimpowder required to fill a 
hollow sphere of 9 inches diameter. 

22. Find the weight of a spherical shell one inch 
thick, the external diameter of which is 10 inclies, com- 
posed of a substance a cubic foot of which weighs 216 lbs. 

23. Find the weight of a spherical shell l| inches 
thick, the external diameter of which is 11 inches, com- 
posed of iron weighing 4 cwt. to the cubic foot. 

24. The external diameter of a shell is 8*4 inches and 
the internal diameter is 7'2 inches: find the weight of the 
shell if it is composed of a substance of which a cubic foot 
weighs 7860 ounces. 

25. Find the weight of a shell If inches thick, the ex- 
ternal diameter of which is 13 inches, composed of metal a 
cubic inch of which weighs 4*4 ounces. 

26. Find the weight of a shell 3i inches thick, the ex- 
ternal diameter of which is 1 foot 5j inches, composed of 
metal a cubic foot of which weighs 480 lbs. 

27. If an iron ball, 4 inches in diameter, weigh 9 lbs., 
find the weight of an iron shell 2 inches thick, whose exter- 
nal diameter is 20 inches. 

28. If a shell, the external and internal diameters of 
which are 5 inches and ^3 inches, weighs 85^ lbs., find the 
weight of a shell composed of the same substance, the ex- 
ternal and internal diameters of which are 7^ inches and 
4^ inches. 

29. Shew that the weight of a cone, 7 inches high on a 
circular base, of which the radius is 2 inches, is equal to 
that of a spherical shell of the same material, of which the 
external diameter is 4 inches and the thickness is 1 inch. 

30. Find the weight of a pyramid of iron, such that 
its height is 8 inches and its base is an equilateral triangle, 
each side being 2 inches, supposing a ball of iron 4 inches 
in diameter to weigh 9 lbs. 



'£3' 



EXAMPLES. XXIX. 173 

31. The radius of the base of a cone is 4 inches: find 
the height, so that the volume may be equal to that of a 
sphere with diameter 4 inches. 

32. The height of a cone is 12 inches: find the radius 
of its base, so that the volume may be equivalent to that 
of a sphere with diameter 6 inches. 

33. The circumference of the base of a cone is 32 feet: 
find the heiglit so that the volume may be equivalent to 
that of a sphere with diameter 10 feet. 

34. A solid is composed of a hemisphere and a cone 
on opposite sides of the same circular base ; the diameter 
of this base is 5 feet, and the height of the cone is 5 feet : 
find the volume of the solid. 

35. Find how many times larger the Earth is than the 
Moon, taking the diameter of the Earth as 7900 miles, and 
that of the Moon as 2160 miles. 

The following examples involve the extraction of the 
cube root : 

3S. Find the length of a cube which shall be equiva- 
lent in volume to a sphere 20 inches in diameter. 

37. Find the diameter of a sphere which shall be equi- 
valent in volume to a cube 20 inches in length. 

38. Find the diameter of a sphere which shall be 
equivalent in volume to a cylinder, the radius of the base 
of which is 8 inches and the height 12 inches. 

89. If 30 cubic inches of gunpowder weigh one lb., 
find the internal diameter of a hollow sphere which will 
hold 15 lbs. 

40. If a leaden ball of one inch in diameter weigh xi-lb., 
find the diameter of a leaden ball which weighs 588 lbs. 

41. If a cubic inch of metal weigh 6'57 ounces, find 
the diameter of a sphere of the metal which weighs 22016 
ounces. 

42. A Stilton cheese is in the form of a cylinder, and 
a Dutch cheese in the form of a sphere. Determine the 
diameter of a Dutch cheese which weighs 9 lbs., when a 
Stilton cheese, 14 inches high and 8 inches in diameter, 
weighs 12 lbs. 



174 ZONE AND SEGMENT OF A SPHERE. 



XXX. ZONE AND SEGMENT OF A SPHERE. 



298. To find the volume of a zone of a sphere. 

Rule. To three times the sum of the squares of the 
radii of the tzoo ends, add trie square of the height ; mul- 
tiply the sum ty the height^ and the product hy '6236 : the 
result icill he the volume. 

299. Examples. 

(1) The radii of the ends arc 8 inches and 11 mches, 
and tlio height is 2 inches. 

64 + 121 = 185; 3x185 = 555; 555 + 4 = 559; 
659 x2x •5236 = 585-3848. 
Thus the Yolume is about 585 cubic inches. 

(2) The radius of each end is 20 inches, and the height 
is 9 inches. 

400 + 400 = 800; 3x800 = 2400; 2400 + 81 = 2481; 

2481 X 9 X -5236 = 11691-4644. 

Thus the volume is nearly 11 691 "5 cubic inches. 

300. The Rule given in Art. 298 will serve to find the 
volume of a segment of a sphere, if we remember that the 
radius of one end of a segment is nothing ; but it may be 
convenient to state the Rul-e for this case explicitly. 

301. To find the volume of a segment of a sphere. 

Rule. To three times the square of the radius of the 
base add tJie square of the height ; m,ultiply the sum by 
the height, and the product by '5236 : the residt will be 
the volume. 



ZONE AND SEGMENT OF A SPHERE. 175 

302. Examples. 

(1) The radius of the base is 5 inches^ and the heigh i 
3 inches. 

3x25 = 75; 75 + 9 = 84; 84x3 x -5236 = 131 9472. 
Thus the volume is nearly 132 cubic inches. 

(2) The diameter of the base of a segment is 3^ feet, 
and the height is 9 inches. 

3 3 9 

9 inches = - of a foot; the square of - = — ; 

7 7 49 

the radius of the base = - feet; the square of - = — ; 

4 4 16 

49_147. ^ 117_11^_§9. 
^16" 16 ' le"*" 16 ~ 16' ~ 4 ' 

3 39 

- X — X -5236 = 3-828825. Thus the volume is nearly 

4 4 , 

3"83 cubic feet. 

303. We will now solve some exercises. 

(1) The height of a segment of a sphere is 3 inches, 
and the diameter of the sphere is 14 inches: find the 
volume of the segment. 

"We must first determine the square of the radius of the 
base of the segment. Using the diagram of Art. 78 we 
have -EZ) = 3 inches, and A'F= 14 inches. By Art. 89 we 
shall find that the square of ^i) = 33. 

33x3 = 99; 99 + 9 = 108; 108 x 3 x •5236 = 169-6464. 

Thus the volume is about 170 cubic inches. 

(2) The radius of the base of a segment is 24 inches, 
and the radius of the sphere is 25 inches : find the volume. 

"We must first determine the height of the segment. 
Using the diagram of Art. 78, we have u4-D = 24 inches, 
and -4(7=25 inches. By Art. 60 we shall find that 
(7Z> = 7 inches. Therefore DE= 18 inches. The square 
of 24 = 576; 576x3 = 1728; the square of 18 = 324; 
1728 + 324 = 2052. 

2052 X 18 X -5236 = 19339-6896. 

Thus the volume is nearly 19340 cubic inches. 



176 EXAMPLES. XXX. 



EXAMPLES. XXX. 



1. The radii of the ends of a zone of a sphere are 

7 inches and 8 inches ; and the height is 3 inches : find 
the volume. 

2. The radii of the ends of a zone of a sphere are 

8 inches and 12 inches ; an.d the height is 6 inches : find 
the Yohime. 

3. The height of a segment of a sphere is 6 feet and 
the diameter of the base is 8 feet : find the vokime. 

^ 4. The height of a segment of a sphere is 2 feet 
8 inches and the diameter of the base is 8 feet : find the 
Yohime. 

5. The height of a segment of a sphere is 4 feet, and 
the diameter of the sphere is 12 feet : find the vohime. 

6. The height of a segment of a sphere is 5 feet and 
the diameter of the sphere is 15 feet; find the vohmie. 

7. The radius of the base of a segment of a sphere 
is 12 feet and the radius of the sphere is 13 feet : find the 
volume of the segment. 

8. The radius of the base of a segment of a sphere 
is 8 feet and the radius of the sphere is 17 feet: find the 
volume of the segment. 

9. The diameter of a sphere is 20 feet : find the 
volumes of the two segments into which the sphere is 
divided by a plane, the perpendicular distance of w^hich 
from the centre is 5 feet. 

10. The diameter of a sphere is IS feet : the sphere is 
divided into two segments, one of which is twice as high 
as the other : find the volume of each. 



EXAMPLES. XXX. 177 

11. The radius of the base of a segment of a sphere 
is 1 inch, and the radius of the sphere is 2j inches : find 
the vokime of the segment. 

12. Find the weight of an iron dumb-bell, consisting 
of two spheres of 4^ inches diameter, joined by a cylindrical 
bar, 6 inches long and 2 inches in diameter ; an iron bail 
4 inches in diameter weighing 9 lbs. 

13. The diameter of a sphere is 9 feet; the sphere is 
divided into three parts of equal height by two parallel 
planes : find the volume of each part. 

14. A sphere, 15 inches in diameter, is divided into 
four parts of equal height by three parallel planes : find 
the volume of each part. 

15. Find the volume of a zone of a sphere, supposing 
the ends to be on the same side of the centre of the sphere, 
and distant respectively 10 inches and 15 inches from the 
centre ; and the radius of the sphere to be 20 inches. 

16. Find the volume of a zone of a sphere, supposing 
the ends to be on opposite sides of the centre of the 
sphere, and distant respectively 10 inclics and 15 inches 
from the centre; and the radius of the sphere to bo 
20 inches. 

17. A bowl is in the shape of a segment of a sphere ; 
the depth of the bowl is 9 inches, and the diameter of the 
top of the bowl is 3 feet : find to the nearest pint the 
quantity of water the bowl will hold. 

IS. Verify by calculating various cases the following 
statement : if the height of a segment of a sphere is three- 
fourths of the radius of the sphere the volume of the seg- 
ment is three-fourths of the volume of a sphere which has 
its radius equal to tlie height of the segment. 



T. M. 12 



178 IRREGULAR SOLIDS. 



XXXI. IRREGULAR SOLIDS, 

304. We will now explain methods by which we may, 
in some cases, determine the volumes of solids that are not 
included in any of the Rules which have been given. 

305. Suppose the solid is one which w^iil sink in water, 
but will not be injured by water. 

Put the solid inside a vessel of convenient shape, such 
as a rectangular parallelepiped or a cylinder. Pour water 
into the vessel until the solid is quite covered ; and note 
the level at which the water stands. Remove the solid 
and note the level at which the water then stands. The 
volume of the solid is of course equal to the volume of the 
water which would be contained in the vessel between the 
tw^o levels j and this can be easily calculated. 

Or we might state the process thus : fill the vessel full 
of water ; put the solid into it gently and measure the 
volume of the water which runs over. 

306. If the solid is composed entirely of the same 
substance we may estimate its volume by means of its 
weight thus : weigh the soKd, also weigh a cubic inch of 
the same substance as the solid ; divide the weight of the 
solid by the weight of the cubic inch, and the quotient ^aIII 
be the number of cubic inches in the solid. If instead of 
ascertaining the weight of a cubic inch of the substance we 
ascertain the weight of any known volume of the substance, 
vv'e can determine by a proportion the volume of the pro- 
posed solid. 

Some examples of the principle that volumes of solids 
of the same substance are in the same proportion as their 
weights, have been given at the end of Chap. xxix. 

307. A Rule resembling that given in Chapter xyiii. 
may be used for finding approximately the volumes of 
certain solids : 



EXAMPLES. XXXI. 179 

Divide the length of the solid into any even nuinhcr 
of equal parts; and ascertain the areas of sections of the 
solid through the points of division perpendicular to tlie 
length of the solid. Add together the first area, the last 
area, ticice the sum of all the other odd areas, and four 
times the sum of all the even areas ; multiply the sum hy 
one-third of the common distance between ttco adjacent 
sections. 

308. Tlie preceding Rule will in general be more ac- 
curate the greater the number of sections that are made ; 
and the rule ought not to be trusted if the solid be very 
irregular in form. The areas required in the Rule may 
themselves be conveniently determined approximately, in 
some cases, by Chapter xviii. The Rule is employed by 
Civil Engineers for calculating quantities of earthwork, and 
by Naval Arclritects for calculating the volumes of water 
displaced by ships. To ensure accuracy, Naval Architects 
often perform the calculation in two vrays ; nainely, from a 
series of horizontal sections, and from a series of trans- 
verse sections. 

EXAMPLES. XXXI. 

1. The radius of the base of a cylindrical vessel is 
10 inches; a block of stone is placed in the vessel and 

.is covered with water; on removing the block the level 
of the water sinks 6 inches : find the volume of the block. 

2. If a cubic foot of marble weighs 2716 ounces, find 
the volume of a block of marble which weighs 4 tons 8 cwt. 

3. A cask full of water weighs 3 cwt. ; the cask when 
empty weighs 40 lbs. : find to the nearest gallon the capa- 
city of the cask. 

4. Five equidistant sections of a solid are taken, the 
common distance being 3 feet ; the areas of these sections 
in square feet are 3*72, 5-28, 6-96, 8-77, and 10-72 : find the 
volume of the solid between the extreme sections. 

5. Five equidistant sections of a solid are taken, the 
common distance being 6 inches ; these sections are all 
circles, and their circumferences are respectively 57 inches, 
63 inches, 69 inches, 76 inches, 83 inches : find the volume 
of the solid between the extreme sections. 

12—2 



180 SIMILAR SOLIDS. 



XXXII. SIMILAR SOLIDS. 

809. Similar solids aro such as are alike in form 
tliougli they may differ in size. 

In common language the fact that one solid is similar 
to another is often expressed by saying that one is a 
model of the other. 

310. All cubes are similar solids. All spheres ai'e 
similar solids. 

311. It is easy in various cases to give tests for deter- 
mining whether two solids, which are called by the same 
name, are similar. For example : 

If the three edges of one rectangular parallelepiped 
which meet at a point are respectively double, or treble, or 
any number of times, the three edges of another which 
meet at a j)oint, the two rectai^gular parallelepipeds are 
similar. 

If the height and the diameter of the base of one 
right circular cone are respectively double, or treble, or 
any number of times, the height and the diameter of the 
base of another, the two riglit circular cones are similar. 
The same test will serve for two right circular cylinders. 

"We may express these statements thus : Two rectan- 
gular parallelepipeds are similar when their edges are pro- 
portionals. Two right circular cones, or two right circular 
cylinders, are similar when their heights and the diameters 
of their bases are proportionals. 

312. The following most important proposition holds 
with respect to similar solids : 

The volumes of similar solids are as the cubes of cor- 
responding lengths : 



SIMILAR SOLIDS. 181 

For example, suppose the diameter of one spnerc to be 
5 inches, and the diameter of anoth.er sphere to be 4 mches ; 
the vohime of the first sphere is to the volume of the second 
as the cube of 5 is to the cube of 4, that is, as 125 is to 64: 
so that the larger sphere is almost double the other. 

Persons "who have not had their attention drawn to 
such a fact, often find a difficulty in reahzing the rapid 
rate at which the volumes of similar solids increase, as 
some assigned dimension of them is increased. 

313. We will now solve some exercises. 

(1) The edge of a cube is 1 foot : find the number of 
feet in the edge of another cube of double tlio volume. 

The cube of the required number is to the cube of 1 as 

2 is to 1 ; so that the required number is the cube root 
of 2 : this will be found to be 1-2599210. Thus we see that 
a cube with its edge 12 6 feet is rather more than double 
a cube with its edge 1 foot. 

(2) The height of a pyramid is 12 feet : it is required 
to cut off a frustum which shall be a fourth of the pyra?.aid. 

Smce the frustum is to bo a fourth of the pyramid, the 
remaining pyramid will be three-fourths of the original 
pyramid ; and these two pyramids are similar. Therefore 
the cube of the height of the remaining pyramid must 
be three-fourths of the cube of the height of the original 
pyramid, that is, | of 1728, that is, 1296; thus the height 
in feet of the remaining pjTamid is the cube root of 1296 ; 
it will be found that this is 10*9027. 

Hence the height of the frustum in feet is 12 — 10'9027, 
that is, 1'0973. 

(3) The diameters of the ends of a frustum of a cone 
are 6 feet and 10 feet, and the height of the frustum is 

3 feet : it is required to divide the frustum into two equal 
parts by a plane parallel to the base. 



182 SIMILAR SOLIDS. 

In the diagram of Art. 210, let AB and CD be the 
diameters of the ends. As in that Article we find 
0K=A'5, and 0M=*J'5. Let OL denote the perpendicular 
distance of the required plane from O. Thus we shall find 
that the cube of OL must be equal to half the sum of the 
cubes of OK and 03f. The cube of 4-5 = 91-125 ; the cube 
of 7-5 = 421-875. Thus the cube of 0L = ^ of 513 = 256-5 ; 
and tlierefore the number of feet in 02/ = the cube root 
of 25G-5 ; it will be found that this is 6-3537. 

Hence the distance of the required plane from the 
smaller end of the frustum in feet = 6-3537 - 4-5 = 1-8537. 

(4) A frustum of a circular cone is trimmed just 
enough to reduce it to a frustum of a pyramid with square 
ends : find how much of the volume is removed. 

This Exercise may be conveniently placed here although 
not strictly connected with the subject of similar solids. 

Suppose that the radius of one end is 2 feet, and the 
radius of the other end 3 feet ; and that the height is 
12 feet. By Art. 268 the volume of the frustum of the cone 

in cubic feet = J x 19 x 3-1416 x 12 = 238-7616. When the 

o 

frustum of the cone is trimmed the ends become squares, 
the diagonals of which are 4 feet and 6 feet respectively : 
by Art. 268 the volume of the frustum of a pyramid in 

cubic feet =- x 38 x 12 = 152. The volume removed is 

therefore 86-7616 cubic feet; and the fractional part of the 

original volume is ^„„ ^^,^ , that is '36...; thus rather 
° 238-7616' 

9 
more than — of the original volume is removed. 

Now on examining this process it will be immediately 
seen that the result will be the same whatever be the 
height of the frustum; and by trial it will be found that 
the result will also be the same whatever be the radii of 
the ends of the frustum of a cone. 

Thus the result is true for any frustum of a right circu- 
lar cone. 



EXAMPLES. XXXII. 183 



EXAMPLES. XXXII. 

1. If i cannon ball, 3 J inches in diameter, weigh 6 lbs., 
find the weight of a balf of the same metal 5^ inches in 
diameter. 

2. If tie model of a steam engine weigh 80 lbs., find 
the weight of the engine itself, supposing it made of the 
same substcnce as the model and of nine times its lineal 
dimensions. 

3. The leights of two similar right circular cylinders 
are 7 inches and 10 inches respectively : shew that a simi- 
lar cylinder 11 '03 inches high is less than the sum of the 
two, and a emilar cylinder 11 '04 inches is greater than the 
sum of the tvo. 

4. The leight of a pyramid is 16 inches, and its volume 
is 400 cubicinches : the'pyramid is divided into two parts 
by a plane )arallel to the base and distant 4 inches from 
it : find thevolumes of the parts. 

The ne* four examples involve the extraction of the 
cube root : 

5. If f cannon ball, 3^ inches in diameter, weigh 6 lbs., 
find the dimeter of a ball of the same metal which weighs 
20 lbs. 

6. Tb height of a right circular cylinder is 4 feet : 
find the eight of a similar cyhnder of nine times the 
volume. 

7. Tb diameters of the ends of the frustum of a cone 
are respetively 20 feet and 16 feet, and the height of the 
frustum 5 5 feet; the frustum is divided into two equal 
parts by- plane parallel to the ends : find the distance of 
the plant from the smaller end. 

8. I the frustum in the preceding Example is di\adcd 
into tln^e equal parts by planes parallel to the ends, find 
the distnces of tlie planes from the smaller end. 



181 



EXAMPLES. XXXII. 



9. A pyramid on a regular hexagonal base is trimmed 
just enough to reduce it to a cone : shew that rather less 

than — of the original volume is removed. 

10. A frustum of a pyramid on a square base is trimmed 
just enough to reduce it to a frustum of a Jone : shew 

that rather more than - of the original volumeis removed. 

o 

11. Every edge of a pyramid on a square kse is 1 foot : 

shew that the volume of the pyramid is ^ of i cubic foot ; 

6 

and that the volume of any pyramid on a squan base which 

has all its edges equal may be obtained by miitiplying the 

cube of an edge by —■ . 



12. Every edge of a pyi-amid on a trian^ilar base is 

1 foot: shew that the volume of the pyramid is -^ of a 

cubic foot ; and that the volume of any pyramd on a tri- 
angular base which has all its edges equal mayje obtained 

by multiplying the cube of an edge by —- . 



185 



FIFTH SECTION. AREAS OF TEE 
SURFACES OF SOLIDS. 

XXXIII. PLANE SURFACES. 

314. Wb now proceed to the measurement of the 
surfaces of solids; this subject is properly connected with 
the second Section of our work, but we have thought it 
more convenient for beginners to treat of the volumes of 
solids before treating of the areas of the surfaces of sohds. 

315. The area of any plane surface of a solid must 
be found by the Rules given in the third Section of the 
work. We will mention the various cases that can arise. 

The faces of a rectangular parallelepiped are all rect- 
angles; the faces of any other parallelepiped are pai-al- 
lelograms, two or four of which may be rectangles. The 
ends of a prism are triangles or other rectilineal figures ; 
the other faces are rectangles or parallelograms accord- 
ing as the prism is right or oblique. The base of a 
pyramid is a triangle or other rectilineal figure, and the 
other faces are triangles. The ends of a prismoid, or of 
a finistum of a pyramid, are triangles or other rectihneal 
figures, and the other faces are trapezoids. Two of the 
faces of a wedge are triangles ; each of the other three 
faces is a trapezoid, or a parallelogram, or a rectangle. 
In all these cases the surfaces are plane rectilineal figures, 
and their areas can be found by Rules already given. 

A Rule has also been given for finding the area of a 
circle ; and the following cases will occur for the applica- 
tion of the Rule: the ends of a circular cyhnder, the base 
of a circular cone, the ends of a friistmn of a circidar cone, 



186 PLANE SURFACES. 

the base of a segment of a sphere, aud the ends of a zone 
of a sphere. 

In fact any Rule given in the third Section of the 
book might find an apphcation in the present Section. 
Thus, for example, the Rule for finding the area of a 
segment of a circle will enable us to find the areas of 
the ends of the segments of cylinders which are considered 
in Art. 255. 

316. Examples: 

(1) Find the area of the whole surface of a cube which 
is 8 inches long. 

Each face of the cube is a square containing 64 square 
inches; and there are six faces; thus the area of the 
whole surface in square inches is 6 x 64, that is 384. 

(2) A pyramid stands on a square base which is 10 
inches long, and each of the four faces which meet at 
the vertex is an equilateral triangle : find the area of the 
whole surface of the pyramid. 

The area of the base is 100 square inches. The area 
of each of the triangular faces is about 43'3 square inches, 
by Art. 206; therefore the area of the four triangular 
faces is about 173*2 square inches. Thus the area of 
the whole surface of the pyramid is about 273*2 square 
inches. 

(3) A vessel is to be made in the form of a rectan- 
gular parallelepiped without a lid; externally the length 
is, 4 feet, the breadth 3 feet, and the height is 2 feet: find 
the area of the whole external surface. 

The surface consists of two rectangles each measuring 
4 feet by 2 feet, two rectangles each measuring 3 feet 
by 2 feet, and one rectangle measuring 4 feet by 3 feet: 
the total area is 40 square feet. 

Suppose that the vessel is to be formed of metal half 
an incli thick; and that we have to find how much metal 
is required. The answer for practical purposes is this: 
we shall require 40 square feet of the assigned thickness. 



PLANE SURFACES. 187 

And in this manner any such question is usually solved 
when the thickness of the metal is small compared with 
the dimensions of the vessel. 

The exact method of solution has been given in Art. 245, 

from which it appears that the exact result is 2814^ 

cubic inches. Now a sheet of metal of 40 square feet 

in area and half an inch thick wnll be found to contain 

28S0 cubic inches. Thus we see that the approximate 

result is slightly in excess of the exact result. The thinner 

the material of vrhich the vessel is composed, the smaller 

will be the difference between the approximate result and 

the true result. 

« 

(4) A vessel is to be made in the form of a rectangiilar 
parallelepiped on a square base without a lid, to hold a 
cubic foot; the height is to be half the length : find the 
area of the whole internal surface. 

A vessel of the same base but of twice the height 
would be in the shape of a cube, and would hold 2 cubic 
feet, that is 3456 cubic inches. Hence the length of a 
side of the base in inches will be the cube root of 3456 : 
thus we shall obtain for a side of the base 15"119... inches. 
Since the height is half the length, the area of the base 
is double that of any of the other four faces; and there- 
fore the area of the whole internal smface is three times 
that of the base; so that in square inches it is three 
times the square of 15'119...: it will be found that this 
is 685'75... 

Hence, as in the preceding Example, if the vessel is 
to be made of metal of an assigned small thickness we 
shall require about QSQ square inches of metal of that 
thickness. 

317. The principle illustrated in the last two Ex- 
amples may be thus stated: in order to construct a 
vessel of material having an assigned small thickness we 
require a sheet of the material equivalent to the ex- 
ternal surface of the vessel. Admitting this principle we 
can give an interesting practical form to some results of 
calculation. Thus, for instance, by comparing the results 
of Examples 37... 41 at the end of the present Chapter, 



188 EXAMPLES. XXXIII, 

and of similar Examples, we can see that the following 
theorem is true : a vessel of given cajmcity is to bo made 
in the form of a rectangular parallelepiped on a square 
base ; if there is to be no lid tlio internal surface will 
be least when the height is half thQ length. Thus, 
"vvitli the view of saving material, the most advanta- 
geous shape is that in which the height is half the 
length. 

In like manner from the results of Examples 42... 46, 
and of similar Examples, we learn that if there is to bo 
a lid the cube is the most advantageous shape. 

And suppose that we have to make a vessel in the 
form of a rectangular parallelepiped on a square base, 
out of a given quantity of material: then if there is 
to be no lid the capacity will be greatest when the height 
is half the length; and if there is to be a lid the capacity 
will be greatest when the vessel is a cube. 



EXAMPLES. XXXIII. 

Find the area of the whole surface of the cubes which 
have the following lengths : 

1. 2 feet 6 inches. 2. 3 feet 8 inches. 

3. 5 feet 10 inches. 4. 6 feet 7 inches. 

Find the area of the whole surfaces of rectangular paral- 
lelepipeds which have the following dimensions : 

5. 2 feet 6 inches, 3 feet, 5 feet. 

6. 2 feet 4 inches, 3 feet G inches, 4 feet. 

7. 2 feet 8 inches, 3 feet 2 inches, 4 feet 10 inches. 

8. 2 feet 11 inches, 3 feet 7 inches, 5 feet 2 inches. 

Find the area of the whole surface of light triangular 
prisms having the foilo^ving dimensions : 

9. Sides of the base, 3, 4, and 5 feet ; height 8 feet. 
10. Sides of the base, 8, 15, and 17 feet; height 10 feet. 



EXAMPLES. XXXIIL 189 

11. Sides of tho base 1 foot 4 inches, 2 feet 1 inch, 

3 feet 3 inches; height 7 feet 6 inches. 

12. Sides of the base 2 feet 1 inch, 2 feet 9 inches, 

4 feet 4 inches ; height S feet. 

13. Find the area of the whole surface of a pyramid 
on a square base; each side of the base is 2 feet 7 inches, 
and the length of the straight line drawn from the vertex to 
the middle point of any side of the base is 3 feet 5 inches. 

14. Find the area of the whole surface of a pyramid on 
a square base ; each side of the base is 3 feet 4 inches, and 
the length of the straight line drawn from tho vertex to 
the middle point of any side of the base is 5 feet 8 inches. 

15. Find the area of the whole surface of a pyramid 
on a square base; each side of the base is 3 feet 4 inches, 
and each of the other edges is 8 feet 5 inches. 

16. Find the area of the vrhole surface of a pyramid 
on a square base; each side of tho base is 28 feet, and each 
of the other edges is 16 feet 1 inch. 

17. Find the area of the whole surface of a pyramid 
on a square base, having its other faces equal ; each side of 
the base is 17 feet 6 inches, and the height of the pyramid 
is 17 feet 4 inches. 

18. Find the area of the whole surface of a pyramid 
on a square base, having its other faces equal ; each side 
of the base is 29 feet 2 inches, and the height of the pyra- 
mid is 24 feet. 

19. Find the area of the whole surface of a frustum of 
a pyramid; the ends are squares, the sides of which are 
2 feet and 3 feet respectively, and tlie distance between 
the parallel sides of each trapezoidal face is 6 inches. 

20. Find the area of the whole surface of a frustum of 
a pyramid ; the ends are squares, tho sides of which are 

2 feet 3 inches and 4 feet 9 inches respectively, and the 
distance between the parallel sides of each trapezoidal face 
is 18 inches. 

21. Find the area of the whole surface of a frustum of 
a pyramid; the ends are squares, the sides of which are 

3 feet 4 inches and 3 feet 10 inches respectively; and each 
of the remaining edges is 5 inclies. 



190 EXAMPLES. XXXIII. 

22. Find the area of the whole surface of a frustum of 
a pyramid; the ends are squares, the sides of which are 
3 feet 2 inches and 4 feet respectively; and each of the 
remaining edges is 13 inches. 

23. Find the area of the whole surface of a prismoid ; 
the ends are rectangles; one measures 7 feet by 6 feet, 
and the corresponding dimensions of the other are 4 feet 
6 inches and 4 feet 10 inches; each of the remaining 
edges is 25 inches. 

24. The four faces of a triangular pyramid are equi- 
lateral triangles, the edge of each being 10 feet : find the 
area of the whole surface. 

25. Find the area of the whole surface of a pyramid 
on a rectangular base which measures 4 feet 6 inches 
by 6 feet 8 inches, each of the remaining edges being 
6 feet. 

26. Tlie area of the whole surface of a cube is 7 
square feet 6 square inches : find the volume. 

27. The dimensions of a rectangular parallelepiped are 
3, 7, and 9 feet : find the edge of the cube of equivalent 
surface. 

28. The edge of a wedge is 12 inches, the length of the 
base is 10 inches, and its breadth 2 inches; eacli of tlie 
other sides of the trapezoidal faces is 25 inches: find the 
area of the whole surface. 

29. The edge of a wedge is 14 inches, the length of the 
base is 6 inches, and its breadth 2 inches; each of the 
other sides of the trapezoidal faces is 9 inches : find the 
area of the whole surface. 

30. Find the area of the whole surface of a right 
prism, the ends of which are regular hexagons, each edge 
of the solid being 2 feet. 

31. The base of a pyramid is a regular octagon, each 
side being 4 feet ; each of the other edges of the pyramid 
is 12 feet 1 inch: find the area of the surface excluding 
the base. 

32. The dimensions of a rectangular parallelepiped arc 
18 inches, 16 inches and 6 inches: find the area of the 
whole surface of a cube of equivalent volume. 



EXAMPLES. XXXIII. 191 

33. The length, breadth, and height of a rectangular 
parallelepiped are respectively 8, 18 and 21 inches : find 
its surface. Also find the surface of a rectangular paral- 
lelepiped of the same height and volume on a square 
base. 

If two rectangular parallelepipeds have the same height 
and volume, and one of them have a square base^ the 
whole surface of this will be less than the whole surface of 
the other : verify this statement by comparing the sur- 
faces of the following rectangular parallelepipeds with the 
surface of others having respectively the same volume, and 
height, and square bases : 

34. Base 3 feet by 4 feet; height 5 feet 

35. Base 3 feet by 7 feet ; height 9 feet. 

36. Base 8 feet by 15 feet; height 19 feet. 

The following examples involve the extraction of the 
cube root : 

A vessel without a lid in the form of a rectangular 
parallelepiped on a square base is to be made to hold 
1000 cubic inches : find in square inches the area of the 
whole external surface in the following cases : 

37. The height equal to the length. 

38. The height equal to twice the length. 

39. The height equal to half the length. 

40. The height equal to three times the length. 

41. The height equal to one- third of the length. 

A vessel with a Hd in the form of a rectangular paral- 
lelepiped on a square base is to be made to hold 1000 cubic 
inches : find in square inches the area of the whole internal 
surface in the folio vfing cases : 

42. The height equal to the length. 

43. The height equal to twice the length. 

44. The height equal to half the length. 

45. The height equal to three times the lengtli. 

46. The height equal to one-third of the length. 



192 RIGHT CIRCULAR CYLINDER. 



XXXIY. RIGHT CIRCULAR CYLINDER. 

318. The surface of a ridit circular cvlincler coiisists 
of two circular ends, and another portion which we shall 
call the curved surface. 

819. Let ABOD be a rect- 
angle. Cut it out of paper or 
cardboard. Then let it be bent — ' 

until the edge AB just comes 
into contact with the edge DC. 
It is easy to see that by proper 
adjustment, we can thus obtain 
a thin cylindrical shell; yli? be- 
comes the height of the slioll, 



and i>(7the circumference of the -^ ^ 

base. Hence it will follow that 

the curved surface of a right circular cylinder is equal to 
a rectangle one dimension of which is the height of the 
cylinder, and the other dimension the circumference of the 
base of the cylinder : thus we obtain the Rule whicli we 
shall now give. 

320. To find the area of the curved surface of a right 
circular cylinder. 

Rule. Ilidtlply the circumference of the hase by the 
height of the cylinder. 

321. Examples. 

(1) The radius of the ba,se of a right circular cylinder 
is 3 feet, and the height is 2\ feet : find the area of the 
curved surface. 

The circumference of the base in feet = 2 x3x3'1416 
= 18-8496; ^ x 18-8496 ==47124. 

Thus the area of the curved surface is about 47"i24 
square feet. 



RIGHT CIRCULAR CYLINDER. 19S 

(2) The diameter of the base of a right circular cylin- 
der is 16 inches, and the height is 25 inches : find the 
area of the whole surface. 

The circumference of the base in inches = 16 x3'1416 
= 50-2656; 25x50-2656 = 1256-64. 

Thus the area of the curved surface is about 1256-64 
square inches. 

The area of the two circular ends in square inches 
= 2 X 64 X 3-1416 = 402-1248, 

1256-64 + 402-1248 = 1658-7648. 

Therefore the area of the whole surface is about 
1658-7648 square inches. 

322. The following inferences may be easily drawn 
from the Rule of Art. 320 : 

If the height of the right circular cylinder be equal to 
the radius of the base, the area of the curved surface is 
equal to that of the two ends of the cylinder; if the height 
be twice the radius, the area of the curved surface is 
twice that of the ends ; if the height be three times the 
radius, the area of the curved surface is three times that 
of the ends : and so on. 

If the height be half the radius the area of the curved 
surface is half that of the ends ; if the height be a third of 
the radius the area of the curved surface is a third of 
that of the ends : and so on. 

We may sum up these inferences thus : the height of a 
right circular cylinder hem's the same proportio7i to the 
radius of the base as the area of the curved surface hears 
to the area of the two ends. 

323.^ By the process of Art. 319 we can easily form 
other right cylindrical shells the bases of which are not 
circles, but various oval curves : and thus we see that the 
area of the curved surface of any right cylinder may be 
found by multiplying the perimeter of the base by the 
height of the cylinder. 

T. M. 13 



194 RIGHT CIRCULAR CYLINDER. 

324. We will now solve some exercises. 

(1) The area of the Avhole surface of a right circular 
cylinder is 20 square feet, and the height of the cylinder 
is equal to half the radius of the base : find the radius of 
the base. 

By Art. 322 the area of the curved surface is equal to 
that of one of the ends ; and thus three times the area of 
an end is 20 square feet : therefore the area of one end = 6f 
square feet = 960 square inches. We must then find the 
radius of the base by Art. 171. Divide 960 by 3-1416 ; the 
quotient is 305'58; the square root of this is 17'48... 
Thus the radius of the base is 17"5 inches nearly. 

(2) The area of the curved surface of a right circular 
cylinder is 30 square feet, and the volume is 120 cubic 
feet : find the radius of the base, and the height of the 
cylinder. 

The product of the height into the area of the base is 
120 ; the product of the height into the circumference of 
the base is SO: hence, by division, the area of tlie base 
divided by the circumference of the base =120-^30 = 4. 
But, by Art. 176, the area of the base divided by the cir- 
cumference of the base is equal to half the radius : thus 
half the radius is 4 feet, and therefore the radius is 
8 feet. 

Hence the circumference of the base in feet = 16 x 31416 
= 50*2656 ; and therefore the height of the cylinder in 
feet = 30 4- 50-2656 = -5967... 

(3) A vessel is to be made in the form of a riglit 
circular cylinder, without a lid, to hold a cubic foot ; the 
height is to be equal to the radius of the base : find the 
area of the whole internal surface. 

Tlie volume is to be 1728 cubic inches. Proceeding as 

in Art. 253, we see that the height in inches will be the 

1728 
cube root of .,.. , ' . : thus we shall obtain for the height 

o i41u ° 

8-1934 inches. The area of the whole internal surf^ice is 
three times the area of one end ; so that in square inches 
it is three times the product of 3'1416 into the square of 
8-1934 ; it will be found that this is 632-70... 



EXAMPLES, XXXIV. 195 

From tliis result and tho last of those obtained in 
Art. 316 we see that less material is required for a vessel 
of the shape here considered tlian for a vessel of tho shape 
there considered. 

325. By comparing the results of Examples 21... 30 at 
the end of the present Chapter, and of similar examples, 
we learn that the most advantageous shape for a right 
circular cylindrical vessel, if there is to be no lid, is that 
in which the height is equal to the radius of the base ; and 
if there is to be a lid, that in which tho height is twice the 
radius of the base. For by adopting these shapes we use 
the least quantity of material to secure a given capacity, 
and we secure the greatest capacity with a given quantity 
of material. 



EXAMPLES. XXXIV. 

Find the area of the curved surface of right circular 
cylinders having the following dimensions : 

1. Height 2 feet 2 inches, circumference of base 4 feet. 

2. Height 2 feet 5 inches, circumference of base 4 feet 
9 inches. 

3. Height 1 foot 10 inches, radius of base 1 foot 
5 inches. 

4. Heiglit 2 feet 6 inches, radius of base 2 feet 
4 inches. 

5. Height 40 feet, radius of base 8 feet. 

Find the area of the whole surface of right circular 
cylinders having the following dimensions : 

6. Height 3 feet, radius of base 2 feet. 

7. Height 5 feet, radius of base 3 feet 6 inches. 

8. Height 1 foot 2 inches, radius of base 8 inches. 

13—2 



196 EXAMPLES. XXXIV. 

9. Height 5 feet 6 inches, circumference of base 
20 feet. 

10. Height 6 feet 3 inches, circumference of base 
24 feet. 

11. The area of the curved surface of a right circular 
cylinder is 6 square feet, and the circumference of the base 
is 3 feet 9 inches : find the height. 

12. The area of the curved surface of a right circular 
cylinder is 5| square feet, and the radius of the base is 
2j feet : find the height. 

13. The area of the wliole surface of a right circular 
cylinder is 14 square feet, and the height of the cylinder 
js equal to the radius of the base : find the radius of the 
base. 

14. The area of the whole surface of a right circular 
cylinder is 24 square feet, and the height of the cylinder is 
twice the radius of the base : find the radius of the base. 

15. The area of the whole surface of a right circular 
cylinder is 30 square feet, and the height of the cylinder is 
half the radius of the base : find the radius of the base. 

* 16. The area of the curved surface of a right circular 
cylinder is 9.\ square feet, and the volume of the cylinder 
is 3^ cubic feet : find the radius of the base. 

17. The area of the curved surface of a right circular 
cylinder is 4 square feet, and the volume of the cylinder is 
5 cubic feet : find the area of one end. 

18. The area of the curved surface of a right circular 
cylinder is 3 square feet, and the volume is 2j cubic feet : 
find the height. 

19. The area of the base of a right circular cylinder is 
314'16 square inches; the volume is 31416 cubic inches : 
find the area of the curved surface. 

20. The area of the base of a right circular cylinder is 
1000 square inches ; the volume is 5 cubic feet : find the 
area of the curved surface. 



EXAMPLES. XXXIV. 197 

The following examples involve the extraction of the 
cube root : 

A vessel without a lid in the form of a right circular 
cylinder is to hold 31416 cubic inches; find in square 
inches the area of the whole internal surface in the fol- 
lowing cases : 

21. The height equal to the radius of the base. 

22. The height double the radius of the base, 

23. The height half the radius of the base. 

24. The height three times the radius of the base. 

25. The height one-third of the radius of the base. 

A vessel with a lid in the form of a right circular 
cylinder is to hold 31416 cubic inches : find in square inches 
the area of the whole internal surface in the following 
cases : 

26. The height equal to the radius of the base. 

27. The height double the radius of the base. 

28. The height half the radius of the base. 

29. The height three times tlie radius of the base. 

30. The height one-third of the radius of the base. 

31. The edge of a cube is 10 inches ; a right circular 
cylinder of the same volume as the cube has its height 
equal to the radius of its base : find the area of the whole 
surface of the cube, and also of the cylinder. 

32. A vessel in the form of a right circular cylinder 
without a lid is to contain 1000 gallons : find the area of 
the whole internal surface supposing the vessel to be of the 
most advantageous shape. 



198 



SEGMENTS OF A CYLINDER. 




XXXV. SEGMENTS OF A RIGHT CIRCULAR 
CYLINDER. RING. 

326. Simple rules can be given for finding the areas 
of the curved surfaces of certain segments of a right 
circular cylinder ; as we will now shew. 

327. Suppose a right circular cy- 
Imder cut into tvro parts by a plane 
parallel to the axis; the surface of 
each part consists of two segments of 
a circle, a rectangle, and another por- 
tion which we shall caU the curved 
surface. 

The area of each of the segments 
of a circle can be found by Arts. 185 
and 186. The area of the rectangle can 
be found by Art. 134. The area of 
the curved surface can be found by the following Rule : 
multijjlij the length of the arc of tJie base by the height of 
the cylinder. 

Or we may find the area of the rectangle and the 
curved surface together by the Rule of Art. 323, multiply 
the perimeter of the base by the height of the cylinder. 

328. Suppose a solid has been 
obtained by cutting a right circular 
cylinder by a plane, inclined to tlie 
axis, which does not meet the base 
of the cylinder. The surface of this 
solid consists of the base which is a 
circle, the other end which is also 
a plane curve, and another portion 
v/hich we shall call the curved 
surface. 

The area of the base can be 
found by Art. IGS. No Rule 
has been given in this work for 
finding the area of the other end 
exactly; but the area might be 
found approximately by Art. 193; 





SEGMENTS OF A CYLINDER. 199 

we may remark that tliis piano curve is called an 
ellipse, and is of great importance in mathematical in- 
vestigations. 

The area of the curved surface can be found by the 
following Rule : multiply the circumference of the base hy 
the height of the solid. 

329. The height of the solid in the preceding Rule is 
to be understood in the same sense as in Art. 256. The 
truth of the Rule may be shewn in the manner of Art. 257. 

330. Suppose a solid ha& been obtained by cutting a 
right circular cylinder by tioo planes inclined to the axis, 
which do not meet each other. The area of the curved sur- 
face will be found by multiplying the circuTnference of the 

hase of the cylinder hy the height of the solid. The height 
of the solid is to be understood as in Art. 259. The Rule 
follows from the fact that the solid may be supposed to 
be the difference of tvro solids of the kind considered in 
Art. 256 or Art. 328. 

331. To find the area of the surface of a solid ring. 

Rule. Multiply the circmnference of a circular sec- 
tion of the ring by the length of the ring. 

The length of the ring is to be understood as in Art. 261. 
The Rule may be illustrated in the manner of Art. 260. 

332. Examples : 

(1) The radius of the circular section of a ring is one 
inch, and the length of the ring is ten inches. 

The circumference of the circular section of the ring is 
2 X 31416 inches ; therefore the area of the surface of the 
ring in square inches is 10 x 2 x 31416, that is 62*832. Thus 
the area of the surface is 63 square inches nearly. 

(2) The inner diameter of a ring is 7 inches, and the 
outer diameter is 8 inches. 

As in Art. 262 we find that the radius of the circular 
section is j of an inch, and the length of the ring is 23*562 
inches; therefore the area of the surface of the ring in 

square inclies = -x 31416 x 23*562 = 37 '01... 



200 EXAMPLES, XXXV, 



EXAMPLES. XXXY. 

1. Find the area of the curved surface of the smaller 
of the two pieces in the diagram of Art. 327, supposing 
the height of the solid to be 4 feet, the radius of the circle 
15 inches, and the chord of the circle equal to the radius. 

2. Find the area of the curved surface of the smaller 
of the two pieces in the diagram of Art. 327, supposing 
the height of the solid to be 4 feet 2 inches, the radius of 
the circle to be 8 inches, and the chord to subtend a right 
angle at the centre of the circle. 

3. The radius of the base of a cylinder is 16 inches ; a 
piece is cut off by two planes inclined to the axis of the 
cylinder, which do not meet each other ; the length of the 
portion of the axis between the two planes is 35 inches : 
find the area of the curved surface. 

Find in square inches the areas of the surfaces of rings 
having the following dimensions : 

4. Length 20 inches; circumference of cross section 
4 inches. 

5. Length 25 inches ; radius of cross section | of an 
inch. 

6. Outer diameter 4*7 inches; inner diameter 4*1 
inches. 

7. Inner diameter 11 inches; diameter of cross sec- 
tion 2 inches. 

8. Outer diameter 26 inches; diameter of cross sec- 
tion 4 inches. 

9. Outer diameter 25 inches; circumference of cross 
section 10 inches. 

10. Inner diameter 20 inches; circumference of cross 
section 12 inches. 

11. The area of the surface of a ring is 100 square 
inches ; the radius of the cross section is 1 inch : find the 
length of the ring. 

12. The area of the surface of a ring is 120 square 
inches ; the length is 20 inches : find the inner diameter. 



BIQRT CIRCULAR CONE, 201 



XXXVI. RIGHT CIRCULAR CONE. 

333. The surface of a right circular cone consists of a 
circular base and another portion which we shall call the 
curved sui'face. 

334. Let ABCD be a sector of a circle. Cut it out of 
paper or cardboard. Then let it be bent until the edge 
u4i? just comes into contact with the 

edge AD. It is easy to see that by 
proper adjustment we can thus ob- 
tain a thin shell, the outside of which 
will correspond to the curved surface 
of a right circular cone : A becomes 
the vertex of the cone, AB becomes 
the slant height of the cone, and 
BCD becomes the circumference of 
the base of the cone. Hence it will 
follow that the curved surface of a 
right circular cone is equal to a sector of a circle, the 
radius of the sector being the slant height of the cone, 
and the arc of the sector being the circumference of the 
base of the cone : thus we obtain the Rule which we shall 
now give. 

335. To find the area of the curved surface of a right 
circular cone. 

Rule. Multiply the circumference of the dase b?j the 
slant height of the cone, and half the product will he the 
area of the curved surface. 

336. Examples : 

(1) The radius of the base of a right circular cone is 
8 inches, and the slant height is 14 inches : find the area 
of the curved surface. 

The circumference of the base in inches = 2 x 8 x 3"1416 
= 60-265G ; ^ X 14 X oO'265G = 35i'8592. 




202 RIGHT CIRCULAR CONE. 

Thus the area of the curved surface is 352 inches 
nearly. 

(2) The radius of the base of a right circular cone is 
4 feet, and the height of the cone is 3 feet: find the 
area of the whole surface. 

"We must first find the slant height of the cone; by 
Art. 55, the slant height in feet is the square root of 16 + 9, 
that is, the square root of 25, that is 5. 

-x5x2x4x 31416 = 20 x 3-1416 = 62-832. 

Thus the area of the curved surface is 62-832 square 
feet. 

The area of the base in square feet = 4x4x 31416 
= 50-2656. Therefore the area of the whole surface in 
square feet = 62-832 + 50-265G = 113-0976. 

337. The following inferences may be easily dray.Ti 
from the llule in Art. 335 : 

If the slant height of the right circular cone be twice 
the radius of the base the area of the curved surface is 
tioice that of the base of the cone; if the slant height bo 
three times the radius of the base, the area of the curved 
surface is three times that of the base of the cone ; and 
so on. 

We may sum up these inferences thus : the slant height 
of a right circular cone hears the same proportion to the 
radius of the base as the ai^ea of the curved surface bears 
to the ai^ea of the base. 

338. We will now solve some exercises. 

(1) The area of the whole surface of a right circular 
cone is 24 square feet, and the slant height is twice the 
radius of the base : find the radius of the base. 

By Art. 337 the area of the curved surface is equal to 
twice the area of the base; and thus three times the area 
of the base is 24 square feet : therefore the area of the 
base is 8 square feet. Wo must then find tbe radius of 



RIGHT CIRCULAR CONE. 203 

the base by Art. 171. Divide 8 by 3'1416; the quotient is 
2'5464...; the square root of this is 1"595S very nearly. 
Thus the radius of the base is 1'595S feet very nearly. 

(2) The volume of a right circular cone is 20 cubic 
feet; the height is twice the radius of the base: find the 
area of the whole surface. 

By Art. 263 we see that the product of the cube of 

2 
the radius of the base into - of 3'1416 is equal to 20; so 

o 

that the cube of the radius of the base 

60 30 ^ _^ 

9-549. 



2x3-141(J 314i6 '"' 

therefore the radius of the base in feet is the cube root of 
9"549...: this we shall find to be 21215... Therefore the 
area of the base in square feet is the product of 3"1416 
into the square of 2 121 : it will be found that this is 
14-140... 

Now if the radius of the base were 1 foot, and the 
height of the cone were 2 feet, the slant height would be 
v/5 feet, by Art. 55; that is, the slant height would be 
a/S times the radius of the base. And thus in the present 
case since the height is tvrico the radius of the base, the 
slant height is Jo times the radius of the base: that is 
the slant height is 2-236... times the radius of the base. 

Hence, by Art. 337, the area of the cuiTed surface in 
square feet is the product of 2-236... into 14*140. .. that 
is 31-617... 

Therefore the whole area of the surface in square feet 
= 14-140 + 31-617 =45-757. 

(3) The volume of a right circular cone is 20 cubic 
feet; the slant height is three times the radius of the base: 
find the area of the whole surface. 

If the radius of the base were 1 foot, and the slant 
height were 3 feet, the height would be JS feet by Art. 60; 
that isj the height woidd be V8 times the radius of the 



204 RIGHT CIRCULAR CONE. 

base. And thus in the present case since the slant height 
is 3 times the radius of the base, the height is ^8 times 
the radius of the base. 

Then, as in the preceding Exercise, we see that the 

3 X 20 
cube of the radius of the base = -7- — ^ . ,,^ ; it will be 

v'8x3'1416 

found that this is 6752...; therefore the radius of the 

base is the cube root of 6752... : it will be found that this 

is 1-890... 

By Art. 337, the area of the whole surface is 4 times 
the area of the base ; so that in square feet it is 4 times 
the product of 3*1416 into the square of l"&jj.. i it will be 
found that this is 44-888... 

The whole surface is less in this case than in the case 
of the preceding Exercise. In fact, it will be found by 
comparing the results of Examples 31... 40 at the end of 
the present Chapter, and of similar Examples, that if the 
whole surface of a right circular cone be given, the volume 
is greatest when the slant height is three times the radius ; 
and if the volume of a right circular cone be given the whole 
surface is least when the slant height is three times the 
radius. 



EXAMPLES. XXXVI. 

Find in square inches the area of the curved surface of 
right circular cones having the following dimensions : 

1. Slant height 2 feet 3 inches, circumference of base 
4 feet 5 inches. 

2. Slant height 3 feet 2 inches, circumference of base 
6 feet 7 inches. 

3. Slant height 2 feet, radius of base 1 foot 9 inches. 

4. Slant height 2 feet 8 inches, radius of base 2 feet 
10 inches. 

5. Slant height 3 feet, radius of base 1 foot 6 inches. 

6. Height 2 feet, radius of base 7 inches. 

7. Height 3 feet 4 inches, radius of base 9 inches. 



EXAMPLES. XXXVI. 205 

8. Height 2 feet 6 inches, radius of base 1 foot 
4 inches. 

9. Height 5 feet, radius of base 11 inches. 

10. Height 4 feet 8 inches, radius of base 2 feet 
9 inches. 

11. Height 5 feet, perimeter of base 62832 feet. 

12. Height 12 feet, perimeter of base 10 feet. 

Find in square feet the area of the whole surface of 
right circular cones having the following dimensions : 

13. Slant height 4 feet, radius of base 2 feet. 

14. Slant height 5"3 feet, radius of base 3'2 feet. 

15. Slant height 6 feet, circumference of base 8 feet. 

16. Slant height 6"4 feet, circumference of base 97 feet. 

17. Height 1 foot, radius of base 5 inches. 

18. Height 1 foot 9 inches, radius of base 1 foot 
8 inches. 

19. Height 18 inches, circumference of base 27 inches. 

20. Height 4 feet, circumference of base 7 feet. 

21. The area of the curved surface of a right circular 
cone is 750 square inches, and the circumference of the 
base is 50 inches : find the slant height. • ' 

22. The area of the curved surface of a right circular 
cone is 800 square inches, and the circumference of the 
base is 64 inches : find the height of the cone. 

23. The area of the curved surface of a right circular 
cone is 12 square feet, and the radius of the base is 1*5 
feet: find the slant height. ^ - 

24. The area of the curved surface of a right circular 
cone is 25 square feet, and the radius of the base is 2'25 
feet : find the height of the cone. 

25. The area of the curved surface of a right circular 
cone is 650 square inches, and the slant height is 25 
inches: find the circumference of the base. 



206 EXAMPLES. XXXVI. 

26. The area of the curved surface of a right circular 
cone is 18 square feet, and the siaut height is 3| feet: 
find the radius of the base. 

27. The area of the whole surface of a right circular 
cone is 15 square feet, and the slant height is three times 
the radius of the base : find the radius of the base. 

28. The area of the whole surface of a right circular 
cone is 19 square feet, and the slant height is four times 
the radius of the base: find the radius of the base. 

29. Find what length of canvass three-quarters of a 
yard wide is required to make a conical tent 12 feet in 
diameter and 8 feet high. 

30. Find what length of canvass two-thirds of a yard 
wide is required to make a conical tent 16 yards in dia- 
meter and 10 feet high. 

The area of the whole surface of a right circular cone 
is 100 square feet; find in cubic feet the volume in the 
following cases : 

31. The slant height t\vice the radius of the base. 

32. The slant height three times tlse radius of the base. 

33. The slant height four times the radius of the base. 

34. The slant height five times the radius of the base. 

35. The slant height six times the radius of the base. 

The following examples involve the extraction of the 
cube root : 

The volume of a right circular cone is 314160 cubic 
inches ; find in square inches the whole surface in the fol- 
lowing cases : 

36. Height equal to the radius of the base. 

37. Height equal to twice the radius of the base. 

38. Height equal to three times the radius of the base. 

39. Height equal to half the radius of the base. 

40. Height equal to a third of the radius of the base. 



FRUSTUM OF A CONE. 



m 



XXXVII. FRUSTUM OF A 

COIS'B. 



EIGHT CIRCULAR 




339. The surface of a frustum of a right circular cone 
consists of two circular ends and another portion which 
we shall call the curved surface. 

340. Let ABGD be a sector 
of a circle. With A as centre 
and any radius less than AB 
describe the arc EFG. Let the 
piece BCDGFE be cut out of 
paper or cardboard. Then let 
it be bent round until the edge 
EB just comes into contact with 
the edge GD. It is easy to see 
that by proper adjustment we 
can thus obtain a thin shell the 
outside of which will correspond to the curved surface of 
a frustum of a right circular cone ; EFG becomes the cir- 
cumference of one end of the frustum, and BCD becomes 
the circumference of the other end; EB becomes the 
slant height of the frustum. Hence it will follow that the 
curved surface of a frustum of a right circular cone is equal 
to the difference of two sectors of circles which have a 
common angle ; the arcs of tlie sectors being the circum- 
ferences of the ends of the frustum, and the difference of 
their radii being the slant height of the frustum : thus 
from the last Rule of Art. 183 we obtain the Rule which 
will now be given. 

341. To fi'.id the area of the curved surface of a frus- 
tum of a right circular cone. 

Rule. Multiply the sura of the cii^cumferences of the 
two ends of the frustum Jyij the slant height of tlie frus- 
tum., and half the product will be the area of the curved 
surface. 



208 FRUSTUM OF A CONE. 

342. Examples. 

(1) The radius of one end of a frustum of a right cir- 
cular cone is 10 inches, and the radius of the other end is 
15 inches; the slant height is 16 inches: find the area of 
the curved surface. 

The sum of the circumferences in inches is the product 
of 3"1416 into the sum of 20 and 30, that is, into 50; thus 
the sum of the circumferences is 50 x 3'1416 inches. 

i X 16 X 50 X 3-1416 = 8 X 50 x 3'1416 = 400 x 31416 -= 1256-64. 

Thus the area of the curved surface is 1256-64 square 
inches. 

(2) The radius of one end of a frustum of a right cir- 
cular cone is 5 feet, and the radius of the other end is 
8 feet; the slant height is 8 feet: find the area of the 
whole surface. 

The area of the curved surface in square feet 
= 13 X 8 X 3-1416 = 104 x 3-1416. 

The area of one end in square feet = 25 x 3-1416, and 
the area of the other end in square feet = 64x 3-1416. 
Hence the area of the whole surface is the product of 
3-1416 into the sum of 104, 25, and 64, that is, into 193 : 
thus the area of the whole surface in square feet 

= 193x3*1416 = 606-3288. 

343. It may be inferred from the Rule of Art. 341 
that the slant height of a frustum of a right circular 
cone hears the same proportion to the difference of the 
radii of the ends as the area of the curved surface hears 
to the difference of the areas of the ends. 

344. We will now solve some exercises. 

(1) The radii of the ends of a frustum of a right circu- 
lar cone are 7 inches and 10 inches, and the height of the 
frustum is 4 inches : find the area of the curved surface. 




FRUSTUM OF A CONE. 209 

Let the diagram re- 
present a section of the 
frustum made by a plane 
containing the axis of the 
cone. We see that the 
slant height is the hypo- 
tenuse of a right-angled 

triangle, of which one side is the height of the frustum, 
and the other side is the difference of the radii of the 
ends. 

In the present case the height of the frustum is 4 inches, 
and the difference of tlie radii is 3 inches ; therefore, by 
Art. 55, the slant height is 5 inches. 

Therefore the area of the curved surface in square 
inches = 5 x 17 x 3-1418 = 267-036. 

(2) The diameters of the ends of a frustum of a right 
circular cone are 16 feet and 24 feet respectively ; the 
height of the frustum is equal to the product of these dia- 
meters divided by their sum : find the area of the curved 
surface, and the area of the two ends. 

The height of the frustum in feet = — —^ = —tt:- = 9 "6. 

40 10 

We must determine the slant height. The square of 

9'6 = 92-16; the difference of the radii of the ends is 

4 feet; the square of 4 = 16; 92-16 4-16 = 108-16; the 

square root of 108-16 = 10-4. Thus the slant height is 

10-4 feet. 

The area of the curved surface in square feet 

= 10-4 X 20 X 3-1416 = 208 x 3-1416 = 653-4528. 

The area of the two ends is the product of 3*1416 into 
the sum of the squares of 8 and 12, that is, the product of 
3*1416 into the sum of 64 and 144, that is, the product of 
3-1416 into 208. 

Henco the area of the two ends is equal to the area of 
the curved surface. It will bo found on trial that this is 
always the case if the height of the frustum of a right cir- 
cular cone is equal to the i)roduct of the diameters of the 
ends divided by their sum. 

T. M. 14 



210 EXAMPLES. XXXVIL 

(3) The radii of the ends of a frustum of a right cir- 
cular cone are 8 inches and 10 inches respectively; the 
Kiant height is 6 inches : if the frustum be divided into 
two of equal curved surfaces, find the slant height of each. 

We must determine the slant height of the whole cone. 

By a method like that of the fourth Exercise of Art. 77, 
we find that the slant height of the whole cone is 30 inches; 
and therefore the slant height of the smaller cone is 24 
inches. 

Then w^e proceed after the manner of Art. 210 to deter- 
mine the slant distance from the vertex of the plane which 
divides the curved surface of the frustum into two equal 
parts. The square of 24 is 576 ; the square of 30 is i) 00 ; 
half the sum of these squares is 738 : the square root of 
738 will be found to be rather more than 27*166. Subtract 
24 from this, and the remainder is 3'1 68. Thus the slant 
height of one part is rather more than 3166 inches; and 
therefore the slant height of the other part is rather less 
than 2*834 inches. 

EXAMPLES. XXXVIL 

Find the area of the curved surface of frustums of right 
circular cones, having the following dimensions : 

1. Circumferences of ends 15 inches and 17 inches, 
slant height 11 inches, 

2. Circumferences of ends 19 inches and 23 inches, 
slant height 13 inches. 

3. Radii of ends 7 inches and 9 inches, slant height 
5 inches. 

4. Radii of ends 2-6 feet and 3*4 feet, slant height 
5 feet. 

5. Radii of ends 11 and 16 inches, height 12 inches. 

6. Radii of ends 4 feet and 3 feet, height 2 feet 
11 inches. 

7. Radii of ends 4 feet and 5 feet, height 3 feet. 

8. Radii of ends h\ feet and e]- feet, height 2i feet. 



EXAMPLES. XXXVIL 211 

Find the area of the whole surface of frustums of right 
coues having the following dimensions : 

9. Circumferences of ends 14 and 16 inches, slant 
height 10 inches. 

10. Circumferences of ends 17 and 21 inches, slant 
height 9 inches. 

11. Radii of ends 2 feet and 3 feet, slant height 
2^ feet. 

12. Radii of ends 3'4: feet and 4*2 feet, slant height 
2 feet. 

13. Radii of ends 12 and 18 inches, height 8 inches. 

14. Radii of ends 12 and 20 inches, height 15 inches. 

Find in square feet the area of the curved surface and 
the area of the two ends, supposing the height of the frus- 
tum equal to the x^roduct of the diameters of the ends 
divided by their sum, for frustums with the following di- 
mensions : 

15. Diameters of ends 6 feet and 4 feet. 

16. Diameters of ends 13 feet and 7 feet. 

17. Diameters of ends 20 feet and 12 feet. 

18. Diameters of ends 25 feet and 40 feet. 

19. The radii of the ends of a frustum are 5 feet and 
8 feet, and the slant height is 4 feet : if the frustum be 
divided into two of equal curved surface, find the slant 
height of each part. 

20. A tent is made in the form of a frustum of a right 
circular cone surmounted by a cone : find the number of 
square yards of canvass required for the tent, supposing 
the diameters of the ends of the frustum to be 28 feet and 
16 feet respectively, the height of the frustum 8 feet, and 
the height of the conical part 6 feet. 



14-2 



212 SPHERE. _,. 

XXXVIII. SPHERE. 

345. To find the area of the surface of a sphere. 
Rule. Multiply the square of the diameter hy 3' 141 6. 

346. Examples. 

(1) The diameter of a sphere is 9 inches. 

9x9x3-1416 = 254-4693. 

Thus the area of the surface is 254-47 square inches 
nearly. 

(2) The diameter of a sphere is Z\ feet. 

3-5 X 3-5x3-1416 = 38-4846. 
Thus the area of the surface is about 38'4846 square feet 

347. Other modes of expressing the Rule in Art. 345 
may be given : multiply the diameter of the sphere by its 
circumference I or, divide the square of the circumfer- 
ence hy 3-1416. By the circumference of the sphere is 
meant the circumference of the circle which will produce 
the sphere in the manner explained by Art. 223, that is, 
the circumference of a great circle of the sphere. 

348. It folloAVS from Arts. 320 and 345, that the area 
of the surface of a sphere is equal to the area of the 
curved surface of a rigid circular cylinder which has its 
height and the diameter of its ends equal to the diameter 
of the sphere. 

349. From Arts. 291 and 345 we may deduce the fol- 
lowing important result : tJie volume of a sp)here is equal 
to one-third of the product of the area of the surface into 
the radius. 

350. The result just given may be easily remembered 
by its resemblance to the Rule for finding the volume of a 
pyramid or a cone. Let denote the centre of a sphere ; 
and let P, Q, R denote three points on the surface of the 
sphere very near to each other. Suppose we cut from the 
sphere the piece bounded by the planes POQ, QOR, ROF, 



SPHERE. 213 

and by the portion of the surface of the sphere which they 
intercept. This piece will resemble a triangular pyramid ; 
so that we may readily admit that the volume of the piece 
is equal to one-third of the product of the radius of the 
sphere into the intercepted portion of the surface of the 
sphere. The whole sphere may be suj)posed to be cut up 
into a very large number of very small pieces like that 
just considered ; and thus we are easily led to the result 
given in Art. 349. The student will observe the resem- 
blance of these remarks to those in Art. 177. 

351. The sphere has the following remarkable pro- 
perty : of all solids of a given volume the sphere is that 
which has the least surface, and of all solids of a given 
surface the sphere is that which has the greatest volume. 
The student may verify this statement by such examples 
as 16... 20 at the end of the present Chapter. 

352. TVe will now solve some exercises. 

(1) The area of the surface of a sphere is 200 square 
inches : find the diameter, and the volume of the sphere. 

The product of the square of the diameter into 31416 
is equal to 200; therefore the square of the diameter 

= — - = 63'6618: the square root of this number will 

3-1416 

be found to be 7-9789. Thus the diameter is 7-98 inches 
very nearly. 

Then, by Art. 349, the volume of the sphere in cubic feet 
= i X 200 X ^ X 7-9789 = 265-96... 

(2) The volume of a sphere is 1000 cubic inches : find 
the area of its surface. 

By Art. 291 tlie cube of the diameter of the sphere 

= 1^ = 1909-855 : therefore the diameter of the sphere 
•5236 ' 

in inches is the cube root of this number : it will be found 

that this is 12-407. Then by Art. 345 we shall obtain for 

the area of the surface 4836 square inches very nearly. 



214 EXAMPLES. XXXVIIL 

EXAMPLES. XXXVIIL 

Find the areas of the surfaces of si)heres haying the 
following dimensions : 

1. Radius 5 inches. 2. Radius 15 inches. 

3. Radius 2*2 feet. 4. Circumference 20 inches. 

6. Circumference 4 feet. 6. Circumference 6*4 feet. 

Find the diameters of the spheres having the following 
superficial areas : 

7. 400 square inches. 8. 64 square feet. 9. 75 square feet. 

Find the volumes of the spheres having the following 
superficial areas : 

10. 20 square feet. ll. 50 square feet. 12. 100 square feet. 

13. Find the volume of a sphere when its surface is 
equal to that of a circle 4 feet in diameter. 

14. Find the volume of a sphere when its surface is 
equal to that of a circle 9 feet in diameter. 

15. A cylinder 5 feet long and 3 feet in diameter is 
closed by a hemisphere at each end : find the area of the 
whole surface. 

16. The radius of the base of a right circular cylinder 
is 10 inches, and the height is 10 inches; the surface of a 
sphere is equal to the whole surface of this cylinder : find 
the volume of each. 

17. The surface of a sphere is equal to that of a cube 
the length of which is one foot : find the volume of each. 

18. The surface of a sphere is equal to that of a right 
circular cylinder the radius of the base of which is one 
foot, and the height two feet : find the volume of each. 

The following examples involve the extraction of the 
cube root : 

1 9. The volume of a sphere is equal to that of a cube 
the length of which is one foot : find the surface of each. 

20. The volume of a sphere is equal to that of a right 
circular cylinder the radius of the base of which is one 
foot, and the height 2 feet : find the surface of each. 



ZONE OF A SPHERE. 215 



XXXIX. ZONE OF A SPHERE. SEGMENT OF A 

SPHERE. 

353. The surface of a zone of a sphere consists of two 
circular ends and another portion which we shall call the 

curved surface. 

The surface of a segment of a sphere consists of a cir- 
cular base and another portion which we shall call the 
curved surface. 

354. To find the area of the curved surface of a zone 
of a sphere or of a segment of a sphere. 

Rule. Multiply the circumference of the sphere Istj 
the height of the zone or segment. 

355. Examples. 

(1) The height of a segment of a sphere is 6 inches, 
and the diameter of the sphere is 18 inches : find the area 
of the curved surface. 

6xl8x3-1416 = 339-2928. 

Thus the area of the curved surface is 339*3 square 
inches nearly. 

(2) The ends of a zone of a sphere are distant 2 feet 
and 4 feet respectively from the centre of a sphere, and 
are on the same side of the centre ; the diameter of the 
sphere is 14 feet : find the area of the whole surface of the 
2;one. 

The area of the curved surface in square feet 

= 2 X 14 X 31416 = 28 x 3-1416, 
By Art. 59 the square of the radius of one end 

= 9x5 = 45; 



210 ZONE OF A SPHERE. 

and the square of the radius of the other end 

= 11x3 = 33: 

thus the area of the two ends in square feet = 78 x 31416. 
Therefore the area of the whole surface in square feet 

= 106x3-1416 = 333-0096. 

356. It appears from Arts. 320 and 354 that the curved 
surface of a zone of a sphere or of a segment of a sphere 
is equal to the area of the curded surface of a right 
circular cylinder which has its height equal to that of 
the zone or segment, and the diameter of its ends equal 
to the diameter of the sphere. 

This remarkable result holds also for the surface of a 
sphere, if by the height of the sphere we understand the 
diameter of the sphere : see Art. 348. 

357. We will now solve some exercises. 

(1) The height of a segment of a sphere is 7 inches, 
and the circumference of the sphere is 64 inches : find the 
area of the wh©le surface of the segment. 

The area of the curved surface in square inches 

= 7x64 = 448. 

The diameter of the sphere is „ , . , - inclies : therefore, 

3-1415 

by Art. 79, the square of the radius of the base of the 

64 
segment is obtained by subtracting 7 from -r-— — -> and 

3-14I6 

^T ». /I J 

multiplying the remainder by 7 ; so that it is ^ ^ ^,^ — 49. 

The area of the base of the segment in square inches is 
the product of this result into 3-1416; therefore it is 
7x64-49x3-1416, that is 448-49x3-1416. Thus the 
area of the whole surface in square inches is 

896-49x3-1416, 

that is 896-153-9384, 

that is 742-0616. 



ZONE OF A SPHERE. 217 

Tlie method which we have adopted in solving this 
exercise may appear rather artificial and difficult to a be- 
ginner ; but it deserves attention. It will be seen that in 
effect we establish the following Rule : the whole sia-face 
of a segment of a sphere is equal to twice the excess of 
tJie curved surface above a circle which has the height of 
the segment for its radius. 

(2) A zone of a sphere is the difference of two seg- 
ments of the heights 1 3 inches and 9 inches respectively, 
and the circumference of the sphere is 82 inches : find 
the area of the whole surface of the zone. 

The area of the curved surface in square inches 

= 4x82 = 328. 
By Art. 79 the square of the radius of one end of the 

zone is obtained by subtracting 9 from "* , and mul- 

9 X 82 
tiplyiug the remainder by 9 ; so that it is r-rrrr-Sl. 

The area of this end of the zone in square inches is the 
product of this result into 3" 141 6 ; therefore it is 

9x82-81x3-1416. 

Similarly the area of the other end ot the zone in 
square inches will be found to be 13 x 82 — 169 x 3'1416. 
Hence the area of the whole surface of the zone is equal 
to the sum of 4x82, 9x82, and 13x82, diminished by 
the sum of 81 x 3*1416 and 169 x 31416, that is, to 

2x13x82-250x3-1416. 
Thus the area of the whole surfaee is 1346-6 square inches. 
It will be seen that in eflfect we establish the following 
Rule for finding the area of the whole surface of a zone of 
a sphere, regarded as the difference of two segments of 
the sphere: from, twice the curved surface of the large 
segment subtract the areas of two circles having for their 
radii respectively the heights of the segments. 

(3) The radius of a sphere is 12 feet; from a point 
which is at the distance of 15 feet from the centre of the 
sphere straight lines are drawn to touch the sphere, thus 
determining a segment of the sphere ; find the area of the 
curved surface of this segment. 



218 



ZONE OF A SPHERE. 




Let A be the centre of the sphere, B the point from 

which straight Mnes are drawn, 
BG one of the straight lines, 
meeting the sphere at G. Draw 
GD perpendicular to AB. Let ^ 
denote the point where AB cuts 
the sphere. Then the surface of 
which we have to find the area is 
the curved surface of the segment 
of which DE is the height. We 
shall now find the length of DE. 

Tiiking all the lengths in feet we have 
AB = 15, AG=AE^12- 
hence, by Art. 60, we obtain BC—2 : and, by Art. 151, we 

shall find that GD = ^-^ = 7-2. By Art. 60 we shall see that 

AD is the square root of 144— 51*84, that is the square 
root of 92*16 ; it will be found that this is 9"6. We may 
obtain this result more easily by similar triangles; by 
Art. 37 we have 

BA : AG :: AG 



that is 
therefore 



Then 



15 : 12 



AD, 
12 : AD; 



.J. 12x12 48 ^^ 
15 5 

Z>^=12-9-6 = 2-4. 



Therefore the area of the curved surface of the segment 
in square feet = 2-4x 2 x 12 x 3-1416 = 18095616. 

An eye placed at B would see exactly that portion of 
the surface of the sphere of which we have just found the 
area. 

If we wish to know merely what fractional part of the 

whole surface of the sphere is visible to an eye placed at 

B, we have only to form the fraction which has the length 

of DE for the numerator, and the length of the diameter 

of the sphere for denominator : in the present case the 

2"4 1 

fraction is ■—-, that is --r . 
24 10 



EXAMPLES. XXXIX, '219 



EXAMPLES. XXXIX. 

Find the areas of curved surfaces of segments of a 
sphere haying the followng dimensions : 

1. Height of segment 10 inches, circumference of 
sphere 85 inches. 

2. Height of segment 2 J feet, circumference of sphere 
20 feet. 

3. Height of segment 9 inches, radius of sphere 
16 inches. 

4. Height of segment 2*4 feet, radius of sphere 3*25 
feet. 

Find the areas of the whole surfaces of segments of 
a sphere having the following dimensions : 

5. Height of segment 2 feet, radius of sphere 7 feet. 

6. Height of segment 8 inches, radius of sphere 
25 inches. 

7. Height of segment 3 feet, circumference of sphere 
27 feet. 

8. Height of segment 11 inches, circumference of 
sphere 90 inches. 

Find the areas of the whole surfaces of zones which 
are the diflFerences of segments having the following 
dimensions : 

9. Radius of sphere 11 feet, heights 3 feet and 10 
feet. 

10. Radius of sphere 15 inches, heights 6 inches and 
9 inches. 



220 EXAMPLES. XXXIX. 

11. Circumference of sphere 40 feet, heights 2 feet 
and 5 feet. 

12. Circumference of sphere 75 inches, heights 3 inches 
and 7 inches. 

Find the areas of the whole surfaces of zones of a 
sphere having the following dimensions : 

13. Radius of sphere 9 feet; distances of ends of zone 
from the centre 2 feet and 3 feet, on opposite sides of the 
centre. 

14. Radius of sphere 16 inches; distances of ends of 
zone from the centre 5 inches and 9 inches, on the same 
side of the centre. 

15. Circumference of sphere 32 feet ; distances of ends 
of the zone from the centre 3 feet and 4 feet, on opposite 
sides of the centre. v ;.-■>■• 

16. Circumference of sphere 90 inches ; distances of 
ends of the zone from the centre 6 inches and 10 inches, 
on the same side of the centre. 

17. A sphere is 80 feet in diameter : find what fraction 
of the whole surface will be visible to an eye placed at a 
distance of 41 feet from the centre, 

18. A sphere is 90 feet in diameter: find what fraction 
of the whole surface will be visible to an eye placed at a 
distance of 8 feet from the surface. 

19. Find at what distance from the surface of a sphere 
an eye must be placed to see one-sixteenth of the surface. 

20. Find at what distance from the surface of a sphere 
an eye must be placed to see one-eighth of the surface. 



221 



SIXTH BEGTIOX. PRACTICAL 
AFPLICA TIONS. 

XL. INTRODUCTIOX. 

358. "We have already given various applica-tions of 
the Rules of Mensuration to matters of practical interest ; 
for instance the Examples of the calculations of the ex- 
pense of carpeting the floors of rooms and papering the 
wails which occur under Chapter xi. Such apphcations 
require only a knowledge of the elements of Arithmetic in 
addition to the principles of Mensuration already ex- 
plained. 

A few more examples will be given at the end of the 
present Chapter. 

359. There are however other applications which re- 
quire the student to know the meaning of certain techni- 
cal terms, or to employ certain approximate Rules admitted 
by custom. We shall consider these cases in the three 
Chapters which immediately follow the present Chapter 



EXAMPLES. XL. 

1. A room is 24 feet 3 inches long, 11 feet 9 inches 
broad, and 11 feet 6 inches high : find the cost of painting 
the walls at Is. 6d. per square foot. 

2. Find the cost of painting the four walls of a room 
which is 32 feet 4 inches long, 15 feet 8 inches broad, and 
11 feet 6 inches high at 3 shillings per square foot. 



222 EXAMPLES. XL. 

3. Find the cost of painting tlic four walls of a room 
whose length is S4 feet 3 inches, breadth 15 feet 8 inches, 
and height 11 feet 6 inches at 4 shillings per square foot. 

4. A room is 24 feet 10 inches long, 16 feet broad, 
and 12 feet 4 inches high : find the cost of painting the 
four wails at 9 pence per square foot. 

5. The length of a room is 7 yards 1 foot 3 inches, the 
breadth is 5 yards 2 feet 9 inches, and the height 4 yards 
6 inches : find the cost of papering the walls, supposing 
the paper to be a yard broad and to cost Qd. per yard. 

6. A cubical box is covered with sheet lead which 
weighs 4 lbs. per square foot j and 294 lbs. of lead are used: 
find the size of the box. 

7. Find the cost of lining the sides and the bottom of 
a rectangular cistern 12 feet 9 inches long, 8 feet 3 inches 
broad, 6 feet 6 inches deep with sheet lead which costs 
^'1. 8^. per cwt., and weighs 8 lbs. to the square foot. 

8. A cistern open at the top is to be lined with sheet 
lead which weighs 6 lbs, to the square foot ; the cistern is 
4 feet 6 inches long, 2 feet 8 inches wide, and holds 42 
cubic feet : find the weight of lead required. 

9. A box with a lid is made of planking Ij inches 
thick ; if the external dimensions be 3 feet 6 inches, 2 feet 
6 inches, and 1 foot 9 inches, find exactly how many square 
feet of planking are used in the construction. 

10. A flat roof is 17 feet 4 inches long and 13 feet 
4 inches wide : find the cost of covering it with sheet lead 
one-sixtccnth of an inch thick, supposing that a cubic inch 
of lead weighs 6^ ounces avoirdupois, and that 1 lb. of it 
costs 3|<:^. 

11. Find the cost of painting the wall of a cylindrical 
room 16 feet high, and 18 feet in diameter, at l^d. per 
square yard. 

12. Find the cost of painting a conical spire 64 feet in 
circumference at the base, and 108 feet in slant height, at 
*J^d. per square yard. 



EXAMPLES. XL. 223 

13. Find how niucli it will cost to gild Llic mncr sur- 
face of a hcmisplierical bowl 2 feet 4 inches in diameter at 
\hd. per square inch. 

14. A circular room has perpendicular walls 15 feet 
high, the diameter of the room being 28 feet; the roof is a 
hemispherical dome : find the cost of plastering the whole 
surface at 9 pence j)er square foot. 

15.^ Find the cost of a string moulding round the 
springing of the dome in the preceding Example at \5d. per 

foot. 

16. A rectangular court-yard is 100 feet long and 90 
feet broad; a footway goes through the length 6 feet broad ; 
the footway is laid with stone at 4^. Qd. per square yard, 
and the remainder is covered with turf at 9d. per square 
yard : find the whole cost. 

17. Required the cost of glazing the windows of a 
house at a shilling per square foot; there being three 
stories and three windows in each story ; the height of the 
windows of the lower story is 8 feet, of the middle story 

7 feet, and of the upper story 5 feet j and the common 
breadth of all the windows is 4 feet. 

18. Find how many square feet of flooring there are in 
a house of three stories, measuring within the walls 58 feet 
by 34 feet, deducting the vacancy for the stairs 15 feet 
3 inches by 8 feet. 

19. A room is 22 feet long, 20 feet wide, and 14 feet 
6 inches high : find the expense of covering the walls with 
paper 30 inches wide at ll^d. a yard; allo^-ing for two doors 
each 8 feet by 5 feet 3 inclies, a fire-place 6 feet 6 inches 
by 6 feet, and a window 12 feet by 5 feet 7 inches. 

20. A square court-yard is 36 feet long; in the mid- 
dle of it is a circular basin 13 feet in diameter; and a 
flower-bed 4 feet wide is left round three sides : find the 
expense of paving the remainder of the court-y^srd at 

8 shillings per square yard. 



224 ARTIFICERS' WORK. 



XLI. ARTIFICERS' WORK. 

360. Artificers compute their work in various ways, 
but in general a foot or a yard is the standard of len<^h, 
and therefore a square foot or a square yard is the stand- 
ard of area, and a cubic foot or a cubic yard is the standard 
of volume. 

361. The work of flooring, roofing, plastering or tiling 
is often estimated by the number of squares, each square 
consisting of 100 square feet. 

362. The measure of tlie roof of a house can bo deduced 
from the measure of the base when the pitch of the roof is 
known. There are three pitches which have received 
names. 

(1) The corainon or true pitch. In this the length of 
the rafters is three-fourths of the breadth of the building; 
and hence the practical rule is to take the flat and half tha 
fiat of the house as the measure of the roof. 

(2) The Gothic pitch. In this the length of the rafters 
is equal to the breadth of the building, and consequently 
the roof is equal to twice the fiat. 

(3) The pedimejit pitch. In this tlie perpendicular 

2 
height is - of the breadth of the building. In this case the 

length of the rafters will be nearly — of the breadth of 

tlie building, so that the roof will be nearly -— of the fiat. 

363. Artificers of every kind have various special 
modes of estimating the charge for workmanship, which 
have the sanction of custom ; but as these modes involve 
no principle of mensuration it is not expedient to detail 
them here. We will give as a specimen the mode in which 
the charge is made for doors. 

It is usual to add the thickness of the door both to 
the length and to the breadth, and to take the product of 



ARTIFICERS' WORK. 225 

the length and the breadth thus increased for the area. If 
the door be panelled on only one side this area is increased 
by its half. If the door be panelled on both sides the area 
is doubled. 

Thus, for example, suppose that a door is 7 feet 5 
inches high, 4 feet 3 inches wide, and 1 inch thick. Then 
the height is taken as 7 feet 6 inches, and the breadth as 
4 feet 4 inches ; and so the area in square feet is taken as 

7| X 4j, that is as -^ x — » ^^'^-^ i^ ^s — - — , that is as 

32J-. If the door be panelled on only one side the charge 
is Tor 45|- square feet. If the door be panelled on both 
sides the charge is for 65 square feet. 

364. Engineers always estimate brickwork by the cubic 
yard; but such brickwork as occurs in connexion with ordi- 
nary house-building is estimated in a peculiar way which 
w^e will now explain. 

365. A brick wall which is a brick and a half thick is 
said to be of the standard thickness. Brickwork of the stand- 
ard thickness is estimated by the number of square yards 
in the area formed by its height and its length, ^ or by the 
number of square rods, each square rod consisting of 30|^ 
square yards, that is of 212^ square feet. Thus a standard 
rod of i3rickwork is a mass of brickwork which has a sur- 
face of a square rod and is a brick anc- a half thick. 

366. To find tlie numher of standard rods of hrick- 
icork in a wall. 

Rule. Find the area of the surface in square feet, 
and divide it l>y 272j ; the quotient will he the nuniber of 
rods if the icall he of the standard thickness; if not, 
multiply the quotient hy the numher of halfhricks in the 
thickness, and divide the product hy 3. 

In practice 272 is generally used instead of 272j. 

367. Examples. 

(1) Find the number of standard rods of brickwork in 
a wall 105 feet long, 8| feet high, and 21 bricks thick. 

105x8^ 5 _ 35 X 5 X S^ _ 35 X 5 X 17 _ ,. ,^ 

— — — X ~ — ^_ ■ — _ , . — O "xu* • • • 

272 3 272 544 

Thus the number of standard rods is nearly 5|. 
T. Ai. 15 



226 EXAMPLES. XLL 

(2) Find the number of yards of brickwork of the 
standard thickness contained in a triangular gable-top 
which is 15 feet high, and the base of which is 20 feet, 
supposing the thickness 2 bricks. 

20x15 50 50 4 200 „„^ 

_ — — • — y i = — 992 

2x9 3 ' 3 3 9 ~ ""' 
Thus the number of standard yards is 22g-. 

368. ^ A common brick is SJ inches long, 4 inches broad, 
and 2^ inches thick ; but, on account of the mortar, when 
laid in brickwork every dimension is to be taken half an 
inch greater: thus the dimensions are to be taken as 9 
inches, 4^r inches, and 3 inches. The standard rod requires 
4500 bricks of the usual size, allowing for Avaste. 

EXAMPLES. XLL 

1. Find the number of standard rods of brickwork in 
a wall 62 feet 6 inches long, 14 feet 8 inches high, and 
2|- bricks thick. 

2. Find the number of standard rods of brickwork in 
a gable-end wall 2 bricks thick, 22 feet long, 27 feet high 
to the eaves, and 36 feet from the ground to the ridge of 
the roof. 

3. Find the cost of a wall with a triangular gable-top 
of 10 feet high, the height of the wall being 36 feet, the* 
breadth 24 feet, and the thickness 2^ bricks, at 345. per 
standard rod. 

4. The end wall of a house is 30 feet long; it is 40 
feet high to the eaves, and there is a triangular gable of 
10 feet high; up to the height of 20 feet the wall is 2^ 
bricks thick, between the height of 20 feet and 40 feet it 
is 2 bricks thick, and the gable-top is 1^ bricks thick : find 
the number of standard yards. . ■,, 

5. Assuming that bricks cost £2 per thousand, that 
4500 are required for a rod, that cartage and mortar cost 
together 22 shillings per rod, and labour £2 per rod : find 
the cost of building a wall 136 feet long, 18 feet high, and 
2 bricks thick. 



EXAMPLES. XLL 227 

6. Find the cost of roofing a house of the common 
pitch at 15 shillings per square; the length being 40 feet, 
and the breadth 35 feet. 

7. Find the cost of roofing a building of the Gothic 
pitch at 25 shillings per square; the length being 120 feet 
and the breadth 40 feet. 

8. A building 30 feet long by 2C feet broad is to be 
covered with lead, so that the roof shall be eleven-tenths 
of the flat : find the cost supposing the lead to weigh 8 lbs. 
to the square foot and to cost 21 shillings per cwt. 

9. A partition measures 45 feet 5 inches, by 8 feet 
2 inches : find the cost at £6. lo5. per square. 

10. The floor of a room measures 44 feet by 24 feet: 
find the cost of flooring at £6. 55. per square, allowing fur 
two hearths each 7 feet by 4 feet. 

11. A room is 34 feet long, 18 feet 6 inches wide, and 
12 feet high : find the cost of wainscoting the room at £10 
per square. 

12. The length of the roof of a house is 50 feet, and 
the length of a string stretched over the ridge from eaves 
to eaves is 60 feet : find the cost of the roof at £2. 7^. ^d. 
per square. 

13. A garden wall is ISO feet long and 7 feet high; 
the wall is 1 brick thick, but there are 18 piers each 1^ 
feet wide, and at these the whole thickness is 1^ bricks : 
find the number of standard vards. 

14. A room is 36 feet long, 18 feet broad, and 12 feet 
high : find the whole cost of plastering the walls at one 
shilling per square yard, and the ceihng at eighteen-pcncc. 

15. Find the whole cost of flooring two rooms at £5 
per square; one room measuring 2S feet by 16 feet, and 
the other 24 feet by 15 feet 6 inches. 

16. A room is 25 feet long, 20 feet broad, and 12 feet 
high ; the walls are to have three coats of paint, each 
costing 10 shillings per square : find the whole cost. 

15-2 



228 TIMBER MEASURE, 



XLII. TIMBER MEASURE. 

369. If a piece of timber be in the shape of any of the 
bodies considered in the Fourth Section, the vohime can 
be determined by the appropriate Rule tliore given. If 
no exact Rule is applicable we may in some cases use with 
advantage the method of equidistant sections which is 
given in Art. 307. In two cases which often occur in 
practice rules are adopted, which although not exact, are 
recommended by their simplicity: these rules v^e shall 
now give. 

370. To find tha volume of squared or four-sided 
tbiiber. 

Rule. Multiply the mean Ireadth hy the mean thick- 
ness^ and the product hy the length; and take the result 
for the volume. 

In order to obtain the mean breadth, the actual breadth 
should be measured at various equidistant points, and the 
sum of the results divided by the number of the measure- 
ments : and in the same way the mean thickness should 
be obtained. 

371. Examples. 

(1) The length of a piece of timber is 24 feet, the mean 
breadth is 1 foot 9 inches, and the mean thickness is 1 foot 
6 inches. 

24xiaxli = 24x^ x| = 63. 
4 2 

Thus we obtain 63 cubic feet for the volume. 

(2) The length of a piece of timber is 16^ feet, tho 
thickness at one end is 1 foot, and at the other end 1 foot 
8 inches : the breadth is 2 feet. 



TIMBER MEASURE. 229 

Here wc take for the raean thickness in feet lialf the 
sum of 1 and 1§, that is Ij. 

qo 2 4 

16^x2xli=:^x-x- = 44. 

Thus the volume is 44 cubic feet. 

872. If the piece of timber tapers regularly from one 
end to the other it is usual to take for the mean breadth 
the breadth at the middle, or, which is the same thing, 
half the sum of the breadths at the ends ; and similarly 
the mean thickness is estimated. But in this case the 
piece of timber is really a prismoid, and so the yolumc 
might be determined exactly by the Rule of Art. 283. The 
approximate Rule has the advantage of being simpler than 
the exact Rule, 

If, as in Example (2) of Art. 371, it tapers regularly as 
to its thickness, and is constant as to its breadth, the Rule 
gives the exact result. The piece of timber is in this case 
a prism, the ends of the prism being trapezoids, and tlio 
height of the prism being the breadth^ of the piece of 
timber. Of course a similar remark aj^plies to the case in 
which the thickness is constant and the breadth tapers 
regularly. 

373. To Jiiid the volume of round or unsquared 
timher. 

Rule. Multiply the square of the mean quarter girt 
by the length, and take the -product for the volume. 

374. Examples. 

(1) The length of a piece of unsquared timber is 32 
feet, and the girt is 6 feet. 

3 3 9 

The quarter girt is ^ fcetj the square of - is -; 

- X 32 = 72. Thus we obtain 72 cubic feet for the volume. 
4 

(2) The length of a piece of unsquared timber is 24 
feet ; the girt at one end is 5 feet, and at the other end 
6 feet. 



230 TIMBER MEASURE. 

5 + 6 
Here we take for the mean girt — r— ; and so for the 

• .11 .1 ^-ll • 121 

mean quarter girt — ; the square oi — is -— ; 



121 ^, 121x3 363 ._3 

— - X 24= — - — = --— =45#. 
64 8 8*^ 

Thus we obtain 45| cubic feet for the volume. 



375. If the piece of timber be exactly a cylinder in 
shape, we can determine its vokime exactly by the Rule of 
Art. 246. We shall find that in the case of a right circular 
cylinder the Rule of Art. 373 gives a result which is rather 
more than three-fourths of the true result; perhaps the 
Rule was constructed with the design of making some 
allowance for the loss of timber which occurs when the 
piece is reduced by squaring in the ordinai-y way ; see the 
last Exercise of Art. 313. 

If the piece of timber be not a circular cylinder the 
result given by the Rule will generally approach nearer to 
the true result. 

376. Dr Hutton proposed to use the following Rule 

instead of the Rule of Art. 373 : Multiply the square of 

one-fifth of the mean girt hy twice the length. Dr Ilutton's 

32 
Rule makes the volume — times as great as the ordinary 

25 

Rule, and gives a result which is very nearly exact when 
the jDicce of timber is exactly a circular cylinder. 

377. If the piece of unsquarcd timber tapers regularly 
from one end to the other it is usual to take as the mean 
girt the girt at the middle, or, which is the same thing, half 
the sum of the girts at the ends. If the ends are exact 
circles and the piece of timber tapers regularly, it is really 
a frustum of a cone, and so the volume might be deter- 
mined exactly by the Rule of Art. 268. The approxnnate 
Rule has the advantage of being simpler than the exact 
Rule. 



EXAMPLES. XLIL 231 



EXAMPLES. XLII. 



Find the number of cubic feet in pieces of timber of 
the follovdns: dimensions : 



*o 



1. Length 22^ feet ; breadth at one end 2 feet 9 inches, 
at the other 2 feet 3 inches ; thickness at the first end 
1 foot 10 inches, at the other 1 foot 6 inches. 

2. Length 27 feet; mean breadth 3j feet, mean 
thickness 1^ feet. 

3. Length 32 feet ; mean breadth 2| feet, mean thick- 
ness 1| feet. 

4. Length 56 feet; mean girt 5 feet. 

5. Length 32 feet; girt at one end 25 inches, at the 
other 35 inches. 

6. Length 24 feet ; mean girt 40 inches. 

7. A piece of timber is 36 feet long and tapers regu- 
larly; its breadth and thickness at one end are 30 inches 
and 20 inches respectively, and at the other end 24 inches 
and 18 inches respectively : find the number of cubic feet 
in the piece by the Rule of Art. 370. 

8. Find the number of cubic feet in the piece of 
timber of the preceding Example by the exact rule of 
Art. 283. 

9. A piece of timber is 40 feet long and tapers regu- 
larly ; one end is a circle 7 feet in circumference, and the 
other end is a circle 4 feet in circumference : find the 
number of cubic feet in the piece by the Rule of Art. 373. 

10. Find the number of cubic feet in the piece of 
timber of the preceding Example by the Rule of Art. 376. 

11. Find the number of cubic feet in the piece of 
timber of Example 9 by the exact rule of Art. 268. 

12. If the piece of timber of Example 9 is squared, 
the ends being made squares as large as possible, find 
the number of cubic feet in the piece produced. [See 
Examples 51 and 52 of Chapter xvi.] 



2.32 GAUGING, 



XLIII. GAUGmG. 



378. By gauging is meant estimating the voluiues of 
casks, that is, the volumes of liquids which the casks 
would hold. 

Casks differ in shape, and various rules havo been s 
given for estimating their volumes according to the shape 
which they take exactly or approximately. For example, 
suppose a cask to ^be formed of two equal frustums of a 
cone united at their bases ; we can determine the volume 
exactly by Art. 268 : and if a cask be very nearly of this 
shape, we may estimate the volume approximately bj mo- 
ceeding as if the cask were exactly of this shape. 

379. But it is found that a Rule may be given wlich 
will serve tolerably well for all the shapes of casks which 
occur in practice. To apply this Rule we must know three 
internal measurements of the cask, namely the length, the 
diameter at one end, which is called the head diameter, 
and tl]e diameter at the middle, which is called the 'bung 
diameter. 

380. The dimensions of a cask will always be taken 
to bo exjoressed in incites. 

381. To find the volume of a cash 

Rule. Add into one sum 39 times the square of the 
"bung diameter, 25 times the square of the head diameter, 
and 26 times the product of those diameters ; and multi- 
ply the sum hy the length of the cash: vnultiply thep>ro- 
dud hy •000031473, and the result may he taken for the 
volume of the cash in gallons. 



GAUGING. 233 

382. Examples. 

(1) The length of a cask is 40 inches, the bung diame- 
ter is 32, and the head diameter is 24. 

39x32x32 = 39936, 

25x24x24 = 14400 

26x32x24 = 13963, 

39936 + 14400 4-19968=74304. 

74304 X 40 X -000031473 = 93-54279. .. 

Thus the volume of the cask is about 93| gallons. 

(2) The length of a cask is 20 inches, the bung diame- 
ter is 16, and the head diameter is 12. 

39x16x16 = 9984, 

25x12x12 = 3600, 

26x16x12 = 4992, 

9984 + 3600 -I- 4992 = 18576, 

18576 X 20 x '000031473 = 11-6928... 

Thus the volume of the cask is about 11*7 gallons. 

383. It is sometimes necessaiy to know the quantity 
of liquid in a vessel which is only partly filled. The word 
ullage means strictly the portion of a partly filled cask 
which is not occupied by the liquid : but the word is now 
applied to the occupied portion as well as to the unoccu- 
pied portion, the former being called the wet ullage and 
the latter the dry ullage. 

384. Two cases may occur ; namely, that of a standing 
cask, and that of a lying cask. We will consider the 
former case. The depth of the liquid is called the wet 
inches; the difference between the wet inches and the 
length of the cask is called the dry indies. 



234 GA UGING. 

385. To estimate the wet ullage of a standing cask 
which is less than half fall. 

Rule. Multijyly the square of the dry inches hi/ the 
difference letween the hung diameter and the head dia- 
meter ; and divide the "product htj the square of tlie 
length: subtract this from the hung diameter, and the 
result may he taken as the mean diameter of the occupied 
portion of the cask. 

Tlieii proceed as in finding tlie volume of a cylinder : 
Multiply the square of the mean diameter hy the wet 
inches, and the product hy •0028326 ; the result may he 
taken for the numher of gallons in the icet ullage. 

386. To estimate the wet idlage of a standing cask 
which is more than half full. 

Apply the method of Art. 385, using wet inches instead 
of dry inches, and dry inches instead of iret inches ; we 
thus obtain the dry ullage : subtract the dry ullage from 
the volume of the cask, and the remainder is the wet 
ullage. 



^o^ 



387. Examples. 

(1) The length of a cask is 40 inches, the bung dia- 
meter is 32, and the head diameter is 24; the number of 
wet inches is 10 : find the wet ullage. 

The number of dry inches is 30 ; the difference between 
the bung diameter and the head diameter is 8. 

30x30x8 9 „„ 9 55 

40 X 40 2 ' 2 2 * 

55 
Thus we take — for the mean diameter. 

^ X ^ X 10 X -0028326 =21-4215... 
Thus the wet ullage is about 21*4 gallons. 



GAUGING. 235 

(2) The length of a cask is 20 iuches, the bung diame- 
ter is 16, and the head diameter is 12; the number of -wet 
inches is 15 : find the wet uHage. 

Here we first find the dry ullage. 

15x15x4 9 'ip_9_^ 

20x20 ~ 4' 4~ 4 ' 

— X — X 5 X '0028326 = 2-6777 Tory nearly. 
4 4 

Subtract this from 11*6928, the volume of the whole 
cask, by Art. 382 ; the remainder is 9'0151. Thus the wet 
ullage is about 9 gallons. 

388. No satisfactory rule can be given for estimating 
the ullage of any lying cask. A rule proposed by Hutton 
amounts to assuming the cask to be a cylinder ; the rule is 
substantially this : find the area of the segment of a circle 
which is obtained by the section of the Jiiiid icith a plane 
perpendicular to the length of the cask; multip)ly this 
area by the length of the cask, and divide the product 
by 277"274; tlie result may be taken for tlie number of 
gallons in the wet idlage, 

389. The business of gauging is practically performed 
by excisemen with the aid of instruments called the 
gauging or diagonal rod, and the gauging or sliding rule. 
These instruments however are recommended not so much 
on account of the accuracy of the results to which they 
lead, as of the expedition with which these results are 
obtained. The construction and mode of using these in- 
struments can be far more readily miderstood by actual 
experience than by any description. 



236 EXAMPLES. XLTII. 



EXAMPLES. XLIIl. 

Find in gallons the volumes of casks having the follow- 
ing dimensions : 

1. Length 60-2, bung diameter Srs, head diameter 22-7= 

2. Length 47*5, bung diameter 28'5, head diameter 21*4. 

3. Length 42-5, bung diameter 32'5, head diameter 26'5. 

4. Length 30*5, bung diameter 26*5, head diameter 23. 

5. Length 46'8, bung diameter 30"5, head diameter 26. 

6. Length 34-5, bung diameter 32-3, head diameter 27*6. 

7. Length 46 "9, bung diameter 3r2, head diameter 26'L 

[The dimensions in the above seven examples are 
stated in Lubbock's tract on Cask-gaging, 1834, to be 
the average dimensions respectively of a Port-pipe, a 
Madeira-pipe, a Sherry-butt, a Sherry-hogshead, a Bour- 
deaux brandy-puncheon, a Rum-puncheon, a Brandy-piece.] 

Find in gallons the wet ullage in the following cases of 
standing casks : 

8. Lerigth 60, bung diameter 36, head diameter 30, 
wet inches 12. 

9. Lengtli 50. bung diameter 32, head diameter 27, 
wet inches 10. 

10. Length 30, bung diameter 27, head diameter 23, 
wet inches 9. 

11. If the length of the cask in Example 1 is increased 
by "1 of an inch, shew that the volume will be increased by 
about '22 of a gallon. 

12. If the head diameter in Example 2 is increased by 
"1 of an inch, shew that the volume will be increased by 
about "27 of a gallon. 

13. If the bung diameter in Example 3 is increased by 
•1 of an inch, shew that the volume will be increased by 
about '43 of a gallon. 

14. If all the dimensions in Example 1 are increased 
by "1 of an inch, shew that the volume will be increased by 
about a gallon. 



237 



SEVENTH SECTION. LAND 
SURVEYING. 

XLIY. USB OF THE CHAIN". 

390. A VERY important application of some of the 
rules of mensuration is furnished iu Land Surveying ; and 
to this we now proceed. 

391. Land is measured with a chain, called Gunter's 
chain, which is 4 poles, that is, 22 yards long ; the chain 
consists of ] 00 equal links, so that each link is y-^ of a 
yard long, that is 7 '92 inches. 

392. K picket is a rod stuck into the gi'ound to mark 
a certain position. 

393. A field-'booJz is a book in which the results ob- 
tained by measurement are recorded. 

394. "We will now explain how a straigld line is mea- 
sured with the chain. 

We will suppose that the straight line which is to be 
measured is the distance between two points each marked 
by a picket. 

Ten small arrows are provided which may be stuck 
in the ground. Two persons engage in the work, one of 
w^liom is called the leader; both place themselves at one 
of the pickets. The leader takes in his hand the ten 
arrows and one end of the chain, and walks towards the 
second picket ; the follower keeps the other end of the chain 
at the first picket. "When the leader has stretched the 
chain to its full length he puts an arrow in the ground to 
mark the spot to which the chain reached ; he then walks 
towards the second picket carrying with him his end of 
the chain as before. The follower now comes up to the 
arrow, and holds his end of the chain at it until the leader 



238 USE OF THE CHAIN. 

has again stretched the chain and stuck the second arrow 
in the ground. Then the follower takes up the first arrow 
and walks towards the second. The process is continued 
until the proposed length is measured. 

Whenever the follower has the ten arrows in his hands 
he records in the field-book that a length of ten chains 
has been passed over; then he gives the ten arrows again 
to the leader, and the work is continued. Thus on arriv- 
ing at the second picket the field-book shews the number 
of tens of chains passed over, the arrows in the follower's 
hands correspond to the number of additional chains, and 
the number of links between the Inst arrow and the second 
picket can bo counted. Thus the required length is 
found. 

395. In measuring with the chain, great care must be 
taken to preserve the proper direction ; and there is in 
general a double test of the accuracy with which this is 
effected. When the leader fixes an arrow he should take 
care that the straight line between this arrow and the first 
picket will pass through the follower's arrow; and at the 
same time the follower should take care that the straight 
line between his arrow and the second picket will pass 
through the leader's arrow. 



"o* 



396. If a field be in the shape of any rectilineal figure, 
and we measure the lengths of the appropriate straight 
lines, we can find the area of the field by the corresponding 
rule in the Third Section. 

397. Examples. 

(1) A rectangular field is 8 chains 95 links long, and 
3 chains 26 links broad. 

8 chains 95 links = 8-95 chains; 3 chains 26 links = 3-26 
chains. We use the rule of Art. 134= 

2-9177 8-9 5 

4 3-2 6 



3-6 7 08 
40 

26-8 3 20 


6370 
1790 

2685 

2 9-177 



USE OF THE CHAIN. 239 

TIiG area of the field is 29*177 square chains, that is 
2"9177 acres; we may reduce the decimal of an acre to 
roods and poles : thus we obtain 2 acres 3 roods 27 poles 
very nearly. See Art. 126. 

(2) The sides of a triangular field are 5*2 chains, 
5"6 chains, and 6 chains respectively. 

^Ye use the rule of Art. 152. 

5-2-l-5-6 + 6 = 16-S, ^ofl6-8-S-4; 

8-4-5-2 = 3-2, 8-4-5-6=2-S, 8-4-6 = 2-4. 

S'4x3-2 X 2-8x2-4 = lS0'6336. The square root of 
180-6336 is 13-44. 

Thus the area is 13-44 square chains, that is, i'344 
acres, that is, 1 acre 1 rood 15-04 poles. 

(3) The radius of a circular grass plot is 2 chains 
50 hnks. 

We use the rule of Art. 168. 

2-5x2-5x3-1416 = 19-635. 

Thus the area is 19-635 square chains, that is, r9635 
acres, that is, 1 acre 3 roods 34-16 poles. 



240 



PERPENDICULARS. 



XLV. PERPENDICULARS. 



398. Some of the rules for finding the areas of recti- 
lineal figures require us to know the length of the perpen- 
dicular from some given point to some given straight 
line. When the situation of such a perpendicular is known, 
the length of it can be measured in the manner explained 
in Art. 394 ; we shall now shew how the situation is deter- 
mined. 

399. To determine the situation of the perpendicular 
to a given straight line from a given p{jint without it. 

Let AB be the given 
straight line, and C the given 
point without it. 

Fold a string into two 
equal parts. Let one per- 
son hold the middle at C; 
let two other persons hold 

the ends, and stretch the two parts, so that the ends may 
lie on the straight line AB, say at Z> and E. Take F the 
middle point of BE. Then CF is the perpendicular re- 
quired. 

400. The straight line -^^ in the preceding Article is 
supposed to be clearly marked on the ground in some 
manner. This may be efi'ected by stretcliing a string or 
chain tightly between A and B, "or by placing pickets at 
short distances in the direction passing through A and B. 
If, however, the straight line AB lias not been thus marked 
out on the gi'ound, a person standing beyond A must take 
care that the end of the string is properly placed at Z>, 
and then standing beyond B he must take care that the 
end of the string is properly placed at E. 




PERPENDICULARS. 



241 




401. To detennine the situation of the straight line 
at rhjht angles to a given straight line from a given 
2^0 Int in it. 

Let AB he the given 
straight line, and F the 
given point in it. 

Take D and E points in 
AB, so that FD and FE 
may be equal Fold a string, 
longer than PE, into two 
equal parts. Let the ends of the string be fixed at Z> 
and E, and let a person take the middle and stretch both 
the parts. Suppose C the point to which the middle of 
the string is thus brought. Then FG is at right angles to 
AB, and is therefore the straight line required. 

402. "We see then that the situation of any required 
perpendicular can be determined witli the aid of a string; 
but an instrument called the Cross is often employed by 
land surveyors for this purpose. 

403. The Cross. This is usually a round piece of ^yood, 
about six inches in diameter, having two fine grooves in it, 
which are at right angles to each other. A staff with a 
pointed end is stuck upright in the ground, and the cross 
is fixed on the top of the staff so as to form a small round 
table. 

404. To detennine tclth the aid of the cross the situ- 
atlon of the 'perpendicular to a given straight line from 
a given point without it. 

Let AB be the given straight 
line, and C the given j)oint with- C 

out it. 

Place i)ickets at A, B, and 
C. Select a x)oint in AB which 
appears by inspection to be at 
or near the intersection of the 
required perpendicular with AB; ^ 
let D denote this point. Fix the 



I> 



3 



stafl: at D, and x>la,ce the cross with one of its groovei» 



T. M. 



16 



242 PERPENDICULARS. 

parallel to AB ; so that in looking along the groove in 
one direction, the picket at A is seen, and in looking along 
the same groove in the other direction the picket at B 
is seen. JLook along the other groove; if the picket at 
G is seen in the line of this groove, then the point D is the 
intersection of the perpendicular from G with AB : but if 
the picket at G is not seen in the line of tlie groove, the 
staff must be taken up and moved along AB, to the right 
or to the left of the assumed position, according as the 
picket at G appeared to the right or to the left of the 
groove. By a little trial the pro^Der position will be found 
for the staff, such that the pickets at A and B can be 
seen in looking along one groove, and the picket at C in 
looking along the other groove; and this position of the 
staff determines the situation of the perpendicular on AB 
from G. 

405. To determine with the aid of the cross the sitic- 
ation of the straight line at right angles to a given straight 
line from a given point in it. 

Let ^i? be the given straight lino, and D the given 
point in it. 

Fix the staff at Z>, and place the cross with one of the 
gi'OOYCS parallel to AB. Then tlie other groove deter- 
mines, the direction of the straight line at risrht andes 
i^AB, 

408. Thus in the preceding Chapter and the present 
we haveexplainedhow the operations are to be performed 
which will furnish the lengths required for ^calculating the 
areas of fields. \Ye will now give some examples of the 
calculations. 

407. Examples. 

(1) The base of a triangle is 13-2 chains, and the 
height is So chains. 

2X13-2 X 8-3 = 54-78. 

The area of the field is 5478 square chains, that is, 
5-478 acres, that is, 5 acres 1 rood 38-48 poleS; 



PERPENDICULARS, 



243 




(2) ABDG is a four-sided 
field ; GP and DQ are perpen- 
diculars on AB. 

The following measureiiients 
are taken in links : j;;^ p q^ Ji 

A.P = \\2, ^Q = 448, yli? = 626, 

(7P = 223, Z)(2 = 295. 

Hence we have PQ-336, Q^=178. 

The follo-^Yiiig will be the areas of the i)arts of tlio field 
in square links : 

the triangle APG= ^ x 1 12 x 223 = 12488, 



the trapezoid PQDG=-y. 336 x 518 = 87024, 



the triangle DQB^-~x 178 x 595 = 26255. 

The sum of these three numbers is 125767; so that 
tlio area of the field is 1*25767 acres, that is, about 1 acre 
1 rood 1 pole. 



16—2 



244 THE FIELD-BOOK. 



XLVI. THE FIELD-BOOK. 



408. Many fields may be conveniently surveyed by 
measuring a straight line from one corner to another, and 
also the perpendiculars on it from the other corners. The 
first straight line is called a hase-line, or a chain-line ; 
and the perpendiculars are called offsets. It is often ad- 
vantageous to take for the base-line the longest straight 
line which can be drawn in the field ; and thus sometimes 
one of the sides of the field may be the base-line, as in the 
second example of Art. 4U7. 

AVe will now explain the method in which the results 
of the measurements are usually recorded in the field- 
book. 

409. _ The field-hook. Each page of the field-book is 
divided into three columns ; the snrveyor begins at the 
bottom of the page and writes upwards. 

In the middle column are entered lengths obtained in 
measuring along the basedine, in the right-hand column 
are entered the lengths of offsets to the right of the base- 
line, and in the left-hand column the lengths of offsets to 
the left of the base-line. The ofisets are entered against 
the corresponding distances of the points on the base-line 
at which they are measured. 

The field-book is used to record not only measured 
lengths, but various particulars, which may be useful in 
drawing a plan of the estate surveyed. Thus a note is 
made in the margin of the point at which the chain-line 
crosses any fence, or foot-path, or ditch, or stream. The 



THE FIELD-BOOK, 



245 



positions of adjacent buildings, or of remarkable trees may 
also be indicated ; and if the chain-line passes near a 
boundary which is rather irregular, the form of the 
boundary may be traced. 

410. Examples. 



(1) 

to D 260 
to B 230 



to J^ 

1125 

750 

625 

300 

From A 



250 to G 




The surveyor begins at A 
and measures towards E\ 
Ab is 300 links, and at & is 
an offset hB to the left of 230 Hiiks ; Ac \% 625 links, and 
at c is an offset cG to the riglit of 250 links : Ad is 750 
links, and at d is an offset dD to the left of 260 links ; 
^^is 1125 hnks. 

TTe can now calculate the areas of the parts ; and we 
shall have the following results in square links : 



the triangle AbB = - x 300 x 230 := 34500 ; 



the trapezoid hdDB = ' x 450 x 490 = 110250 ; 



the triangle dED = - x 375 x 260 - 48750 ; 



the triangle AEG=- x 1125 x 250 = 140625. 



Thus the whole area is 334125 square links, that is, 
3-34125 acres, that is, 3 acres 1 rood 14"6 poles. 



246 



THE FIELD-BOOK. 



(2) 





ioG 




1020 


to F 470 


990 




610 


to D 320 


685 


to (7 70 


440 




315 




From^ 




50 to E 



350 to B 



The measurements are 
taken in links ; and we shall 
have the following results for the areas of the parts in 
square links : 

the triangle yl&5 = -x 315x350^55125, 

the trapezoid h3EB = ~x 295 x 400 ==59000, 
the triangle eGE=~ x 410 x 50 = 10250, 
the triangle GfF= ^ x 30 x 470 = 7050, 
the trapezium/JZ>i^=:- X 405 x 790 = 159975, 
the trapezium ckCD^~ x 145 x 390 = 2S275, 

the triangle cAC^ ^ x 440 x 70 = 15400. 

Thus the whole area is 335075 square links, that is, 
3-35075 acres, that is, 3 acres 1 rood 16-12 poles. 

411. The ends of a chain-line are called stations ; they 
are frequently denoted in the field-book thus : ®, ®, ©,... 

The situations of the chain-lines with respect to tlio 
points of the compass are often recorded. The record 
from ® range E, indicates that the chain-line commenc- 
ing at the first station proceeds towards the East. So the 
record jS". 50" W., indicates that the chain-line com- 



THE FIELD-BOOK, 247 

menciiig at the second station proceeds in tlie direction 
■which makes an angle of 50 degi-ees with the North direc- 
tion reckoned towards the "West. 

Sometimes the situation of the successive chain-lines is 
sufficiently shewn by the words right and left. Thus the 
record froim ® on Z, indicates that on arriving at the 
second station, the surveyor turns to the left hand of the 
direction in which he has been walking. 

If occurs in either offset column, it indicates that the 
chain-r.ne meets at the corresponding point the boundary 
of the knd which is beins: surveyed. 



'a 



412. In order to provide themselves with a test of the 
accuracy of their work, surveyors always measure more 
lengths than would be theoretically sufficient. Thus, for 
exanple, suppose that a field bounded by_ four straight 
sides is to be surveyed ; it would be theoretically sufficient 
to measure the four sides and one diagonal : for the area 
of each of the two triaiiglcs into which this diagonal di- 
vides the figure can then be calculated. But the surveyor 
will also measure the second diagonal. He will draw a 
plan of the figm-e from the measured lengths of the four 
sides and of the first diagonal, and draw the second dia- 
gonal on the plan : he will then examine if the length of 
the second diagonal as found from the plan by the known 
scale on which the plan was drawn corresponds with the 
measured length. If these two lengths do correspond, the 
surveyor gains confidence in the accuracy of the work : 
but if the two lengths do not correspond there is an error 
in the operations with the chain or in the drawing of the 
plan, and this error must be discovered and corrected. 

If the field to be surveyed be in the form of a triangle, 
the sides will be measured, from which the area can be 
found, and a plan can be drawn. To test the accm'acy of 
the work, either the perpendicular from an angle to the 
opposite side, or the straight line drawn from a deSuito 
point of one side to a definite point of another will be 
measured; and this measured length will be compared 
with the length obtained from the plan. 

A length which is measured for the purpose of testing 
the accuracy of the work is called a ^;roQ/'-lino, or a check- 
line, or a ^^Mine, 



248 



THE FIELD-BOOK. 



413. Ill surveying a field or a number of fields, we 
shall have a series of operations and records like those 
exemplified in Art. 410 ; namely, one for each chain-line. 
As an example, we will take the case of a field which ap- 
proximates to the form of a triangle, so that three chain- 
lines will occur in the survey. 





© A 






1650 







1300 


30 Z 


D 1232 


d 726 






500 


OA^ 




260 


10 H 












© a 




turn 


to the 


left 




© G 







1430 




i^lO 


820 






600 







270 


40 J^ 












© B 




turn 


to the 


left 




© D 






1540 







960 


30 G 




300 


10 B 










from 


© A 


go Nort 



hU 




The ofi'sets are much ex- 
aggerated in the figure for 
the sake of distinctness. 



The sides of the triangle ADG are respectively 1540 
links, 1430 links, and 1650 links ; hence it will be found by 
Art. 152 that the area of this triangle is 1016400 square 
links. We proceed to calculate the areas of the small 
pieces lying between the sides of this triangle and the 
boundary of the field. 

Along AD there are offsets to B a,nd to C; thus we 
have to estimate a triangle, a trapezoid, and another tri- 
angle ; the areas arc the following in square links : 



THE FIELD-BOOK. 24<9 

the first triangle = ^ x 300 x 10 = 1500, 
the trapezoid = ^ X 660 x 40 = 13200, 

the second triangle = ^ x 580 x 30 = 8700 ; 

the total is 23400. 

Along DG there is an offset to E^ and an inset to F ; 
thus there are two corresponding triangles, the latter of 
which is to be subtracted: 

the first triangle = ^ x 600 x 40 = 12000, 

the second triangle = - x 830 x 10 = 4150 \ 

the balance is 7850, to be added. 

Along GA there are off*sets to H and to Z, and the 
boundary meets the chain-line at K\ thus there are two 
corresponding triangles : 

the first triangle = ^ x 500 x 20 = 5000, 

the second triangle = - x 1150 x 30 = 17250 ; 

the total is 22250. 

101 6400 + 23400 + 7850 + 22250 = 10G9900. 

Thus the area of the whole field is 10*699 acres. 

The perpendicular is measured as a proof-line, and 
found to be 1232 links, while Gd is 726 links. 

414. Instead of the field-book, another method of re- 
cording the results of measurement is sometimes adopted. 
A plan is drawn resembling the field to be surveyed, and 
the lengths are noted in the plan as they are found against 
the corresponding parts of the figure. 



250 



THE FIELD-BOOK. 



415. Wo have hitherto supposed that the boundary of 
a field which is to be surveyed may be regarded practi- 
cally as composed of a moderate number of straight lines. 
But if the boundary is so irregular that this supposition is 
not admissible, vt^e must employ the principle of adjust- 
ment which has been explained in Art. 202 : a plan of the 
field must be drawn and the boundary changed into a rec- 
tilineal boundary enclosing an equal area. We will ex- 
plain a convenient method of applying the princii3le. 

416. Let ABDKEG be the plan of the field. 
Draw on the plan equidistant parallel straight lines; 
thus dividing the figure into strips of equal width. Con- 
sider one of these strips, as BDEG. Draw tlie straight 
lino Id at right angles to the parallels, so that the area of 



A 




7>S/f "'t 


^xd c 


^X 




yd. 
A 


nr~ 




the strip may be the same whether BD or hd be regarded 
as its end ; if BD can be regarded as a straight hue, td 
will pass through its middle point ; if BD be not a straight 
line the position of hd must be determined as well as pos- 
sible by the eye. Similarly, draw ce at the other end of 
the strip, so as to leave the area uncliangcd. Then tho 
area BDEO is cqmYalent to the rectangle hdec^ 



EXAMPLES. XL VI. 



251 



Proceeding in tins way we obtain a series of rectangles 
whicli are together equivalent to the original figure. The 
area corresponding to all these rectangles can be easily 
ascertained, and therefore the area of the original figure. 
Suppose, for example, that the parallel straight lines are 
drawn an inch apart ; and that the sum of the lengths of 
all the rectangles is 29 inches: the area of the original 
figure is 29 square inches. Now suppose that the plan has 
been drawn on the scale of three chains to an inch; then 
a square inch of the plan corresponds to nine square chains 
of the field; therefore the area of the field is 9 x 29 square 
chains, that is, 261 square chains. 

In practice the process of forming the sum of the 
lengths of the rectangles is performed by an instrument 
called the commttatmi scale. 



EXAMPLES. XLYI. 

Draw plans and find the areas of fields from the follow- 
ing notes, in which the lengths are expressed in links : 



toDlOO 
to X? 155 




toClSG 



110 to G 



to G 50 
toi>'lGO 



to^ 
300 
243 
162 

from A 



ISO to D 
122 to B 



toE 

500 

220 

100 

from A 



to B 200 



140 to D 



toE 

450 

290 

150 

from A 



80 to D 
90 to G 



252 



EXAMPLES. XLVI, 



6. 



tojE;i20 
toZ) 70 

toj5l50 



to F 
800 
650 
400 
350 
180 
from A 



110 to G 



to i^ 140 

to ^150 

to (7 50 



to G 
600 
660 
480 
470 
380 
100 
from A 



to B 170 
to B 150 



8. 





78 






102 




8 


53 


4 


9 


75 


8 


14 


36 


8 


4 


40 


12 


4 


21 

© 


5 


17 


12 

© 


7 



9. 





120 


19 


100 


26 


80 


27 


60 


25 


40 


18 


20 




© 





10. 






130 






110 


22 


26 


90 






50 


40 


28 


30 

© 





11. 



12. 











1394 










270 
220 


1112 
940 




60 


380 


20 




4 


260 


100 


184 


614 




4 


ISO 


76 




368 


235 


60 


100 


10 




160 


62 


20 


80 


60 




38 


42 







© 


50 






© 





EXAMPLES. XL VI. 



253 



13. Make a rough sketch of the field ABC, and calcu- 
late its area from the accompauyiug field-book; the chain- 
lines are all within the field. 



10 


250 A 




50 


200 









© 







300 a 




40 


200 




30 


100 




10 









© B 







6G0 B 




30 


100 









N. 52nv. 

A 





14. Make a rough sketch of the field ABGD, and cal- 
culate its area from the accompanying field-book. The 
side AD was uot measured, but it was a straight line 
without offsets. 



150 D 
400 


N. 66° E. 
© G 



lOUO 

SOO 

660 

600 

420 

240 

200 

S. 84" E. 

© B 



1000 B 
600 

S. 3G« AY. 
© A 



lOj 



254 



EXAMPLES. XLVL 



15. Lay down the field ABGDEFG, ar.d find its area 
from the following dimensions : 





to © i> 






1560 






854 


20 E 


(?G90 


618 
from © F 






to ® G 






1305 




i>G90 


363 
from © G 






to © (7 






1650 




5 362 


1230 






405 


390 G 


Begin 


at © A 


range Ec 



16. Make a rough sketch of a field from the following 
field-book : find the area of the wliole field, assimiing that 
the triangle BCD contains 416732 square links, and that 
the piece at G between the boundary and the off"scts to 
GD and GB respectively contains 300 square links. 





© B 






9-^4 







830 


60 


(/o 


510 




480 






120 










20 




© G 






1024 G 


10 




640 


30 












© D 






1292 D 


10 




1040 


20 




680 


80 


Ul 


312 5 







40 


Eogin at 


© A 





\ 



PROBLEMS. 



255 



XLYII. PROBLEMS. 

417. In our account of Land Surveying wo have con- 
fined ourselves to illustrating tlie use of the chain and the 
cross. In surveys of great extent or of extreme accuracy, 
instmnients are employed for measuring angles, and the 
calculations are effected by the aid of the science of tri- 
gonometry. With these resources also problems relating 
to the distance of an inaccessible object are usually solved. 
Nevertheless some of these problems may be treated suffi- 
ciently for practical purposes in a simpler manner : vre will 
give examples. 

418. To find the bread in of a river. 

Let A be an object close to the 
river; B an object on the other side, 
directly opposite to A, and also close 
to the river. 




Draw a straight line AG at riglit 
angles to AB, of any convenient 
length, and fix a picket at G. Pro- 
duce the straight line AG io ^ point 
D, such that GD is equal to' AJJ. 
From D draw "a straight line at right angles to AD, an 1 
in it find the point E so that BGE may be in a straight 
line. 



Then the triangles GAlB and GDE are equal in all 
respects, and Z>^is equal to AB. We can measure DE, 
and thus we find the length of AB, that is, the breadth of 
the river. 

419. The preceding Article requires us to be able to 
draw a straight line at right angles to another straight 
line; we have shewn in Chapter xly. how this may be 
done. We will however uoav solve the problem by another 
method Avhich does not require a right angle. 

420. T'o f.nd tJui distance heticecn two points, one of 
ichich is inaccessible. 



256 



PROBLEMS. 




Let A and B be the two points, 
B being iuacccssible on account 
of a river, or some otlicr obstacle. 

From A measure any straight 
line AG. Fix a picket at any 
point D in the direction BG. 
Produce GA to F, so that AF 
may be equal to AG', and pro- 
duce DA to E so that AE may 
be equal to AD. Fix pickets at 
i^and E. Then find the point G at 
which the directions oi BA and FE intersect; that is, find 
the point from which A and B appear in one straight line, 
and E and F appear in another straight line. 

Then the triangles DAB and EAG are equal in all 
respects, and GA is equal to AB. ^ye can measure GA^ 
and thus we find the length of AB. 

421. In Articles 418 and 420 we assume that we have 
the command of sufficient ground to enable us to trace a 
straight line of the same leilgth as that which we wish to 
nieasm-e. These methods would be practically inapplicable 
if the inaccessible objects were at a considerable distance. 
We shall therefore give a solution w^iich could be used in 
such a case. 

422. To find the distance hetwecn two 'polrds, one <>/ 
which is inaccessible and remote. 

Let A and B bo the two points, 
B being inaccessible and remote. 

Measure any length AG in the 
direction of BA ; and from G in any 
convenient direction measure GD equal 
to GA. Take two strings, each equal 
to GA ; fix an end of one at A, and an 
end of the other at D. Then stretch 
the strings out, so that they form 
straight lines, and so that their other D 

ends shall meet at a point; let E be this point. Then 
AGDE is a rhombus. Place a picket at F, the point at 
which the directions AE 2i\\6. BD intersect. 

The triangles FAB and FED are similar. Therefore 
^i^is to ED as A F is to AB. Thus if we measure EF 
and FA, we can find AB from this proportion. 




DUODECIMALS. 



257 



XLVIII. DUODECIMALS. 



423. Examples relating to square measure aucl to 
solid measure are sometimes worked by a method which is 
called Cross Multiplication or Duodecimals. This method 
is found convenient in practice, and the theory of it is 
instructive ; so that the explanation which we shall now 
give deserves attention. We shall first consider the case 
of square measure. 

424. The student of course knows perfectly well what 
is meant by a square foot and what is meant by a square 
inch; he must now become familiar with another area, 
namely, a rectangle which is twelve inches long and one 
inch broad : we will call this a superficial prime. 

Thus we have the following addition to the Table of 
square measiu'e : 

12 square inches make 1 superficial prime, 

12 superficial primes make 1 square foot. 

425. Any number of square inches greater than 12 can 
be separated into superficial primes and square inches. 
Thus, for example, 

17 square inches = 1 superficial prime 5 square inches, 

32 square inches = 2 superficial primes 8 square inches, 

54 square inches = 4 superficial primes 6 square inches. 

Any number of superficival primes greater than 12 can 
be separated into square feet and superficial primes. Thus, 
for example, 

19 superficial primes = 1 square foot 7 superficial 
primes, 

45 superficial primes = 3 square feet 9 superficial 
primes, 

54 superficial primes = 4 square feet 6 superficial 
primes. 

T, M. 17 



258 



DUODECIMALS, 



~^ B 



426. A rectaiiglo wliicli measures 1 foot by 1 incii is a 
supoi'ficial prime; lience a rectangle which measures 2 feet 
by 1 iiicli contains 2 superficial primes, a rectangle which 
measures 3 feet by 1 inch coutaius 3 superficial primes, 
and so on. 

Again, a rectangle which measures 7 feet by 1 inch 
contains 7 superficial primes ; hence a rectangle which 
measures 7 feet by 2 inches contains 14 superficial primes, 
a rectangle which measures 7 feet by 3 inches contains 
21 superficial primes, and so on. Hence we arrive at a 
general result which is expressed briefly thus : the product 
of feet into inches gives primes. 

^ 427. Required the area of a rectangle which is 8 feet 
9 inches long, and 5 feet 6 inches broad. 

Let ABDG represent the 
rectangle; AB being the 
length, and AG the breadth. 

tSupposo ^^ to be 8 feet, 
and ^4i'' to be 5 feet ; so that 
EB is 9 inches, and FG is 
6 inches. Through E draw 
Ell parallel to AG\ and 
through F draw FK parallel 
to AB : let G be the x^oint of intersection of tlieae 
straight lines. 

Thus the whole rectangle ABDG is divided into four 
parts, namely : 

the rectangle EBKG, which measures 5 feet by 9 
inclies, and contains therefore 45 superficial primes, that is 

3 square feet 9 superficial primes; 

tlie rectangle AEGF, which measures 8 feet by 5 feet, 
and contains therefore 40 square feet ; 

the rectangle GKDII, which measures 9 inches by 
6 inches, and contains therefore 54 square inclies, that is 

4 superficial primes 6 square inches ; 

the rectangle FGHG, which measures 8 feet by fi inches, 
and contains therefore 48 superficial primes, that is 4 
S(juare feet. 




K I) 



DUODECIMALS. 259 

The sum of the first two rectangles is 43 square feet 
9 superficial primes ; tlie sum of the second two rectangles is 

4 square feet 4 superficial primes 6 square inches : the su^ii 
of the four rectangles is therefore 4S square feet 1 supef- 
ficial prime 6 square inches. 

428. The work of the preceding exam- 8 9 
pie is recorded thus : 5 6 

The length is written in one line, and 43 9 
the breadth in another ; feet under feet, 4 4 6 

and inches under inches. 43 \ g 

We first multiply by the 5 which stands for feet : 

5 times 9 are 45 ; 45 superficial primes are 3 square feet 
9 superficial primes : set down 9 and carry 3 ; 5 timea 8 are 
40 ; 40 and 3 are 43 ; set down 43 to the left of the place 
in which the superficial primes "were recorded. 

Then we multiply by the 6 which stands for inches. 
6-times 9 are 54 ; 54 square inches are 4 superficial primes 

6 square inches ; set down 6 to the right of the column 
in which superficial primes are recorded, and cany 4 ; 
6 times 8 are 48 j 48 and 4 are 52 ; 52 superficial primes 
are 4 square feet 4 superficial primes j set down these in 
their proper columns. 

Then add together the two lines which have been 
obtained. 6 square inches are brought down ; 4 and 9 are 
13; 13 superficial primes are 1 square foot 1 superficial 
prime ; set down 1 and carry 1 ; 1 and 4 and 43 are 48. 

Tims the result is 48 square feet 1 superficial prime 
6 square inches. 

429. "VYe may express the result in other forms. 

Thus 1 superficial prime 6 square inches are 18 square 
inches; therefore the result is 48 square feet 18 square 
inches. 

Again, 1 superficial prime is — of a square foot, 6 square 

n 

inches are ■—- of a square foot: therefore the result in 
144 

17—2 



260 DUODECIMALS. 



1 ^ 11 

square feet is 48 + — + -t^ , that is 48 + — + - , that is 



4S,fi, that is 48|. 

These forms of the result agree with those which we 
should obtain without the aid of cross multiplication. 

Thus : 8 feet 9 inches = 105 inches, 5 feet 6 inches = 66 
inches. 105x66 = 6930; 

6930 square inches = 48 square feet 18 square inches. 

35 
Or thus : 8 feet 9 inches = 8| feet= '— feet, 



4 

11 

2 



11 

5 feet 6 inches = 5J feet = — feet, 

35 11 385 ,^, 

4 2 8 ^ 

430. We will now briefly consider the extension of the 
method to examples relating to solid measure ; we must 
introduce two new terms, solid primes and solid seconds, 
the meanings of which are thus assigned : 

12 cube inches make 1 solid second, 
12 solid seconds make 1 solid prime, 
12 solid primes make 1 solid foot. 

431. By proceeding as in Art. 426 we shall easily 
arrive at results which, in conjunction with two results 
already known, may be expressed briefly thus : 

the product of feet into superficial feet is solid feet, 
the ijroduct of feet into superficial primes is solid 
primes, 

the product of feet into square inches is solid seconds ; 

the product of inches into superficial feet is solid 
primes, 

the product of inches into superficial primes is solid 
seconds, 

the product of inches into square inches is solid inches. 



DUODECIMALS. 261 

432. Required the volume of a rectangular paral- 
lelepiped wliich is 8 feet 9 inches long, 5 feet 6 inches 
broad, and 4 feet 3 inches high. 

We have found in Art. 428 that the area of the base is 
48 square feet 1 superficial prime 6 square inches : we will 
now give the remainder of the process : 

48 1 6 
4 3 



192 6 
12 4 6 

204 6 4 6 

We first multiply by the 4 which stands for feet. 
4 times 6 are 24 ; 24 solid seconds are 2 solid primes ; set 
down and carry 2 ; 4 times 1 are 4 ; 4 and 2 are 6 ; set 
down 6 : 4 times 48 are 192; set down 192. 

Then we multiply by the 3 which stands for inches. 
3 times 6 are 18 ; 18 solid inches are 1 solid second 6 solid 
inches; set down 6 to the right of tho column in which 
solid seconds are recorded, and carry 1 ; 3 times 1 are 3 ; 

3 and 1 are 4 ; set down 4 ; 3 times 48 are 144 ; 144 solid 
primes are 12 solid feet; set down in the column of solid 
primes and 12 in the cohimn of solid feet. 

Then add together the two lines which have been 
obtained. Thus the result is 204 solid feet 6 solid primes 

4 solid seconds 6 solid inches. 

433. As in Art. 429 we may express the result in 
other forms, and shew that it agrees with what we should 
obtain without the aid of cross multiplication. If we ex- 
press the result in terms of solid feet it becomes 

204 + A^A 6 



12 144 1728' 
that is, 204 + i + i + ^ , that is 204^^- 



26-2 DUODECIMALS. 

431:. The inclios on a cai-penter's rule are divided 
into ticelfths ; and consequently examples relating to areas 
and volumes may occur in practice in which the dimen- 
sions involve twelfths of an inch. These examples are 
similar in principle to those we have already considered. 
"We have to introduce other new terms ; so that the whole 
series of terms for areas will be assigned by the following 
table : 

a superficial prime is one-twelflh of a square foot, 

a superficial second is tlie same as a square inch, 
namely, oue-twelftli of a superficial prime, 

a superficial third is one-twelfth of a superficial second, 

a superficial fourth is one-twelfth of a superficial third. 

435. Required the area of a rectangle which mea- 
sures 8 feet 9 inches iO tv^'cifths by 5 feet 6 inches 
7 twelfths. 



8 


9 


10 






5 


6 


7 






44 


1 


2 




4 


4 


11 









5 


1 


8 


10 


48 


11 


2 


8 


10 



Tlie residt is 48 square feet 11 superficial primes 
2 superficial seconds 8 superficial thirds 10 superficial 
t'ourtiis. 



EXAMPLES. XLVIII. 2G3 



EXAMPLES. XLYIII. 

Find by duodecimals tlie areas of rectangles having tbe 
following dimensions : 

1. 4 feet, 2 feet 3 inches. 

2. 5 feet, 3 feet 4 inches. 

3. 3 feet 8 inches, 2 feet 6 inches. 

4. 4 feet 5 inches, 3 feet 9 inches. 

5. 5 feet 7 inches, 4 feet 10 inches. 

6. 5 feet 11 inches, 4 feet 7 inches. 

7. 4 feet 3 inches 4 twelfths, 3 feet 3 inches. 

8. 4 feet 8 inches 5 twelfths, 3 feet 4 inches. 

9. 5 feet 4 inches 8 twelfths, 2 feet 7 inches 3 
twelftlis. 

10. 6 feet 8 inches 7 twelfths, 2 feet 4 inches 5 
twelfths. 

Find by duodecimals the volumes of rectangular paral- 
lelepipeds having the following dimensions : 

11. 3 feet, 3 feet, 1 foot 6 inches. 

12. 5 feet, 3 feet, 2 feet 3 inches. 

13. 4 feet, 3 feet 4 inches, 3 feet 3 inches. 

14. 5 feet, 4 feet 8 inches, 3 feet 2 inches. 

15. 6 feet 3 inches, 5 feet 3 inches, 3 feet 9 inches. 

16. 7 feet 5 inches, 6 feet 7 inches, 3 feet 10 inches. 



264 METRICAL ISYSTEM. 



XLIX. METRICAL SYSTEM. 

436. The French system of measures called the metri- 
cal system is frequent -y used in English scientific works \ 
go that we shall here explain the system, 

437. The standard of length is the tnetre which is 
equal to 39"37079 English inches The metre was intended 
to be one-ten-millionth part of the distance from the pole 
to the equator measured on the Earth's surface ; recent in- 
vestigations shew that the metre is about — — of an inch 

shorter than it should have been to correspond to this in- 
tention. 

The standard of area is the are,^ wliich is 100 square 
metres. 

The standard of volume is the stere^ which is a cubic 
metre. 

All the multiples and the sub-divisions of any measure 
are decimal and are formed in the same manner; the mul- 
tiples by syllables derived from the Greek, and the sub- 
divisions by syllables derived from the Latin. Thus 

Myrjameter= 10000 metres. 
Kilometre = 1000 metres, 



Hectometre = 


100 metres, 


Decametre = 


10 metres. 


Decimetre - 


,g metre, 


Centimetre — 


rob "'*''' 


Millimetre = 


1000 "'*'"• 


a hectare = ] 00 ares, a cen 



Similarly a hectare = ] 00 ares, a centiare = YTrT are, a 
dekastere ~ 1 stores, a decistere - — stere. 



EXAMPLES. XLTX. 265 

For liquid measures the standard is a litre, which is a 
cubic decimetre. 

For weight the standard is a gramme, wliich is the 
weiglit of a cubic centimetre of water : it is equal to 15"432 
English grains. 



EXAMPLES. XLIX. 

1. The diameter of a circle is 15 metres : find the 
circumference, 

2. Find the area of a rectangle which is 407*75 metres 
long, and 304 metres wide. 

3. The parallel sides of a trapezoid are 157*6 metres 
and 94 metres; and the perpendicular distance between 
them is 72 metres : find the area. 

4. Find in cubic metres the volume of a wall which is 
48 metres long, 3*4 metres high, and "45 of a metre thick. 

5. Find to the nearest cubic decimetre the volume of a" 
right cone, the height being 2'4 metres, and the radius of 
the base '4 of a metre. 

6. A vessel when empty weighs 1*67 kilogi-ammes, and 
when full of water weighs 6*9 kilogrammes : find the 
capacity of the vessel in cubic decimetres. 

7. A cistern is 8*5 metres long, 6 nietres wide, and 
9'2 metres deep : find bow many hectolitres of water it 
will hold. 



8. The weight of water which a certain cylinder will 
hold is 36 kilogrammes; the radius of the cylinder is 15 
centimetres: find the height of the cylinder to the nearest 
centimetre. 

9. A telegrapli wire is 35 kilometres long, and 2\ 
millimetres in diameter : find the volume in cubic deci- 
metres. 



2GG EXAMPLES. XLIX, 

10. The diameter of an iron ball is "2 of a metre : find 
its Tveiglit in kilogrammes, supposing that any volume of 
iron is 7 "5 times as heavy as an equal volume of water. 

11. Find to the nearest square metre the area of the 
"wliole surface of a right cone, the radius of the base being 
3 metres, and the slant height 8 metres. 

12. Find to the nearest square metre the area of the 
surface of a sphere of which the diameter is 9 metres. 

13. Shew that an acre contains about 40*47 ares. 

14. Shew that a cubic yard contains about 765*4 cubic 
decimetres. 

15. Shew that a gallon contains about 4*54 litres. 

The following exam^Dles involve the extraction of the 
cube root : 

16. A hollow sphere holds a litre: find the radius of 
the sphere. 

17. A vessel in the form of a right circular cylinder 
with its height equal to the diameter of its base holds a 

' litre : find the height. 

18. The volume of a right circular cone with its height 
equal to the radius of its base is a cubic decimetre ; find 
the height. 



2G7 



MISCELLANEOUS EXAMPLES. 



1. The top of a may-pole, being broken off by a blast 
of wind, struck the ground at a distance of 15 feet from 
the foot of the pole : tind the height of the whole may-pole, 
supposing the length of the broken piece to be 39 feet. 

2. Find how many square feet there are in 1200 square 
inches. 

3. The area of two squares is 100 acres, and the side 
of one is three tunes as long as the side of the other : find 
the area of each. 

4. A square contains 2533 feet G4 inches: find its side. 

5. The perimeter of a rectangle is 144 yards, and the 
leDgth is three times the breadth : find the area. 

6. Find the expense of carpeting a room 21 feet long 
and 20 feet broad, with carpet 27 inches wide at 45. Qd. 
per yard. 

7. A room is 16 feet 2|- inches long, 15 feet Z\ inclies 
broad, and 12 feet high: find the expense of covering the 
walls with paper, 9 inches wide, at 2^<^. per yard. 

8. A building has 63 windows: 40 of them contain 
12 panes each 20 inches by 16; the others contain 9 panes 
each 16 inclies square : find the cost of glazing the whob 
at 25. 3d per square foot. 

9. The sides of a triangle are 890, 990, and 1000 links: 
find the area. 

10. The area of a trapezoid is 475 square feet; tlie 
pen^endicular distance between the two parallel sides is 
19 feet: find the two parallel sides, their difference behig 
4 feet. 

11. Two roads cross at right angles ; two men start at 
the same time from the point where the roads meet, one 
man Avalking along one road at the rate of 4 miles an hour. 



268 MISCELLANEOUS EXAMPLES. 

and the other man walking along the other road at the rate 
of 3 miles an hour : find how far apart the men are ten 
minutes after starting. 

12. Find how many square yards there are in A of a 
square mile. 

13. The perimeter of oiio square is 748 inches, and 
that of anotlier is 33G inches: find tlie perimeter of a 
square which is equal in area to the two. 

14. A square contains 3690 feet 81 inches: find its 
side. 

15. Find how long it will take to walk round a field 
containing 13 acres 1089 yards in the form of a square at 
the rate of 2^ miles an hour. 

^ 16. Find the expense of carpeting a room 16 feet 
8 inches long, and 12 feet broad, with carpet a yard wide 
at 6 sJiilliugs a yard. 

17. A room is 24 feet long, 20 feet broad, and 14 feet 
3 inches high : find the expense of covering the walls with 
paper 30 inches wide at Hid. per yard; allowing 8 feet by 
5 feet 3 inches for each of 4 doors, 10 feet by 6 feet 8 inches 
for each of 3 windows, and 6 feet 6 inches by 4 feet for a 
fire-place. 

18. Tl'.e length of a room is double the breadth; the 
cost of colouring the ceiling at ^d. per square yard is 
£2. 12,?. Id., and the cost of painting the four walls at 
2s. 4d. per square yard is ^35 : find the height of the 



room. 



19. The sides of a triangle are 848, 900, and 988 links: 
find the area. 

20. A rectangle is 41 yards 1 foot 3 inches by 10 yards 
10 inches: find how many circles of one inch radius are 
equivalent in area to this rectangle; assuming that the 
area of a circle of one inch radius is f^| square inches. 

21. Construct a diagonal scale; explain its use; shew 
where the points of the compasses should be placed to 
measure ofi" a length 1-37, 

22. Determine the excess of 15 feet square over 
15 square feet. 



MISCELLANEOUS EXAMPLES. 269 

23. There are two rectangular fields equal in area; 
the sides of one field are 945 yards and 1344 yards in 
length, and the longer side of the other field is 1134 yards: 
find the length of the shorter side, and express the area 
of each field in acres, roods, poles, and square yards. 

24. Find the side of a square which contains 7367 
square feet 52 inches. 

25. The area of a rectangular field whose length is 
three times its breadth is 6 acres 960 yards : find its peri- 
meter. Find also the distance from corner to corner. 

26. Find the expense of carpeting a room 18 feet 
9 inches long, and 17 feet 6 inches broad, with carpet 
27 inches wide at 5s. 2>d. per yard. 

27. A room is 15 feet long, 10 feet broad, and 9 feet 
9 inches high : find the expense of painting the walls and 
the ceiling at Is. 9d. per square yard. 

28. A room three times as long as it is broad is car- 
peted at 4s. 6d. per square yard ; and the walls are 
coloured at 9d. per square yard; the respective costs 
being £S. 5s. ihd. and 4 guineas : find the dimensions of 
the room. 

29. The perimeter of an isosceles triangle is 306 feet, 
and each of the equal sides is five-eighths of the third side : 
find the area. 

30. ABODE is a five- sided figure, and the angles at 
B, O, and D are right angles: if AB = 20 feet, BC= 18 feet, 
6'Z> = 32 feet, and DE— 13 feet, find the area of the figure, 
and the length of AE. 

31. The span of a bridge, the form of which is an arc 
of a circle, being 96 feet, and the height being 12 feet, find 
the radius. 

32. A square yard is divided into 576 equal squares : 
find the length of a side of each. 

33. Find the number of acres in the area of the base 
of the great pyramid of Egypt. See page 154. 

34. Find the side of a square whose area is equal to a 
rectangle 81 feet long and 60| inches wide. 



270 MISCELLANEOUS EXAMPLES. 

35. A rectangular field is 300 yards lonj? and 200 broad : 
find the distance from corner to corner. If a belt of trees 
30 yards wide bo planted round the field, find the area of 
the interior space. 

36. In a rectangular court which measures 96 feet by 
84 feet, there are four rectangular grass-plots measuring 
each 22^ feet by 18 : find the cost of paving the remaining 
part of the court at 8|d per square yard. 

37. A room is 30 feet long, 15 feet broad, and 15 feet 
high: find the expense of covering the walls with paper 
4^ feet wide at 4^^^. per yard. 

Also find the cost for a room twice as long, twice as 
broad, and tv»ice as high, the paper being half as wide and 
coating half as much per yard as before. 

38. A room whose height is 11 feet, and length twice 
its breadth, takes 165 yards of paper 2 feet wide for its 
four walls : find how much carpet it will require. 

39. The sides of a triangle are 25, 39, and 56 feet 
respectively : find the areas of the two triangles into which 
it is divided by the perpendicular from the angle opposite 
the largest side on that side. 

40. Make a rougli sketch and find the area of a field 
ABCD from the following measures taken in links : 

BM t\\Q perpendicular from B on yl(7=740, 
i>iVtlie perpendicular from D on AC=Sl% 
^C=1220, ^Ji^-532, AN=im. 

41. The chord of an arc is 24 feet, and the height of 
the arc is 5 feet : find the length of the arc. 

42. A room is 15 feet 9 inches long: find its breadth, 
that it may contain 21 square yards. 

43. The area of a rectangle is 372 feet 32f inches 
square measure; and one side is 20 feet 5^ inches: find 
the other side. 

44. Find the length of the side of a square which is 
equal in area to a rectangle 972 yards long and 363 yards 
broad. 



MISCELLANEOUS EXAMPLES. 271 

45. If a rectangular piece of building" land S75 feet 
6 inches long, and 75 feet 6 inches broad, cost £ll8. 2.9. 6]r/., 
find the price of a piece of similar land 278 feet 9 inches 
long and 157 feet broad. 

46. Find the cost of paving a street half a mile long 
and 47 feet broad, at the cost of *lld. per square yard. 

47. A room is 20 feet 6 inches long, 15 feet 6 inches 
broad, and 16 feet high: find the expense of covering the 
walls with paper, 30 inches wide, at lid. per yard; allow- 
ing for two doors each 8 feet by 3 feet 9 inches, one win- 
dow 5 feet by 7 feet, two other windows each 5 feet by 
4 feet, and a fire-place 4 feet 8 inches by 3 feet. 

48. The carpeting of a room twice as long as it was 
broad at 5 shillings per square yard cost £Q. 2s. 6d.; and 
the painting of the walls at 9^. per square yard cost 
£2. 12s. 6d. : find the dimensions of the room. 

49. ABCD is a four-sided figr.re; BC is parallel to 
AD; AB=BC=CD = ^2^ feet; and ^i>=-733 feet: find 
the area. 

50. The diameter of a circle is 12 feet: find the area 
of a square inscribed in it. 

51. The sides of one end of a frustum of a triangular 
pyramid are 12, 15, r'nd 20 inches respectively ; the longest 
side of the other end is 25 inches : find the other sides of 
this end. 

52. The expense of paving a rectangular court which 
measures 48 feet by 24 feet is ^42 : find the expense of 
paving another rectangular com't which measm^es 60 feet 
by 32 feet. 

53. The area of a rectangle is 3075 feet 70f- inches 
square measure, and one side is 81 feet 9^ inches : find the 
other side. 

54. Assuming that three hectares contain 35881 square 
yards, and that one hectare contains 10000 square metres, 
find the length of a metre in terms of a yard. 

5.5. Find how many planks, each 13^ feet long and 
10| inches wide, will be required for the constraction of a 
platform 54 yards long and 21 yards broad. Find the cost 
of the wood at b\d. per square foot. 



272 MISCELLANEOUS EXAMPLES. 

56. Find the difference in expense of carpeting a room 
17 feet 9 inciies long and 12 feet 6 inches broad, with 
Brussels carpet | of a yard wide at 45. Qd. per yard, and 
with Kidderminster carpet f yard wide at 3^. Qd. per yard. 

57. A room is 15 feet 9 inches long, 9 feet 3 inches 
broad, and 10 feet high: find how many postage stamps 
will be required to cover the walls, allowing for two win- 
dows, each 5 feet by 4 feet, and three doors, each 6 feet 
8 inches by 3 feet. A postage stamp is \% inch long, and 

1 inch wide. 

58. A person has a triangular-shaped garden, the base 
of which contains 200 yards, and divides it into two equal 
parts by a hedge parallel to the base : find the length of 
the hedge. 

59. Verify the following statement by examples : the 
area of the space between two concentric circles is equal 
to the area of a circle which has for its diameter a chord of 
the outer circle which touches the inner circle. 

60. Find the expense of paving a circular court 30 feet 
in diameter, at 2^. ^d. per square foot, leaving in the centre 
a space in the form of a regular hexagon, each side of v/hich 
measures 2 feet. 

61. Find how many bricks there are in a wall which is 
120 bricks long, 15 bricks high, and 2 bricks thick. 

62. Find the thickness of a solid whose length is 

2 yards, breadth l^ yards, and solid content 1 cubic yard 
6 cubic feet and 1296 cubic inches. 

63. A room 18 feet 9 inches long and 13 feet 4 inches 
broad is flooded with water to a depth of 2 inches: find 
the weight of the water, supposing a cubic foot of water to 
weigh 62^ lbs. 

64. If' gold can be beaten so thin that a grain will form 
a leaf of 56 square inches, find how many of these leaves 
will make an inch thick, the weight of a cubic foot of gold 
being 10 cwt. 95 lbs. 

65. A cube contains 2'376 cubic yards : find how many 
linear feet there are in an edse. 



MISCELLANEOUS EXAMPLES. 273 

^^. A closed vessel formed of metal one inch thick, 
and whose external dimensions are 7 feet 3 inches, 6 feet 
5 inches, and 4 feet 3 inches, weighs 2 cwt. 2 qrs, 7 lbs. : 
find the weight of a solid mass of the metal of the same 
dimensions. 

67. A monolith of red granite in the Isle of IMiill is 
said to be about 11.5 feet long, and to have an average 
transverse section of 113 square feet. If shaped for an 
obehsk it would probably lose one-third of its bulk and 
then weigh about 639 tons. Determine the number of 
cubic yards in such an obelisk, and the weight in pounds 
of a cubic foot of granite. 

68. The top of a circular table is 7 feet in diameter, 
and 1 inch thick: find how many cubic feet of wood it 
contains, and what will be the cost of polishing its upper 
surface at sixpence per square foot. 

69. Find how many gallons of water can be held in a 
leathern hose 2 inches in bore and 40 feet long. 

70. The ends of a frustum of a pyramid are right- 
angled triangles ; the sides containing the right angle of 
one end are 2 feet and 3 feet; the smallest side of the 
other end is 8 feet; the height of the frustum is 7 feet: 
find the volume. 

71. Find how many bricks of which the length, breadth, 
and thickness are 9, 4|, and 3 inclies will be required to 
build a wall of whicli the length, height, and thickness 
are 72, 8, and 1^ feet. 

72. Find the length of a solid Avhose thickness is 1 foot, 
breadth 18 inches, and solid content 3 cubic feet 216 cubic 
inches. 

73. A box is 4 feet long, 2 feet 6 inches wide, and 
1 foot 6 inches deep ; in it are packed 252 books each 
8 inches long, 5 inches wide, and \\ inches thick: find how 
many more books each 6 inches long 3 inches wide and 
\\ inches thick can be packed in the box. 

74. Find the height of a parallelepiped which contains 
659 feet 1243 inches, and vvhose base is 26 feet 6 inches. 

75. Find the edge of a cube which contains 
10970-645048 cubic inches. - 

T. M. 18 



274 MISCELLANEOUS EXAMPLES. 

76. The surface of a cube contains 3936G square feet: 
find the length of an edge and the cubical content. 

77. Find the volume of a cone, the height of which is 
one yard and the radius of the base one foot>, 

78. A garden roller of iron is half an inch thick, the 
length is 30 inches, and the diameter of the inner surface 
is 20 inches : find the weight, supposing a cubic inch of iron 
to weigh 4'562 ounces. 

79. A square tower 21 feet on each side is to have 
either a fiat roof covered with sheet lend which costs 
6 pence per square foot, or a pyramidal roof whose vertical 
height is 10 feet, covered with slates which cost 18^. 9c/. 
per hundred, and each of which has an exposed surface of 
12 inches by 9 inches. Find the cost in each case. 

80. The surface of a certain solid is three times as 
great as the surface of a similar solid : find the proportion 
which the volume of the first solid bears to the volume of 
the second. 

81. The three conterminous edges of a rectangular pa- 
rallelepiped are 3G, 75, and SO inches respectively : find the 
QiCigo of a cube wlrlch will be of the same capacity. 

82. A river 30 feet deep and 200 yards wide is flowing 
at the rate of 4 miles an hour : find how many tons of 
water run into the sea per minute. 

83. Find the number of inches in the side of a cube 
whose solid content is 5'359375 cubic feet. 

84. The content of a cistern is the sum of two cubes 
whose edges are 10 inches and 2 inches, and tlie area of 
its base is the difference of two squares whose sides are 1^ 
and Ig feet: find the depth of the cistern. 

85. The cost of a cube of metal at ^3. 105. Ad. per cubic 
inch is £1206. 45. 4(j?.: find the cost of gilding it oyer at ^d. 
per square inch. 

86. If gold can be beaten out so thin that a gi'ain will 
form a leaf of .56 square inches, find how many of those 
leaves will be required to make up the thickness of a sheet 
of paper; a cubic foot of gold weighing 1215 lbs. Avoirdu- 
pois, and 400 sheets of paper making a book one inch thick. 



MISCELLANEOUS EXAMPLES. 27 o 

87. A hemispherical basin 15 feet in diameter -will hold 
120 times as much as a cylindrical tub, the depth of ^vhich 
is 1 foot 6 inches : find the diameter of the tub. 

88. A right-angled triangle of "vvhich tlie sides are 3 
and 4 inches in lengtli is made to turn round on the longer 
side : find the volume and the area of the whole surface of 
the cone thus formed. 

89. The height of a frustum of a pp'amid is 4 inches; 
tlic lower end is a rectangle which is 9 inches by 12 ; tlio 
upper end is a rectangle of which the longer side is 8 inches ; 
find the volume of the frustum. 

90. The area of the surface of a sphere is 25 square 
inches: find the vohime of the sphere. 

91. Find how many superficial feet of inch plank can 
be sawn out of a log 20 feet 4 inches long, 1 foot 10 inches 
wide, and 1 foot G inches deep. 

92. Find the weig'nt of water in a bath 6 feet long, 
3 feet wide, and 1 foot 9 inches deep. 

93. Find the length of the side of a cube which con- 
tains 344324701729 cubic inches. 

94. Find how many cubical packages each having 4i 
inches in an edge will fill a box whose length, breadth, and 
depth are 2 feet 2 inches, 3 feet 3 inches, and 4 feet 4 inches. 

95. The flooring of a room 14 feet 3 inches long, and 
13 feet 4 inches broad is composed of half-inch planks, cacli 
8 inches wide and 10 feet hnig : find how manv v.ill bo 
required, and the weight of the whole if one cubic inch of 
wood weighs half an ounce. 

96. Supposing a cubical room to contain 4665G cubic 
feet, find the expense of carpeting it v.ilh carpet 27 inches 
wide at As. 6d. per yard. 

97. A pyi'amidal roof 16 feet liigh, standing on a 
square base which is 24 feet on each side, is covered with 
sheet lead -^q of an inch thick : find the weight of the lead, 
supposing a cubic inch to weigh 7 ounces. 

If the lead is stripped ofi* and cast into bullets, each of 
which is in tlie form of a cylinder, ^ inch long and YV ^''^^^ 
in diimetcr, terminated at one end by a cone of the same 

IS— 2 



276 MISCELLANEOUS EXAMPLES 

diameter and f inch liigli, find how many bullets there 
will be. 

98. A sphsrical cannon ball, 9 inches in diameter, is 
melted and cast into a conical mould tiie base of which is 
18 inches in diameter : find the height of the cone. 

99. A cylinder, 24 feet long and 4 feet in diameter, 
is closed by a hemisphere at each end: find the area of the 
whole surface. 

100. A tin funnel consists of two parts; one part is 
conical, the slant height is 6 inches, the circumference at 
one end 20 inches, and at the other end ]| inches; the 
other part is cylindrical, the circumference being Ij inches 
and the length 8 inches : find the number of square inches 
of tin. 

101. Find how many cubes whose edges are 2| inches, 
can be cut out of a cube of which the edge is 22 inclies. 

102. The breadth of a room is two-thirds of its length 
and three halves of its height ; the content of the room is 
5832 cubic feet : find the dimensions of the room. 

103. Find the length of tlie side of a cube which con- 
tains 733626753859 cubic inches. 

104. One solid contains 30| cubic feet; another solid 
contains 4^ cubic yards : find what multiple the latter is 
of the former. 

105. Find the number of gallons of water which pass 
in 10 minutes under a bridge 17 feet 8 inches wide ; the 
stream being 10 feet 11 inches deep, and its Telocity 
4 miles an hour. 

106. A cubic inch of metal expands so that each face 
is increased by "0201 of its former area: find the increase 
of volume. 

107. A sphere and a cube have the same surface: shew 
that the volume of the sphere is 1*3820 times that of the 
cube. 

108. A sphere has the same surface as a right circular 
cylinder with its height equal to twice the radius of its 
base: shew that the volume of the sphere is 1*2247 times 
that of the cylinder. 



MISCELLANEOUS EXAMPLES. 277 

109. A sphere and a cube have the same vokime : shew 
that the surface of the cube is 1-2407 times that of the 
sphere. 

110. A sphere has the same volume as a right circular 
cylinder witlf its height equal to twice the radius of its 
base : shew that the surface of the cylinder is ri-t47 times 
that of the sphere. 

ill. Find to the nearest lb. the weight of the w^ater 
which a cistern will hold, whose length, breadth, and depth 
are 5 feet 6 inclies, 3 feet 9 inches, and 1 foot 3 inches 
respectively. 

11-2. A bed of gravel 4 feet 6 inches in deptli extends 
over the whole of a field of 3 acres 3 roods : find the value 
of the gravel at sixpence per cubic yard. 

113. The volume of a cube is 5 cubic feet 821 cubic 
inches : find the length of a diagonal of the cube. 

114. Supposing a brick to be 9 inches long, 4|- inches 
wide, and 3 inches tiiick, and to weigh 5 lbs., find the 
weight of a stack of bricks 10 feet high, 6 feet wide, and 
3 feet thick. 

115. A pyramid on a square base has four equilateral 
triangles for its four other faces, each edge being 20 feet : 
find the volume. 

116. The radius of the base of a cylinder is 3 feet, and 
the height is 20 feet : find the volume. 

117. A ditch of a certain length is 4 feet deep, 16 feet 
broad at the top, and 12 feet broad at the bottom : if the 
ditch be half filled with v>'ater, find the depth of the 
water. 

118. The radius of one end of a frustum of a cone is 
18 inches: the radius of the other end is 12 inches; the 
height is 15 inches : find the volume. 

119. If the weight of an iron ball, 4 inches in diameter, 
is 9 lbs., find the weight of a shell whose external and in- 
ternal cUameters are 7 inches and 3 inches respectively. 

120. The area of the whole surface of a right circular 
cone is 32 square feet, and the slant height is three times 
the radius of the base : find the volume of the cone. 



278 MISCELLANEOUS EXAMPLES. 

121. The area of tlie base of a pa,rai]elepiped is a sqiir^re 
yard, and the hoiglit is 2 feet 6 ineiies : tind the volume. 

122. A layer of coal 5 feet thick imdcdies the vviiole 
of an estate of 120 acres : find the value of the coal at 12 
shillings per ton, supposing a cubic yard of the coal to 
weigh a ton. 

123. Find the number of cubic inches in a piece of 
plate glass 5 feet long, 3 feet 4 inches wide, and fths of an 
inch thick. 

124. The area of the coal field of South Wales is 1000 
square miles, and the average thickness of the coal is GO 
feet. If a cubic yard of coal weigh a ton, and the annual 
consumption of coal in Groat Britain be 70U00000 tons, 
find the number of years for which this coal field alone 
would supply Great Britain ^vith coal at the present rate 
of consumption. 

125. A pyramid is cut into two pieces by a plane 
parallel to the base, midway between the vertex and the 
base : shew that one piece is equal to seven times the 
other. 

126. A cylinder is 5 feet in diameter, and its volume 
is 785"4 cubic feet : find the height of the cylinder in feet. 

127. The ends of a prismoid are rectangles, the corre- 
sponding dimensions of which are 8 feet by 7 and 10 feet 
by 6, and the height is 4 feet : find the volume of the 
prismoid. 

128. The height of the frustum of a cone is 7 feet, aiid 
th.e radii of the two ends are 4 i'eet and 5 feet respectively ; 
the frustum is cut into two pieces by a plane parallel to 
the ends and distant 3'8S4 feet from the smaller end : shew 
that the two pieces are of equal volume. 

129. A solid ball 4 inches in radius of a certain 
material weighs 8 lbs. : find the weight of a s]>herical shell 
of that material, the internal diameter of which is 8 inches, 
and the external diameter 10 inches. 

130. The circumference of a great circle of a ball is 
15*708 feet : find the whole surface of the ball. 



MISCELLANEOUS EXAMPLES, 279 

131. A stone wall is built 10 feet 6 inches high and 

2 feet 3 inches thick; the excavation from which the stone 
was taken measures in length, breadth, and depth, 30 feet, 
28 feet, and IS feet respectively; find the length of the 
wall. 

132. A cistern measures in length, depth, and breadth 
5 feet, 4 feet, and 3 feet respectively; the cistern after 
being filled with water can be emptied by a pipe in an 
hour and a half: find how many gallons are discharged 
through the pipe in a minatc. 

133. Find how many cubic feet of deal are contained 
in 200 planks each 15 feet long, 10 inches wide, and l| 
inches thick. 

134. A pond whose area is 4 acres is frozen over with 
ice to the uniform thickness of 6 inches : if a cubic foot of 
ice weigh 896 ounces Avoirdupois, find the weight of ice 
on the pond in tons. 

135. Suppose that the coal consumed in a month in 
England were formed into a square pyi-amid on a base 
equal to that of the great pyramid of Egypt : find the 
height of the p^i-amid which would be thus formed. See 
pages 154 and 278. 

136. A trench is dug 8 feet deep, 14 feet wide at 
the top and 10 feet wide at the bottom; and the earth 
removed is thrown up by the side of the trench so as to 
form a bank sloping on each side at the same angle to the 
horizon, the height of the bank being three-fourths of its 
base : find the height of the bank. 

137. Find the volume of a cyhnder 40 feet high, the 
radius being 8 feet. 

138. A bucket is in the form of a frustum of a cone ; 
the diameter at the bottom is 1 foot, and at the top 1 foot 

3 inches ; the depth is 1 foot 6 inches : find to the nearest 
pound how much more the bucket weighs when full of 
water than v*hen empty. 

139. If 30 cubic inches of gimpowder weigh 1 lb., 
find the diameter of a hollow sphere which will hold 
11 lbs. 



280 MISCELLANEOUS EXAMPLES. 

140. The height of a zone of a sphere is 2^ feet, and 
the diameter of the sphere is G^ feet : find the area of the 
curved surface. 

141. A field contains 1 acre, 2 roods, 16 poles : find 
how many cubic yards of earth will be required to raise 
the surface of the field 18 inches. 

142. A cubic inch of gold is beaten into gold leaf 
sufiicient to cover 7 square feet : find the thickness of the 
gold leaf. 

143. Find how many cubic feet of air a room will con- 
tain which is 24 feet 9 inches long, 18 feet 4 inches broad, 
and 10 feet 8 inches high. 

144. In laying the foundation of a house an excavation 
is made 40 feel long, 30 feet broad, and 6 feet deep ; the 
earth removed is spread uniformly over a field containing 
lialf an acre : find how much the surface of the field will 
be raised supposing that each cubic foot of earth is in- 
creased ^ of a cubic foot by the removal. 

145. Find the volume of a pyramid the height of 
which is 12 inches, and the base an equilateral triangle 
each side of which is 10 inches. 

146. Find what length of wire "08 of an inch in 
diameter can be formed out of a cubic inch of metal. 

147. A gutter is formed by joining two equal planks 
so as to have two of their longest edges in contact, the 
planks are 5 inches wide, and they are fastened together 
so that the extreme breadth of the gutter is 8 inches : if 
the gutter is 4 yards long, find how many cubic inches of 
water it will hold. 

148. The radii of the ends of a frustum of a right 
circular cone are 4 inches and 5 inches respectively; the 
lieight of the frustum is 3 inches: find the volume of the 
whole cone. 

149. If a 12 lb. shot have a diameter of 4'4 inches, find 
the weight of a shot the diameter of which is 3*96 inches. 

150. The area of the curved surface of a right circular 
cylinder is 600 square inches; and the height of the 
cylinder is 25 inches : find tlie radius of the base. 



MISCELLANEOUS EXAMPLES. 281 

151. A rectangular field is 440 yards long, and 154 
yards ^vide : find its area in acres. Also lind the aieas of 
the portions into which it is divided by a straight line 
drawn from the middle point of one side to one of the 
opposite corners. 

152. The walls of a room 21 feet long, 15 feet 9 inches 
wide, and 11 feet 8 inches high, are painted at the expense 
of i>'9. Vis. 8d. : find the additional expense of painting the 
ceiling at the same rate. 

153. A parallelogram has two sides which are 8 feet 
9 inches long, and two ^vhich are 7 feet 4 inches long, and 
a diagonjd which is 11 feet 7 inches long : determine 
whether the parallelogram is a rectangle. 

154. From a point in the circumference of a circle two 
chords are drawn at right angles, and their lengths are 13 
and 17 inches respectively : find the area of the circle. 

155. The area of the Yorkshire coal field is 937^ square 
miles, and the average thickness of the coal is 70 feet. Ii 
a cubic yard of coal weigh a ton and the annual consump- 
tion of coal in England be 70000000 tons, find the number 
of years for which this coal field alone would supply Great 
Britain with coal at the present rate of consumption. 

15b'. If the coal consumed in one year in England, 
were piled up into a rectangular stack having for base an 
area of ten acres, find the height of the stack to the nearest 
yard. 

157. A cubic foot of gold is extended by hammering 
so as to cover an area of 6 acres : find the thickness of the 
gold in decimals of an inch, correct to the first two signi- 
ficant figures. 

158. Find to three decimal places the number of 
gallons in a cubic foot. 

159. A cubic inch of brass is to be drawn into a wire 
2^ of an inch in diameter : find the length of the wu-e to 
the nearest inch. 

160. The sides of a right-angled triangle are 3 inches 
and 4 inches respectively : find the volume of the double 
cone formed by the revolution of this triangle round its 
hypotenuse. 



282 MISCELLANEOUS EXAMPLES. 

161. Tlio base of an afinarium is a square, the height 
is lialf a side of the base, and there is no lid ; the glass 
cost ^1. lis. od. at 15.?. a square yard: find the number of 
gallons the aquarium will hold. 

162. Supposing a cubic foot of brass to weigh 8500 
ounces, find the weight of a yard of brass wire the thick- 
ness of wliich is -g^Q of an inch. 

163. A right-angled triangle of which tlie sides are 
6 and 12 inches in length is made to turn round the hypo- 
tenuse : find the voluaie and the surface of the double cone 
thus formed. 

164. The weights of two globes are as 9 to 25; the 
"weigiits of a cubic inch of the substances are as 15 to 9 : 
compare the diameters of the globes. 

165. A gold wire of "01 of an inch in thickness is bent 
into a ring of one inch internal diameter : if the area en- 
closed by the ring be gilded with a weight of gold equal to 
the weight of the ring, find the thickness of the gilding. 

166. A solid is composed of a cone and a hemisphere 
on opposite sides of the same circular base, the diameter 
of which is 2 feet, and the vertical angle of the cone i;^ 
a right angle : the sulid is immersed in a cylinder full of 
water, whose circular section also has a diameter of 2 feet, 
so that the vertex of the cone rests on tlie centre of tiie 
cylindrical base, while the highest part of the hemisjthero 
just coincides with the surface of the water : find the quan- 
tity of water remaining in the cylinder. 

167. A hemispherical ball of 6 feet in diameter is 
partially buried wIlIi its mouth downwards, and in a hori- 
zontal position, so that only one- third of the height appears 
above the ground : find what quantity of earth nuist be 
dug out in order to leave the ball entirely uncovered, and 
just surromided by a cylindrical wall of earth. 

168. The frustum of a right cone is 6 feet high, the 
radius of the smaller end is 2 feet, and the radius of tlie 
larger end is 3 feet : find the position of the plane parallel 
to the ends which will divide the frustum into two equal 
parts : find also the volume of each jjart. 



MISCELLANEOUS EXAMPLES. 283 

1G9. Find to the nearest square inch tlto quantity of 
leather required to cover a spherical foot bail which mea- 
sures 23 inciies iu circumference. 

170. Find tlie vohime of a cask in gallons, the length 
boing 47"5 inches, the bung diameter iS"6, and the h.cad 
diameter •2G"5. 

171. An area in the form of an equilateral triangle is 
paved at the rate of dd, per square foot, and it is fenced 
at the rate of 5 shillings per foot : shew that the cost of 
fencing is to the cost of paving as 80 Jo is to three times 
the number of feet iu a side. 

172. Find the side of an equilateral triangle, supposing 
it cost as much to pave the area at ^d. per square foot as 
to fence the three sides at 5 shillings per foot. 

173. A rectangle is '202 of an inch longer than a certain 
square, and "2 of an inch narrower; but contains the 
same area : shew by a figure that the number of inches iu 
the side of the square is the product of '202 into '2, divided 
by their difference. 

174. In measuring th.e edges of a cubical box to ascer- 
tain its conten.t, an error of "202 of an inch is made in excess 
for the length, and of '2 of an inch iu defect for the 
breadth, the height being properly measured ; the cal- 
culated volume agrees with the true volume : find the 
volume in cubic inches. 

175. The radius of a circle is J2 inches; two parallel 
straight lines are drawn in it, each an inch from the 
centre : find the area of the part of the circle between 
the straight lines. 

176. A square hole 2 inches wide is cut through a 
solid cylinder of which the radius is v''2 inches, so that 
the axis of the hole cuts at right angles the axis of the 
cjlinder : find how much of the material is cut out. 

177. A vessel is to be made in the form of a rect- 
angular parallelepiped on a square base, and another 
vessel of the same capacity in tlie form of a right circular 
cylinder ; the vessels are to have no lids, and each vessel 
Ls to be made of the most advantageous shape for the sake 



284 MISCELLANEOUS EXAMPLES. 

of saving material : shew that the material in the cylindrical 
vessel is about *92 of the material in the other vessel 

178. Shew that the same result as in the preceding 
Example holds if the vessels are to have lids. 

179. A pyramid on a square base has every edge 
100 feet long : find the edge of a cube of equal volume. 

180. Verify by calculating various cases the following 
statement : a right circular cone is divided into a cone and 
a frustum of a cone, and the frustum is trimmed just enough 
to reduce it to a right circular cylinder ; if the height of the 
frustum is one-third of the height of the original cone the 
volume of the cylinder is greater than in any other case, 
and is four-ninths of tlie volume of the original cone. 



285 



ANSWERS. 

[The answers when not exact are given to the nearest 
figure ; so that sometimes they are a little too great and 
soiDietimes a little too small] 

V. 1. 557 feet. 2. 8545 feet. 3. 382 ft. 10 in. 
4. 945 Yards 1 foot. 5. 554-92. 6. 5858-66. 

7. 388-69. 8. 1840-78. 9. 333 feet. 10. 8225 feet. 

11. 98 ft. 9 in. 12. 259 yds. 2 ft. 13. 482'54. 
14. 3270-31. 15. 321-77. 16. 1824-14. 
17. 19488 + 5687 ft. 18. 12637,12012 ft. 19. 7 ft. 
20. .32 + 24 ft. 21. 14 + .30 ft 22. 1-4142135624 in. 
23. 155-56 ft. 24. 84-32 ft. 25. 98 vds. 

26. 9-64 ft. 27. 5-66 ft. 28. 8-485 ft. 

29. 5-74 ft. 30. 21-91 in. 

3J. 11-832, 11-314, 10-392, 8-944, 6-633- 32. 9-75 ft. 

VI. 1. 5-6 indies. 2. 25*98 in. 3. 40 ft. 
4. 67^ ft. 5. 5 ft. 2^ in. 6. A quarter of an 
inch to a mile. 7. 68 miles. 8. 30 in. 9. 9|in. 
10. 4f in. 11. 25 ft. 12. 10 ft., 12 ft. 

YII. 1. 16^ ft. 2. 22-42 ft. 3. 5^ in. 

4. 1-74 ft. 5. 8 ft. 9 in. 6. 9-65 ft. 7. 29 ft. 

8. 25-03 ft. 9. 8-23 in. 10. 44-72 in. 11. 4 ft. 

12. 4-75 ft. 13. 50-7 ft. 14. 12563 ft. 

VIII. 1.44 ft. 2. 271yds. 1ft. 3. 672 yds. 8 in. 

4. 4 furlongs. 5. 848232 ft. 6. 58ri96 ft. 

7. 5235-4764 ft. 8. 942-48 yds. 9. 21 yds. 

10. 70 yds. 11. 238 vds. 12. 560 yds. 

13. -3183 of a foot. 14. 7*9577 feet 15. 103-4505 ft 
16. 70028 yds. 17. 306. 18. 360. 

19. 190985 feet 9,0. 467 feet 



286 ANSIFERS, 

IX. 1. 12-5GG4 in. 2, 42-656 in. 3. 35°-81. 

4. 67"-3. 5. 38§ in, 6. 56-19 in. 

7. 6-il7in. 8. 17S54in. 9. 47-7602 in. 

10. 5-1416 in. 11. 19-45 in. 12. 141897 in. 



XL 1. 196. 2. 576. 3. 756 J. 4. 91.5^^. 

5. 113 yds, 7 ft. 6. 152 yds. 1ft. 7. 348 yds. 4 ft, 

S. 413 yds. 4 ft. 9. 14 yds. 2 ft. 64 in. 

10. 34 yds. 6 ft. 16 in. 11. 73 yds. 6 ft. 9 in. 

12. 213 yds. 4 ft. 52 in. 13. 2 acres 4 poles. 

14. Sac. 1 ro, 1 po. 15. 15 ac. 2 ro, *04po. 

16. 70 ac. 2 ro. 6-9376 po. 17. 32512-5 sq. ft. 

18. 3938 sq. yds. 2 ft. 76-5 in. 19. 7 ac. 2ro. 'Spo. 

20. 16 ac. 3 ro. 16-7168 1)0. 21. 42 yds. 22. 85 yds. 

23. 273 yds. 24.' 440 yds. 25. 880 yds, 

26. 110 yds. 27. 8-004 ft. 28. 127 yds. 

29. 10-954. 30. 16*941. 31, 65-597. 

32. 68-823. 33. 2556169. 34. 346-107. 

35. 3-742 in. 36. 1^ in. 37. 280. 

38. 432. 39. 279. 40. 374|. 

41. 34 yds. 42. 63 yds. 5 ft. 43, 127 yds. 4 ft. 

44. 180 yds. 4 ft. 45. 7 yds. 8 ft. 108 in. 

46. 16 yds. 96 in. 47. 24 yds. 1 ft. 80 in. 

48. 59 yds. 87 in. 49. 3 ac. 34 po. 

50. Sac. 2ra. 34-6363 po. 51. 9 ac. 2 ro. ir6224po. 

52. 13 ac. Iro. 22-72 po. 53. 32 ft. 54. 44 yds. 

.55. 352 yds. 56. 1100 yds. 57. 110 yds. 

58. 36^\ yds. 59. 78 yds. 1 ft. 60. 2 yds. 

5 

61. -125. 62. - . 63. 660. 64. 3 chains. 

o 

65. 53040 sq. ft. 66. 11 feet. 67. 10-488 ft. 

68. 49. 69. 4320. 70. 324. 71. 816. 

72. 48. 73. S')]^. 74. 40. 75. 84. 

76. 6:)6960. 77. 3200. 78. 3 84. 79. 20. 

80. 30780. 81. 3456. 82. 162. 8.3. As 2 is to 5. 

84. 280 ft. 85. ISlOf. 85. 1444 sq. ft. 

64 

87. 21, 63 ft. 88. — - of an ounce. 91. £6. 6s. 
' 13o 

92. ^1102. 105. 93. ^3. 155. G^d. 94. £2. 105. 

95. ^18. 145. 5H. 96. i;245. 18.9. U^. 

97. £25. 75. 2cl 98. £02. l5. 4c/. 99. 43 ft. 



ANSWERS, 2S7 

100. 71 vds. 101. ^33. 102. 32. 103. 44. 

104. 46 yds. 2ft. 105. 24yds. 33in. lOG. 8S yds. 2ft. 

107. 81 vds. 27 in. \0%. £\Q.Os.l\d. 109. ^30. Is. 3d 

110. £:^. 1 Is. 1 IjV^. 1 1 1. -£23. 1 15. 9c/. 

112. £i'2.\ds.df^d. 113. £4. 65. 11 r/. 114. ^5. 25.9 J^. 

115. £6.\os.Wid. 116. -£8.185. 9d 1 1 7. £8. 45. 0|</. 

118, 9 ft. il9. 2Sjsq.yds., £S.Ss.dd., 20a|sq.yds. 

120. 109yds. 1ft. 121. 1 80 yds. 1 ft. 4 m. 122."£10. 105. 

123. £11. 45. Qd. 124. £d. Is. Md. 



XTI. 1. 70sq.ydg. 2. 177 sq. yds. 5 ft. 

3. 249 sq. yds. 3 ft, 72 in. 4. 13 ac. 1 ro. 277888 po. 

6. 25 ft. 6, 70 yds. 7. 5 ft. 8 in. 8. 35ft. Sin. 
9. 4 ft. 10. 9 ft., 4 ft. 6 in. 11. 498-8 sq. ft. 

12. 886-8 sq, ft. 

SIII. 1. 72 sq.ft. 2. 212-0 sq.ft. 

3. 40 sq.yds. 1 sq. ft. 81 sq. in. 

4. 8 ac. 2^0. 25-568 po. 5. 6090. 6. 54264. 

7. 24-995. 8. 42-214. 9. 12. 10. 1848. 
11. 2772. 12. 69300. 13. 2310. 14. 3570. 
15. 6006. 16. 9240. 17. 18060. 18. 66990. 
19. 223860. 20. 5515650. 21. 2-905. 22. 20-976. 
23. 24-249. 24. 109-982. 25. 379473. 

26. 463-757. 29. 577-5. 1732-5. 30. 1026S, 3080, 5133|. 

31. 12ft. 32. 45 feet; 540, 630 sq. ft!' 

33. 2400, 2600, 1800, 3200 square feet. ^ 

|ft.. AB.^-^, na-^, AcJ-l 

147| sq.ft. 35. £6.\\s.G'id. 36. £2.155. 

37. 2116. 38. 10000, 2000, 8000 sq. vds. 
39. 750 sq. ft. 40. 2100 sq. ft. 41. £10. 4.9. ^d. 
42. 293. 43. 141|. 44. 12505. 45. 27r45. 



34. DB= — ft., AB = rr^ , DC= — , AC^ — ; area 



XIV. 1. 463-7408. 2. 1134 sq. ft. 

3. 72 6531 sq. cli. 4. 63765 sq. ch. 5. 1372 sq.ft. 

6. 29-88 eh. *J. 40 sq. ft. 8. 44 sq. ft. 

9. 204 sq.yds. 10. 5 sq. cli. 11. 147 sq. ch. 

12. 1.52075 so. ch. 13. 125 yds. 14. 280^ yds. 

15. 17i, 22^ sq. ft 16. 60, QS, 76 sq. yds. 



288 



ANSWERS. 



17- 312 sq.ft. 18. An acre. 19. 1152 sq. yds., £19. 4^. 

20. 421 ft. 21. 1800 sq. ft. 22. 12G0 sq. ch. 

23. 839-553 sq. ft. 24. 16-825 sq. ft. 25. 688 sq. ft. 

26. 506430 sq. ft. 27. 7200. 7200 sq. ft. 

28. 10296 sq. ft. ; 125, 82-368 ft. 

29. 1054, 625, 566-6304 ft. 

XY. 1. 134 sq.ft. 2. 110"865sq. ft. 3. 150-6 sq.ft. 

4. 6813-52 sq.ft. 5. 142-557 sq. ft. 6. 1039-23 sq. ft. 

7. 6495-2 sq. ft. 8. 259*81 sq. ch. 9. 2^/2 sq. ft. 

10. 6 X -51764 sq. ft. : see Art. 99. 



1386 



XYI. 1. 

4. 1963-5. 

7. 6-64. 8. 

11. 301-79. 

14. 3769-92 sq. ft. 

17. 15-116 feet. 

20. 554622 sq. ft. 

23. 5-657 ft. 

26. 539-3057 sq. ft, 

29. 407-01. 

32. 5857 sq. ft. 

35. 54-9376 sq. ft. 

38. 104-2 ft. 39 

41. 71-62 sq.ft. 
52.- 2-48 sq. ft. 



2. 7857^. 
5. 3091.535-46. 
58-86. 9. 283-53, 

12 2978-9. 
15. 236-24832 so. ft 
18. 8-956 ft. 
21. 523-16 ft. 
24. 8-05 in. 
27. 136. 
30. iS'47. 2-48.9. 
33. 141-Sft. 
36. 184-9392 sq. ft. 
1-8426 sq. ft. 40. 
42. 140-374 sq. ft. 



3. 1369028f 

S. 547392384. 

10. 12616. 

13. 188-496 sq. ft. 

16. 1.5-094 in. 

19. 38993 sq.ft. 

22. 1134-4 ft. 

25. .5-77, 8-16 in. 

28. £141. 7-445. 

31.^19.05. IM 

34. 4.5-1 ft. 

37. 2024-8 sq.ft. 

1296883896 sq.ft. 

51. '81 sq.ft. 



4. 

/. 
11. 
14. 

16. 
18. 
20. 
2.3. 



XYII. 1. 125-664 
43-63 sq. ft. 



8. 
12. 



2. 150-7968. 
5. 97-74 sq. ft. 
80^57. 9. 11-75 ft. 
15-41 ft. 13. 2942 sq. in. 
15. 1-180 sq. ft. 
see Art. 99. 17. 

28-05 sq. ft. ' 19. 398-25 sq. ft. 

20-382, 686-478 sq. ft. 21. 61'464. 



69-26 ft. 
112^-5. 
27-53 sq. 
61-42 so. 



m. 
ft. 



3. 562-9747. 
6. 18-54 ft. 

10. 7-4375 ft. 

; see Art. 122. 

; see Art. 167. 

296-04 sq. ft. 

22. 28-546. 



9-059. 



24. 1-180. 



25. -1489 sq. ft. 



xYiri. 

5. 56-907. 
9. 70-641. 



1. 



162. 

6. 8-403. 
10. 22-0957 



2. 192. 



3. 

7. 263-9. 
11. -7817. 



288. 



12. 



4. 27. 
8. 59-307. 
•6931. 



ANSWERS. 28.9 

XIX. 1. 324. 2. 653-4 sq. in. .3. 1 in. to 36. 

4. 1 in. to G60. 5. 7'2 iu. 6. 6-33G inches to a mile. 

7. 11832, 17-748 ft. 8.221,238,255. 

9. 50-96, 109-20, 145-60. 

10. The square is 1-299 times the triangle. 

11. The hexagon is 1-1547 times the square. 

12. The circle is 1-2732 times the square. 

13. The circle is 1*1028 times the hexagon. 

14. 15-197 ft. 15. 61-237 ft. 16. 13-236 ft. 
17. 6-7082, 9-4868, 11-6190, 13-4164 ft. 18. The 

perimeter of the triangle is 1-14 times that of the 
squai-o. 19. 2-638 ft. 20. 3-114 ft. 

XXII. 1. 18 ft. 1664 inches. 2. 216 ft. 

3. 107 ft. 297 in. 4. 4492 ft. 216 in. 5. 38ft. 192 in. 

6. 78 ft. 810 iu. 7. 200 ft. 200 iu. 8. 3399 ft. 1008 iu. 

9. 68 ft. 10. 20 ft. 1440 in. 11. 32 ft. 752 in. 

12. 66 ft. 1296 iu. 13. 9 in. 14. 4*32 in. 

15. 5 ft. 16. 5 ft. 10 in. 17. 20 sq. ft. 

15. 4 sq.ft. 48 in. 19. 17sq.ft. 2iu. 20. 34 sq.ft. 120 in. 
21. 1346. 22. 1777. 23. 2314. 24. 3605. 

25. 70. 26. 89. 27. 141. 28. 365. 
80. 31-255875. 31. 24 ft. .32. 15360. 33. 384. 
34. -000004 of an iuch. 35. 35'314. 36. 14 lbs. 
37. 116109 ounces. 39. 314^. 40. 3-8 in. 
41. 291G00. 46. 6196. 47. 6810|. 48. 2^1^., |-lb. 
49. 9 cubic ft. 50. 3 ft. 6 in. 52. 1*26 ft. 
63. 14-42 ft. 54. 103-51 in. 55. 3-297 ft. 
56. 20 m. 

XXIII. 1. 15 It. 1050 in. 2. 62 ft. 729 in, 
3. 109 ft. 133 iu. 4. 194 ft. 363 in. 5. 1ft. 162 in, 
6. 3 ft. 1056 m. 7. 5 ft. 408 iu. 8. 1 1 ft. 1452 iu, 
9. 43-9824. 10. 83-44875. 11. 221*28645, 

12. 569-6768. 13. 33 iu. 14. 44 iu. 15. 47 in. 

16. 53 in. 17. 7-979 m, 18. 1*1732 ft 
19. 1-7-208 ft. 20. 2-336 ft. 21.23. 22.489. 
23. 3916. 24. 11013. 25. 1800 cub. ft 

26. 117333333^. 27. 144000. 28. 13461. 
29. 1562-5 ft. 30. 89. 31. 21206. 32. 55-4. 
33. 1745-3. 34. 489. 35. £6, As. 9d, 



T. M. 



19 



290 ANSWERS. 

36. £5. Is. 37. £Q. 135. Qd. 38. 10-fil in. 

39. 8S006lin. 40. 3520244 ft. 41. 168 cub. ft. 

42. 7-068Q cub. ft. 43. 17-6715 cub. ft. 44. 373 lbs. 

45. 654 lbs. 46. £4. 175. 3d 47. 8*2467 lbs. 

48. 433 cub. in. 49. 2-598 cub. ft. 50. ^^ of an inch. 

51. 90 cub. ft. 52. 33 45 in. 53. 6-46 ft. 

54. 15-55 in. 55. "927 ft. 56. 5*61 in. 



XXIV. 3. 49-3. 4. 15-2. 5. 1 very nearly. 

6. 391-6. 7. 410-5, 8. 27. 9. 637 in. 

10. 384 in. 

XXY. 1. 6 ft. 362 in. 2. 17 ft. 896 in. 

3. 32 ft. 704 in. 4. 45 ft. 1121 in. 5. 19-596. 

6. 83-785. 7. 9S7798. 8. 2378-571. 9. 16755. 

10. 64-141. 11.97-905. 12.1047-2. 13. 17 ft. 3 in. 

14. 21 ft. 9 in, 15. 30 ft. 16. 32 ft. 3 in. 

17. 7-979in. 18. 2-6846 ft. 19. 32612 ft. 

20. 4-2853 ft. 21. 407293-5 cub. ft. 22. 6G6G66| cub. ft. 

23. 206-448 cub. ft. 24. 192000 cub. ft. 25. 3494-4cub.ft. 

26. 1164-9 cub. ft. 27. 3640 cub. ft. 23. 3147'18cub.ft. 

29. 1231-5 cub.ft. 30. 391-8 cub.ft. 31. 3392-928 cub. ft. 

32. 55461-8064 cub.ft. 33. 62652. 34. 3-1416. 

35. 101-86 cub.ft. 36. 116-224 cub.ft. 37. 10-472 ft. 

38. 3637307. 39. 16G|cub.ft. 40. 2708 cub. in. 

41. 3466145. 42. 1173-46 cub.ft. 



XXVI. 1. 12-2,5. 2. 11-227. 3. 38-2475. 
4. 48-216. 5. 203-4186. 6. 50r7292. 7. 714-1736. 
8. 1038-4362. 9. 917-3472. 10. £60. 4s. 

11. 73-323 cub. ft. 12. 7600 cub. ft. 13. 4824*32 cub.ft. 

14. 65. 15. 1608-5 cub. ft. 16. 247-8, 283-1 cub. ft. 

17. 278873-5, 305413-8, 3331G0-4. 

18. 11276, 1408-2 cub. ft. 

XXVII. 1. 1620 cub. in. 2. 21 cub. ft. 

3. 3240 cub. in. 4. 3464 cub. in. 5. 688 cub. in. 

6. 4000 cub. in. 7. 995-9 cub. in. 

8. 462, 1155 cub. in. 9. 1008, 728 cub. in. 

10, 420, 480 cub. in. 



ANSWERS. 291 

XXVIII. 1. 744000 cub. ft. 2. 832000 cub. ft. 

3. 125067 cub. in. 4. 103110. 5. 336600. 

6. 1262, 19S?i cub. ft. 7. 473, 371, 281 cub. ft. 

8. 480, 720 cub. ft. 9. 261^, 485.^ cub. in. 

10. 250, 670, 970 cub. in. 12. 1144, 584 cub. ft. 



XXIX. 1. 696-9 cub. in. 2. 2681 cub. ft. 

3. 7238-2 cub. ft. 4. 17974-2 cub. ft. 5.3-65. 

6. 8-65. 7. 16-89. 8. 2918. 9. 3r94. 

10. 154-99. 11. 34401. 12. 1239-88. 13. 20-73. 

14. 272 minutes. 15. 197-9208 cub. ft. 

16. 6494 lbs. 17. 43-412 cub. in. 18. 29-688 lbs. 

19. 432-227 ounces. 20. 5986 lbs. 21. 12 72 lbs. 

22. 31-94 lbs. 23. Iiri8lbs. 24. 522-67 ounces. 

25. 2926 2 ounces. 26. 611-12 lbs. 27. 549 lbs. 

28. 28-6875 lbs. 30. 1-24 lbs. 31. 2 in. 32. 3 in. 

33. 19-28 in. 34. 65-45 cub. ft. 35. Nearly 49 times. 

36. 16 12 in. 37. 24*814 in. 38. 1664 in. 

39. 95076 in. 40. 14 in. 41. 4 in. very nearly. 

42. 10-03 in. 



XXX. 1. 546-6384 cub. in. 2. 2073 456 cub. in, 

3. 263-8944 cub. ft. 4. 7695 cub. ft. 5. 234-57 cub. ft. 

6. 458-15 cub. ft. 7. 2077-64 cub. ft. 8. 205-25 cub. ft. 

9. 654-5, 3534-3 cub. ft. 10. 791-6832, 2261952 cub. ft. 

11. -375 of a cub. in. 12. 30-5 lbs. 

13. Two of 98-96 cub. ft., one of 183*78 cub. ft. 

14. Two of 3351 cub. in., two of 737-2 cub. in. 

15. 3796-1 cub. in. 16. 268345 cub. in. 17. 143. 



XXXI. 1884-96 cub. in. 2. 5806 cub. ft, 3.30. 
4. 84-56 cub. feet. 5. 9333 cub. in. 



XXXII. 1. 20-25 lbs. 2. 58320 lbs. 

4. 168-75, 231-25 cub. in. 5. 5-228 in. 6. 8-320 ft. 

7. 2-7744 ft. 8. 1-9265, 3-5635 feet 

19—2 



292 ANSWERS. 

XXXin. 1. 37 sq.ft. 72 in. 2. 80 sq.ft. 96 m, 

3. 20-isq. ft. 24in. 4. 2G0sq.ft. G in. 6. 70 sq.ft. 

6. 63 sq. ft. 7. 73 sq. ft. 40 in. 8. 88 sq. ft. 10 in. 

9. 108 sq.ft. 10. 520 sq.ft. 11. 51 sq. ft. 96 in. 

12. 77 sq. ft. 132 in. 13. 24 sq. ft. 47 in. 

14. 48 sq.ft. 128 in. 15. 66 sq. ft. 16 in. 

16. 1227 sq. ft. 48 in. 17. 985 sq. ft. 120 in. 

18. 2488 sq. ft. 128 in. 19. 18 sq. ft. 

20. 43 sq. ft. 90 in. 21. 30 sq. ft. 84 in. 

22. 40 sq. ft. 52 in. 23. 104 sq. ft. 84 in. 

24. 173-2 sq.ft. 25. 88-2888 sq. ft. 26. 2197 cub. in. 

%1. 6-083 ft. 28. 619-52 sq. in. 29. 19ri3 sq. in. 

30. 44-7846 sq. ft. 31. 190 sq. ft. 96 in. 32. 6 sq.ft. 

S3. 1380, 1296 sq. in. 37. 500. 88. 566-96. 

39. 476-22. 40. 624-97. 41. 485-35. 42. 600. 

43. 629-96. 44. 634-96. 45. 673-05. 46. 693-36. 



XXXI Y. 1. 1248 sq. in. - 2. 1653 sq, in. 

3. 234992 sq.in. 4. 6277-89 sq.in. 5. 2010*624 sq.ft. 

6. 62-832 sq.ft. 7. 186-9252 sq.ft. 8. 1105-84 sq.in. 

9. 173-662 sq.ft. 10. 241-673 sq.ft. 11. 19-2 in. 

12. 4-775 in. 13. 12-666 in. 14. 13-541 in. 

15. 21-409 in. 16. 2-6 ft. 17. 19-635 sq. ft 

18. 3-S2in. 19. 628-32 sq. in. 20. 968-54 sq.in. 

21. 4375. 22. 4593. 23. 4630. 24. 4907. 

25. 5055. 26. 5833. 27. 5512. 28. 6944. 

29. 5608. 30. 8088. 31. 600, 586 sq. in. 

32. 18683 sq. in. 



XXXY. 1. 753-98 sq. in. 
3. 3518-59 sq. in. 4. 80. 

7. 256-61. 8. 868-53. 9. 

11. 15-92 in. 12. 446 in. 



2. 


628-32 


sq. in. 


5. 117-81. 


6. 


13-0.3. 


685-4. 


10. 


897-98. 





XXXVI. 


1. 715-5. 


2. 


1273. 


3. 


1583-37. 


4. 


3418-06. 


5. 2035-76. 


6. 


549-78. 


7. 


1159-25. 


8. 


1709-03. 


1). 2108 01. 


10. 


6738-73. 


11. 


2306-74. 


12. 


8715-7. 


n. 37-699. 


14. 


85-452. 


15. 


29-093. 


16. 


38-527. 


17. 1-9635. 


18. 


21-3803. 


19. 


2-1378. 



ANSWERS. 



293 



20. 
24. 

27. 
SO. 



18-4322. 

2-729 ft. 
1-0925 ft. 
827 yds. n^ 
65-147, 



21. 



37. 28701. 



arly. 
34. 



30 in. 22. 22831. 

25. 52 m. 26. 

28. 1-0998 ft. 29. 

31. 62-688. 
62-688. 35. 60-074. 



J8. 2817-2. 39. 



47335. 



23. 2-5465 ft. 

1-8335 ft. 

28 yds. nearly. 

32. 66-490. 

3G. 33989. 



40. 



60154. 



3. 

6. 

9. 
12. 
15. 
19. 



XXXVII. 1. 

251-328 sq. in. 
9764-09 sq. in. 
185-97 sq. in. 
139-49 sq. ft. 
40-84. IG. 



176 sq. in. 

4. 94-248 sq. ft. 
7. 89-41 sq. ft. 
10. 229-09 sq. in. 
13. 2412-75 sq. in. 
171-22. 17. 427-26, 



2. 273 sq. in. 

5. 1102-70 sq. in. 

8. 96-35 sq. ft. 
11. 80-11 sq. ft. 
14. 3418-06 sq. in. 
18. 1747-515. 



2-228, 1*772 feet. 



20. 105 nearly. 



1. 



3. 

6. 

9. 
12. 
14. 
16. 
17. 
18. 
19. 
20. 



314-16 sq. in. 2. 2827-44 sq. in. 

4. 127-32 sq. in. 5. 5-093 sq. ft. 

7. 11-284 in. 8. 4-5135 ft. 

8-4104 cub. ft. 11. 33-215 cub. ft, 

13. 44888 cub. ft. 
15. 75-3984 sq. fc. 
cylinder 3141-6 cub. in. ; sphere 418S-3 cub. in. 
cube 1 cub. ft. ; sphere r382 cub. ft. 
cylinder 6-282 cub. ft. ; sphere 7-695 cub. ft. 
cube 6 sq. ft. ; sphere 4-836 sq. ft. 
cylinder 18-8496 sq. ft. ; sphere 16-4666 sq. ft. 



XXXYIII. 

60-821 sq.ft. 
13 038sq. ft. 
4 886 ft. 10. 

94-0315 cub. ft. 
47713 cub. ft. 



3. 

6. 

9. 

11. 

13. 
15. 
18. 



XXXIX. 1. 850 sq. in. 



2. 45 sq. ft. 



90478 sq.m. 
2312-22 sq. in. 
1039-8696 sq. ft 
308-8936 sq. ft. 
750-8424 sq. ft. 
308-4.343 sq. ft 
19 



"k 



4. 49 009 sq. ft. 5. 163-3G32 sq. ft. 
7. 133-725G sq. ft. 8. 1599*87 sq. in. 
10. 1328-897 sq. in. 
12. 86779 sq. in. 
14. 16776144 sq. in. 
16. 1221894 sq. in. 17. ?^V 

20. ^ of the radius. 



I of the radius. 



XL. 1. £62.25. 
4. i*37. 155. 6d. 
7. X37. 165 A^d. 



2. £165.125. 
5. £A. 35. Ad. 
8. 373 lbs. 



3. £183. \1s.4d. 

6. 3-^ ft. long. 

9. 34|. 



^* 



294 ANSWERS. 

]0. £12. 65. o^d 11. £3. 26\ lOd very nearly. 12.^12 

13. £1. 145. nearly. 14. £95. VM. 3d very nearly 

15. £5. 105. very nearly. 16. £50. 17. £12 

16. 5550. 19. £G. 85. 6d 20. £33. 185. 6d 

XLI. 1. 5-62. 2. 3-4. 3. £10. 55. 4. 216^. 

5. £145. 45. 6. £15. 155. 7. £l20. 8. £49. 105. 

9. £25. 05. Sd 10. £62. 105. 11. £126. 

12. £71. 55. 13. 100^. 14. £12. 125. 15. £41. 

16. £16.45. 

XLII. 1. 93|. 2. 117. 3. 161. 4 87^ 

5. 12^. 6. 16f. 7. 128^. 8. 128^. 9. 75|. 

10. 96|. 11. 98-676. 12. 628. 

XLIII. 1. 110-87. 2. 88-18. 3. 108-54. 

4. 54-2. 5. 1087. 6. 9003. 7- 112-43. 
8. 35-15. 9. 23-49. 10. 15-98. 

XLVI. 1. -7975 of an acre. 2. •626ofanacra 

3. '56283 of an acre. 4. 7165 of an aero. 

5. 1-1445 acres. 6. I'l 165 acres. 7. 766 sq. links. 
8. 1511 sq. links. 9. 2300 sq. links. 10. 5640 sq. links. 

11. -311 of an acre. 12. 3-61135 acres. 13. '662 of an acre. 

14. 8-606127 acres ; it will be found that the angle ABC 

is 60*, and the angle ACD is 90". 

15. 16-24425 acres. 16. 5-02922 acres. 

XLVIII. 1. 9 sq. ft. 2. 16 sq. ft. 8 pr. 3. 9 sq. ft. 2 pr. 

4. 16 sq. ft. 6 pr. 9 in. 5. 26 sq. ft. 11 pr. 10 in. 

6. 27 sq. ft. 1 pr. 5 in. 7. 13 sq. ft. 10 pr. 10 in. 
8. 15 sq. ft. 8 pr. in. 8 th. 9. 14 sq. ft. pr. 4 in. 10 th. 

10. 22 sq. ft. 7 pr. 4 in. 10 th. 11 fo. 11. 13 cub. ft. 6pr. 

12. 33 cub. ft. 9 pr. 13. 43 cub. ft. 4 pr. 

14. 73 cub. ft. 10 pr. 8 sec. 15. 123 cub. ft. Opr. 6 sec. in. 

16. 187 cub. ft. 2pr. Oscc. 2 in. 

XLIX. 1. 47-124 metres. 2. 12 hectares 39*56 ares. 
3. 90-576 ares. 4.73-44. 5.402. 6.5-23. 7.4692. 8.51. 
9.172. 10.31-416. 11.104. 12.254. 16. '062 of a metre. 

17. '108 of a metre. 18. '098 of a metre. 



ANSWERS. 295 



MISCELLANEOUS EXAMPLES. 

1. 75 ft. 2. Si. 3. 10 ac; 90 ac. 4. 50 ft. 4 in. 

5. 972sq.vd3. 6.^14. 7. £3. 10s. 8.^161.85. 9. 3*96 ac. 
10. 23 ft. ; 27 ft. 11. | of a mile. 12. 193G00. 13. 820 in. 
14. 60 ft. 9 in. 15. 13-8 minutes. 16. .£6. 135. 4f/. 

17. £o. Is. 6d. IS. 6 yds. 19. 3-5568 acres. 20. 175602. 
22. 210 sq. ft. 23. 1120 yds.; 262 ac. 1 rd. 26 po. Sh sq. yds. 
24. 85ft. lOin. 25. 800yds.; 316228yds. 26. XU.ISs.l^d. 
27. £6. 3s. nhd. 28. Length 10^ yds., breadth 3^, height 4. 
29. 346S sq. ft. 30. 546 sq. ft. ; 13 ft. 31. 102 ft. 

32. l.Un. 33. Rather more than 13|. 34. 243 in. 

35. 360-555 yds. ; 33600 sq. yds. 36. £25. 75. 2d. 

37. £1. 176". 6c^. Four times as much. 38. 50 sq. yds. 

39. 150 sq.ft.; 270 sq.ft. 40. 9-4916 ac. 41. 26|ft. 
42. 12 ft. 43. 18 ft. 2i in. 44. 594 vds. 

45. £182. 65. llfrZ. 46. £430^1 65. 8d 47. £4.35.''7f^. 
48. 21ft. long, lOi ft. broad, 10ft. high. 49. 133837 sq.ft. 
50. 72 sq.ft. 51. 15 in.; ISf in. 52. £70. 53. 37 ft. 7^iD. 
54. 1-0936 yds. 55. 864; £233. 175. 9(/. 56. £2. 95. 3§fi?. 
57. 819-20. 58. 141-42 yds. 60. £78. 75. 61. 3600. 
6-2. 15 in. 63. 26043- lbs. 64. 275625. 65. 4 feet. 
66. 3349.Ubs. 67. 320^2; 165. 68. 3-20705; IDs.M. 69. 5*44. 
70. 147cub.ft. 71.12288. 72. 25 in. 73.480. 74. 25 ft. 4 in. 
75. 22-22 in. 76. 8-1 ft., 53r441cub.ft. 77. 31416 cub. ft. 
78. 275-4 lbs. 79. £11. 05. 6d. ; £7. 125. Sd. 80. 3 ^3 to 1. 
81.5 ft. 8-2. 176786. 83. 21. 84. 27in. 85. 125.3d 
86. 689. 87. 2ft. 6 in. 88. 37*6992 cub. in. ; 75-3984 sq.in. 
89. 304 cuU in. 90. 11-754 cub. in. 91.671. 9-2. 1969 lbs. 
93. 7009 in. 94. 648. 95. 28.^ ; 427^ Ib.s. 96. £43. 45. 
97. 3780 lbs.; 591.59. 98. 4. ^n. 9'9. 3518592 sq.ft. 100. 73|. 
101. 512. 102. 27, 18, 12 ft. 103. 9019 in. 104. Four times. 
105. 4230794. 106. '030301 of the former yolume. 

111.1611. 112. £680. 125. 6d7. 113. 21V3in. 114. 12800 lbs. 
115. 1885-6 cub. ft. 116. 565-488 cub. ft. 117. 2-142 ft. 
118. 10744-272 cub. in. 119. 44lbs. 7oz. 120. 12036 cub. ft. 
121. 22^ cub. ft. 122. £580800. 123.900. 124.885. 126.40. 
127. 233^ cub. ft. 129. 7| lbs. 130. 78-54 sq. ft. 131. 640 ft, 
132.4-15. 133. 312i 134.2178. 13.5. 809 ft. 136.12 ft. 
137. 8042-5 cub. ft. 138.94. 139. 8-574 in. 140. 5r051 sq.ft. 
141.3872. 142. yo^ of an inch. 143.4840. 144. y\ of a foot. 
145. 173-205 cub. in. 146.199in. 147. 1728. 148. 392-7 cub. in. 



296 ANSWERS. 

149. 8-748 lbs. 150. 382 in. 151. 14, 10|, 3^ 

152. £3. 145. 36?. 153. No. 154. 359'7 sq. in. 155. 968. 
156. 144G. 157. -000046. 158. 6'232. 159. 796. 

160. 30'16 cub. in. 161. 487. 162. -1545 of an oz. 

163. 290 cub. in. ; 246 sq. in. 164. As 3 to 5. 

165. -00032 of an in. 166. 31416 cub, ft. 167. 8-3776 cub. ft. 
168. 3"577feet from smaller end ; 59-69 cub. ft. 169. 108. 
170.1034. 172. 46-188 ft. 174. 8242-408 cub. in. 

175. 5-1416 sq. in. 176. 10-2832 cub. in. 179. 6r77 ft. 



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